Degenerate Diffusion Operators Arising in Population Biology (AM-185) [Course Book ed.] 9781400846108

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Annals of Mathematics Studies Number 185

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Degenerate Diffusion Operators Arising in Population Biology

Charles L. Epstein and Rafe Mazzeo

PRINCETON UNIVERSITY PRESS PRINCETON AND OXFORD 2013

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c 2013 by Princeton University Press Copyright  Published by Princeton University Press, 41 William Street, Princeton, New Jersey 08540 In the United Kingdom: Princeton University Press, 6 Oxford Street, Woodstock, Oxfordshire OX20 1TW press.princeton.edu All Rights Reserved Library of Congress Cataloging-in-Publication Data Epstein, Charles L. Degenerate diffusion operators arising in population biology / Charles L. Epstein and Rafe Mazzeo. pages cm. – (Annals of mathematics studies ; number 185) Includes bibliographical references and index. ISBN 978-0-691-15712-2 (hardcover : alk. paper) – ISBN 978-0-691-15715-3 (pbk. : alk. paper) 1. Elliptic operators. 2. Markov processes. 3. Population biology–Mathematical models. I. Mazzeo, Rafe. II. Title. QA329.42.E67 2013 577.8801519233–dc23 2012022328 British Library Cataloging-in-Publication Data is available This book has been composed in LATEX. The publisher would like to acknowledge the authors of this volume for providing the camera-ready copy from which this book was printed. Printed on acid-free paper ∞ Printed in the United States of America 10 9 8 7 6 5 4 3 2 1

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Out here in the perimeter there are no stars Out here we is stone Immaculate. Jim Morrison and the Doors, “The Wasp (Texas Radio and the Big Beat)” (L.A. Woman)

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Contents

Preface 1

xi

Introduction 1.1 Generalized Kimura Diffusions . . 1.2 Model Problems . . . . . . . . . . 1.3 Perturbation Theory . . . . . . . . 1.4 Main Results . . . . . . . . . . . 1.5 Applications in Probability Theory 1.6 Alternate Approaches . . . . . . . 1.7 Outline of Text . . . . . . . . . . 1.8 Notational Conventions . . . . . .

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1 3 5 9 10 13 14 16 20

I Wright-Fisher Geometry and the Maximum Principle

23

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Wright-Fisher Geometry 2.1 Polyhedra and Manifolds with Corners . . . . . . . . . . . . . . . . . 2.2 Normal Forms and Wright-Fisher Geometry . . . . . . . . . . . . . .

25 25 29

3

Maximum Principles and Uniqueness Theorems 3.1 Model Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Kimura Diffusion Operators on Manifolds with Corners . . . . . . . . 3.3 Maximum Principles for the Heat Equation . . . . . . . . . . . . . .

34 34 35 45

II Analysis of Model Problems 4

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The Model Solution Operators 4.1 The Model Problem in 1-dimension . . . . 4.2 The Model Problem in Higher Dimensions . 4.3 Holomorphic Extension . . . . . . . . . . . 4.4 First Steps Toward Perturbation Theory . .

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51 51 54 59 62

Degenerate H¨older Spaces 5.1 Standard H¨older Spaces . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 WF-H¨older Spaces in 1-dimension . . . . . . . . . . . . . . . . . . .

64 65 66

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CONTENTS

6 H¨older Estimates for the 1-dimensional Model Problems 6.1 Kernel Estimates for Degenerate Model Problems . . . . . . . . . . . 6.2 H¨older Estimates for the 1-dimensional Model Problems . . . . . . . 6.3 Properties of the Resolvent Operator . . . . . . . . . . . . . . . . . .

78 80 89 103

7 H¨older Estimates for Higher Dimensional Corner Models 7.1 The Cauchy Problem . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 The Inhomogeneous Case . . . . . . . . . . . . . . . . . . . . . . . . 7.3 The Resolvent Operator . . . . . . . . . . . . . . . . . . . . . . . . .

107 109 122 135

8 H¨older Estimates for Euclidean Models 8.1 H¨older Estimates for Solutions in the Euclidean Case . . . . . . . . . 8.2 1-dimensional Kernel Estimates . . . . . . . . . . . . . . . . . . . .

137 137 139

9 H¨older Estimates for General Models 9.1 The Cauchy Problem . . . . . . . . . 9.2 The Inhomogeneous Problem . . . . . 9.3 Off-diagonal and Long-time Behavior 9.4 The Resolvent Operator . . . . . . . .

143 145 149 166 169

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III Analysis of Generalized Kimura Diffusions 10 Existence of Solutions 10.1 WF-H¨older Spaces on a Manifold with Corners 10.2 Overview of the Proof . . . . . . . . . . . . . . 10.3 The Induction Argument . . . . . . . . . . . . 10.4 The Boundary Parametrix Construction . . . . 10.5 Solution of the Homogeneous Problem . . . . . 10.6 Proof of the Doubling Theorem . . . . . . . . . 10.7 The Resolvent Operator and C0 -Semi-group . . 10.8 Higher Order Regularity . . . . . . . . . . . .

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181 182 187 191 194 205 208 209 211

11 The Resolvent Operator 11.1 Construction of the Resolvent . . . . . . . . . . . . . . . . . . . . . 11.2 Holomorphic Semi-groups . . . . . . . . . . . . . . . . . . . . . . . 11.3 Diffusions Where All Coefficients Have the Same Leading Homogeneity

218 220 229 230

12 The Semi-group on Ꮿ0 (P) 12.1 The Domain of the Adjoint . . . . . . . . . . . . .  12.2 The Null-space of L . . . . . . . . . . . . . . . . 12.3 Long Time Asymptotics . . . . . . . . . . . . . . 12.4 Irregular Solutions of the Inhomogeneous Equation

235 237 240 243 247

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CONTENTS

A Proofs of Estimates for the Degenerate 1-d Model A.1 Basic Kernel Estimates . . . . . . . . . . . . . A.2 First Derivative Estimates . . . . . . . . . . . . A.3 Second Derivative Estimates . . . . . . . . . . A.4 Off-diagonal and Large-t Behavior . . . . . . .

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251 252 272 278 291

Bibliography

301

Index

305

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Preface

This lange megillah is concerned with establishing properties of a mathematical model in population genetics that some might regard, in light of what is being modeled, as entirely obvious. Once written down, however, a mathematical model has a life of its own; it must be addressed in its own terms, and understood without reference to its origins. The models we consider are phrased as partial differential equations, which arise as limits of finite Markov chains. The existence of solutions to these partial differential equations and their properties are suggested by the physical, economic, or biological systems under consideration, but logically speaking, are entirely independent of them. What is far from obvious are the regularity properties of these solutions, and, as is so often the case, the existence of solutions actually hinges on these very subtle properties. Using a Schauder method, we prove the existence of solutions to a class of degenerate parabolic and elliptic equations that arise in population genetics and mathematical finance. The archetypes for these equations arise as infinite population limits of the WrightFisher models in population genetics. These describe the prevalence of a mutant allele, in a population of fixed size, under the effects of genetic drift, mutation, migration and selection. The formal generator of the infinite population limit acts on functions defined on [0, 1] (the space of frequencies) and is given by L W F = x(1 − x)∂x2 + [b0 (1 − x) − b1 x + sx(1 − x)]∂x .

(1)

Processes defined by such operators were studied by Feller in the early 1950s and used to great effect by Kimura et al. in the 1960s and ’70s to give quantitative answers to a wide range of questions in population genetics. Notwithstanding, a modern appreciation of the analytic properties of these equations is only now coming into focus. In this monograph we provide analytic foundations for equations of this type and their natural higher dimensional generalizations. We call these operators generalized Kimura diffusions. They act on functions defined on generalizations of convex polyhedra, which are called manifolds with corners. We provide the basic H¨older space-type estimates for operators in this class with which we establish the existence of solutions. These operators satisfy a strong form of the maximum principle, which implies uniqueness. The partial differential operators we consider are degenerate and the underlying manifolds with corners are themselves singular. This inevitably produces significant technical challenges in the analysis of such equations, and explains, in part, the length xi

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xii

PREFACE

of this text. The Markov processes defined by these operators provide fundamental models for many Biological and Economic situations, and it is for this reason that we feel that these operators merit such a detailed and laborious treatment. A large portion of this book is devoted to a careful exploration of operators of the form n m   L b,m = [x i ∂x2i + bi ∂xi ] + ∂ y2l , (2) i=1

⺢n+

l=1

⺢m .

× Here the coefficients {bi } are non-negative acting on functions defined in constants. These operators are interesting in their own right, arising as models in population genetics, but for us, they are largely building blocks for the analysis of general Kimura diffusions. These are the analogues, in the present context, of the “constant coefficient elliptic operators” in classical elliptic theory. A notable feature of this family is that, because the coefficient of ∂x2i vanishes at x i = 0, we need to retain the first order transverse term. The value of b = (b1 , . . . , bn ) has a pervasive effect on the behavior of the solution. Much of our analysis rests upon explicit formulæ for the solutions of the initial value problems: ∂t v − L b,m v = 0 with v(x, y, 0) = f (x, y).

(3)

Using these models, we have succeeded in developing a rather complete existence and regularity theory for general Kimura diffusions with H¨older continuous data. This in turn suffices to prove the existence of a C0 -semi-group acting on continuous functions, and shows that the associated martingale problem has a unique solution. The existence of a strong Markov process, which in applications to genetics describes the statistical behavior of individual populations, follows from this. In special situations, such results have been established by other authors, but without either the precise control on the regularity of solutions, or the generality considered herein. As long as it is, this text just begins to scratch the surface of this very rich field. We have largely restricted our attention to the solutions with the best possible regularity properties, which leads to considerable simplifications. For applications it will be important to consider solutions with more complicated boundary behavior; we hope that this text will provide a solid foundation for these investigations. ACKNOWLEDGMENTS We would like to acknowledge the generous financial support and unflagging personal support provided by Ben Mann and the DARPA FunBio project. It is certainly the case that without Ben’s encouragement, we would never have undertaken this project. We would like to thank our FunBio colleagues1 who provided us with the motivation 1 Phil Benfey, Michael Deem, Richard Lenski, Jack Morava, Lior Pachter, Herbert Edelsbrunner, John Harer, Jim Damon, Peter Bates, Joshua Weitz, Konstantin Mischaikow, Gunnar Carlsson, Bernd Sturmfels, Tim Buchman, Ary Goldberger, Jonathan Eisen, Olivier Porquie, Thomas Fink, Ned Wingreen, Jonathan Dushoff, Peter Nara, inter alios, and research staff: Mark Filipkowski, Shauna Koppel, Rachel Scholz, Matthew Clement, Traci Kiesling, Traci Pals, inter alios.

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PREFACE

xiii

and knowledge base to pursue this project, and Simon Levin for his leadership and inspiration. CLE would like to thank Josh Plotkin, Warren Ewens, Todd Parsons, and Ricky Der, from whom he has learned what little he knows about population genetics. We would both like to thank Charlie Fefferman for showing us an explicit formula for kt0 (x, y), which set us off in the very fruitful direction pursued herein. We would also like to thank Dan Stroock for his help with connections to Probability Theory. We would like to thank Camelia Pop for providing a long list of corrections. CLE would like to acknowledge the financial support of DARPA through grants HR0011-05-1-0057 and HR0011-09-1-0055, the support of NSF through grant DMS0603973, and the hospitality and support provided by Leslie Greengard and the Courant Institute, where this text was completed. He would also like to thank his wife Jane for doing the hard work to get us there and back. RM would like to acknowledge the financial support of DARPA through grants HR0011-05-1-0057 and HR0011-09-1-0055, and the support of NSF through grants DMS-0805529 and DMS-1105050. He is also immensely grateful to his wife Laurence for her general forbearance and support, and to his daughters Sabine and Sophie for making it all worthwhile.

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Chapter One Introduction In population genetics one frequently replaces a discrete Markov chain model, which describes the random processes of genetic drift, with or without selection, and mutation with a limiting, continuous time and space, stochastic process. If there are N + 1 possible types, then the configuration space for the resultant continuous Markov process is typically the N-simplex ᏿ N = {(x 1 , . . . , x N ) : x j ≥ 0 and x 1 + · · · + x N ≤ 1}.

(1.1)

If a different scaling is used to define the limiting process, different domains might also arise. As a geometrical object the simplex is quite complicated. Its boundary is not a smooth manifold, but has a stratified structure with strata of codimensions 1 through N. The codimension 1 strata are 1,l = {xl = 0} ∩ ᏿ N for l = 1, . . . , N,

(1.2)

1,0 = {x 1 + · · · + x N = 1} ∩ ᏿ N .

(1.3)

along with Components of the stratum of codimension 1 < l ≤ N arise by choosing integers 0 ≤ i 1 < · · · < i l ≤ N and forming the intersection: 1,i1 ∩ · · · ∩ 1,il .

(1.4)

The simplex is an example of a manifold with corners. The singularity of its boundary significantly complicates the analysis of differential operators acting functions defined in ᏿ N . In the simplest case, without mutation or selection, the limiting operator of the Wright-Fisher process is the Kimura diffusion operator, with formal generator: L Kim =

N 

x i (δi j − x j )∂xi ∂x j .

(1.5)

i, j =1

This is the “backward” Kolmogorov operator for the limiting Markov process. This operator is elliptic in the interior of ᏿ N but the coefficient of the second order normal derivative tends to zero as one approaches a boundary. We can introduce local coordinates (x 1 , y1 , . . . , y N−1 ) near the interior of a point on one of the faces of 1,l , so that 1

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2

CHAPTER 1

the boundary is given locally by the equation x 1 = 0, and the operator then takes the form N−1 N−1   x 1 c1l ∂x1 ∂ yl + clm ∂ ym ∂ yl , (1.6) x 1 ∂x21 + l=1

l,m=1

where the matrix clm is positive definite. To include the effects of mutation, migration and selection, one typically adds a vector field: V =

N 

bi (x)∂xi ,

(1.7)

i=1

where V is inward pointing along the boundary of ᏿ N . In the classical models, if only the effect of mutation and migration are included, then the coefficients {bi (x)} can be taken to be linear polynomials, whereas selection requires at least quadratic terms. The most significant feature is that the coefficient of ∂x21 vanishes exactly to order 1. This places L Kim outside the classes of degenerate elliptic operators that have already been analyzed in detail. For applications to Markov processes the difficulty that presents itself is that it is not possible to introduce a square root of the coefficient of the second order terms that is Lipschitz continuous up to the boundary. Indeed the best one can hope for is H¨older- 12 . The uniqueness of the solutions to either the forward Kolmogorov equation, or the associated stochastic differential equation, cannot then be concluded using standard methods. Even in the presence of mutation and migration, the solutions of the heat equation for this operator in 1-dimension was studied by Kimura, using the fact that L Kim + V preserves polynomials of degree d for each d. In higher dimensions it was done by Karlin and Shimakura by showing the existence of a complete basis of polynomial eigenfunctions for this operator. This in turn leads to a proof of the existence of a polynomial solution to the initial value problem for [∂t − (L Kim + V )]v = 0 with polynomial initial data. Using the maximum principle, this suffices to establish the existence of a strongly continuous semi-group acting on Ꮿ0 , and establish many of its basic properties, see [38]. This general approach has been further developed by Barbour, Etheridge, Ethier, and Griffiths, see [17, 2, 16, 25]. As noted, if selection is also included, then the coefficients of V are at least quadratic polynomials, and can be quite complicated, see [9]. So long as the second order part remains L Kim , then a result of Ethier, using the Trotter product formula, makes it possible to again define a strongly continuous semi-group, see [18]. Various extensions of these results, using a variety of functional analytic frameworks, were made by Athreya, Barlow, Bass, Perkins, Sato, Cerrai, Cl´ement, and others, see [1, 4, 6, 7, 8]. For example Cerrai and Cl´ement constructed a semi-group acting on Ꮿ0 ([0, 1] N ), with the coefficient ai j of ∂xi ∂x j assumed to be of the form ai j (x) = m(x)Ai j (x i , x j ).

(1.8)

Here m(x) is strictly positive. In [1, 4, 3], Bass and Perkins along with several collaborators, study a class of equations, similar to that defined below. Their work has

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3

INTRODUCTION

many points of contact with our own, and we discuss it in greater detail at the end of Section 1.5. We have not yet said anything about boundary conditions, which would seem to be a serious omission for a PDE on a domain with a boundary. Indeed, one would expect that there would be an infinite dimensional space of solutions to the homogeneous equation. It is possible to formulate local boundary conditions that assure uniqueness, but, in some sense, this is not necessary. As a result of the degeneracy of the principal part, uniqueness for these types of equations can also be obtained as a consequence of regularity alone! We illustrate this in the simplest 1-dimensional case, which is the equation, with b(0) ≥ 0, b(1) ≤ 0, ∂t v − [x(1 − x)∂x2 + b(x)∂x ]v = 0 and v(x, 0) = f (x).

(1.9)

If we assume that ∂x v(x, t) extends continuously to [0, 1] × (0, ∞) and lim x(1 − x)∂x2v(x, t) = lim x(1 − x)∂x2 v(x, t) = 0,

x→0+

x→1−

(1.10)

then a simple maximum principle argument shows that the solution is unique. In our approach, such regularity conditions naturally lead to uniqueness, and little effort is expended in the consideration of boundary conditions. In Chapter 3 we prove a generalization of the Hopf boundary point maximum principle that demonstrates, in the general case, how regularity implies uniqueness. 1.1 GENERALIZED KIMURA DIFFUSIONS In his seminal work, Feller analyzed the most general closed extensions of operators, like those in (1.9), which generate Feller semi-groups in 1-dimension, see [20]. He analyzes the resolvent kernel, using methods largely restricted to ordinary differential equations. In [10], Chen and Stroock use probabilistic methods to prove estimates on the fundamental solution of the parabolic equation, ∂t u = x(1 − x)∂x2u. Up to now very little is known, in higher dimensions, about the analytic properties of the solution to the initial value problem for the heat equation ∂t v − (L Kim + V )v = 0 in (0, ∞) × ᏿ N with v(0, x) = f (x).

(1.11)

Indeed, if we replace L Kim with a qualitatively similar second order part, which does not take one of the forms described above, then even the existence of a solution is not known. In this monograph we introduce a very flexible analytic framework for studying a large class of equations, which includes all standard models, of this type appearing in population genetics, as well as the SIR model for epidemics, see [19, 38], and models that arise in Mathematical Finance, see [21]. Our approach is to introduce non-isotropic H¨older spaces with respect to which we establish sharp existence and regularity results for the solutions to heat equations of this type, as well as the corresponding elliptic problems. Using the Lumer-Phillips theorem we conclude that the Ꮿ0 -graph closure of this operator generates a strongly continuous semi-group.

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4

CHAPTER 1

The approach here is an extension of our work on the 1-d case in [15], which allows us to prove existence, uniqueness and regularity results for a class of higher dimensional, degenerate diffusion operators. While our methods also lead to a precise description of the heat kernel in the 1-dimensional case, this has proved considerably more challenging in higher dimensions. It is hoped that a combination of the analytic techniques used here, and the probabilistic techniques from [10] will lead to good descriptions of the heat kernels in higher dimensions. Our analysis applies to a class of operators that we call generalized Kimura diffusions, which act on functions defined on manifolds with corners. Such spaces generalize the notion of a regular convex polyhedron in ⺢ N , e.g., the simplex. Working in this more general context allows for a great deal of flexibility, which proves indispensable in the proof of our basic existence result. Locally a manifold with corners, P, can be described as a subset of ⺢ N defined by inequalities. Let { pk (x) : k = 1, . . . , K } be smooth functions in the unit ball B1 (0) ⊂ ⺢ N , vanishing at 0, with {d pk (0) : k = 1, . . . , K } linearly independent; clearly K ≤ N. Locally P is diffeomorphic to K 

{x ∈ B1 (0) : pk (x) ≥ 0}.

(1.12)

k=1

We let k = P ∩ {x : pk (x) = 0}; suppose that k contains a non-empty, open (N − 1)-dimensional hypersurface and that d pk is non-vanishing in a neighborhood of k . The boundary of P is a stratified space, where the strata of codimension n locally consists of points where the boundary is defined by the vanishing of n functions with independent gradients. The components of the codimension 1 part of the b P are called faces. As in (1.4), the codimension-n stratum of b P is formed from intersections of n faces. The formal generator is a degenerate elliptic operator of the form L=

N  i, j =1

Ai j (x)∂xi ∂x j +

N 

b j (x)∂x j .

(1.13)

j =1

Here Ai j (x) is a smooth, symmetric matrix valued function in P. The second order term is positive definite in the interior of P and degenerates along the hypersurface boundary components in a specific way. For each k N 

ai j (x)∂xi pk (x)∂x j pk (x) ∝ pk (x) as x approaches k .

(1.14)

i, j =1

On the other hand, N  i, j =1

ai j (x)v i v j > 0 for x ∈ int k and v = 0 ∈ Tx k .

(1.15)

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INTRODUCTION

The first order part of L is an inward pointing vector field V pk (x) =

N 

b j (x)∂x j pk (x) ≥ 0 for x ∈ k .

(1.16)

j =1

We call a second order partial differential operator defined on P, which is non-degenerate elliptic in int P, with this local description near any boundary point a generalized Kimura diffusion . If p is a point on the stratum of b P of codimension n, then locally there are coordinates (x 1 , . . . , x n ; y1 , . . . , ym ) so that p corresponds to (0; 0), and a neighborhood, U, of p is given by U = {(x; y) ∈ [0, 1)n × (−1, 1)m }. (1.17) In these local coordinates a generalized Kimura diffusion, L, takes the form L=

n 

[aii (x; y)x i ∂x2i +  bi (x; y)∂xi ] +

i=1 m n  

x i bik (x; y)∂x2i yk +

i=1 k=1

 1≤i = j ≤n

m 

x i x j ai j (x; y)∂x2i x j +

ckl (x; y)∂ y2k yl +

k,l=1

m 

dk (x; y)∂ yk ;

(1.18)

k=1

(ai j ) and (ckl ) are symmetric matrices, the matrices (aii ) and (ckl ) are strictly positive definite. The coefficients { b(x; y)} are non-negative along b P ∩ U, so that first order part is inward pointing. Let P be a compact manifold with corners and L a generalized Kimura diffusion defined on P. Broadly speaking, our goal is to prove the existence, uniqueness and regularity of solutions to the equation (∂t − L)u = g in P × (0, ∞) with u( p, 0) = f ( p),

(1.19)

with certain boundary behavior along b P × [0, ∞), for data g and f satisfying appropriate regularity conditions. These results in turn can be used to prove the existence of a strongly continuous semi-group acting on Ꮿ0 (P), with formal generator L. This is the “backward Kolmogorov equation.” The solution to the “forward Kolmogorov equation,” (∂t − L ∗ )m = ν, is then given by the adjoint semi-group, canonically defined on a (non-dense) domain in [Ꮿ0 (P)] = ᏹ(P), the space of finite Borel measures on P. 1.2 MODEL PROBLEMS The problem of proving the existence of solutions to a class of PDEs is essentially a matter of finding a good class of model problems, for which existence and regularity can be established, more or less directly, and then finding a functional analytic setting

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6

CHAPTER 1

in which to do a perturbative analysis of the equations of interest. The model operators for Kimura diffusions are the differential operators, defined on ⺢n+ × ⺢m , by L b,m =

n  j =1

[x j ∂x2j + b j ∂x j ] +

m 

∂ y2k .

(1.20)

k=1

Here b = (b1 , . . . , bn ) is a non-negative vector. We have not been too explicit about the boundary conditions that we impose along b ⺢n+ × ⺢m . This condition can be defined by a local Robin-type formula involving the value of the solution and its normal derivative along each hypersurface boundary component of b P. For b > 0, the 1-dimensional model operator, L b = x∂x2 + b∂x , has two indicial roots β0 = 0, β1 = 1 − b, (1.21) that is

L b x β0 = L b x β1 = 0.

(1.22)

If b = 1, then the boundary condition, lim [∂x (x b u(x, t)) − bx b−1u(x, t)] = 0,

x→0+

(1.23)

excludes the appearance of terms like x 1−b in the asymptotic expansion of solutions along x = 0. In fact, this condition insures that u is as smooth as possible along the boundary: if g = 0 and f has m derivatives then the solution to (1.19), satisfying (1.23), does as well. This boundary condition can be encoded as a regularity condition, that is u(·, t) ∈ Ꮿ1 ([0, ∞)) ∩ Ꮿ2 ((0, ∞)), with lim x∂x2 u(x, t) = 0

x→0+

(1.24)

for t > 0. We call the unique solution to a generalized Kimura diffusion, satisfying this condition, or its higher dimensional analogue, the regular solution. The majority of this monograph is devoted to the study of regular solutions. In applications to probability one often seeks solutions to equations of the form Lw = g, where w satisfies a Dirichlet boundary condition: w  P = h. Our uniqueness results often imply that these equations cannot have a regular solution, for example, when g ≥ 0. In the classical case the solutions to these problems can sometimes be written down explicitly, and are seen to involve the non-zero indicial roots. Usually these satisfy the other natural boundary condition, a la [20]. In 1-dimension, when b = 0, 1 it is: lim [∂x (xu(x, t)) − (2 − b)u] = 0, (1.25) x→0+

and allows for solutions that are O(x 1−b ) as x → 0+ . These are not smooth up to the boundary, even if the data is. The adjoint of L is naturally defined as an operator on ᏹ(P), the space finite Borel measures on P. It is more common to study this operator using techniques from probability theory, see [40].

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INTRODUCTION

For a generalized Kimura diffusion in dimensions greater than 1, the coefficient of the normal first derivative can vary as one moves along the boundary. For example, in 2-dimensions one might consider the operator L = x∂x2 + ∂ y2 + b(y)∂x .

(1.26)

If b(y) is not constant, then, with the boundary condition lim [∂x (xu(x, y, t)) − (2 − b(y))u(x, y, t)] = 0,

x→0+

(1.27)

one would be faced with the very thorny issue of a varying indicial root on the outgoing face of the heat or resolvent kernel. As it is, we get a varying indicial root on the incoming face, a fact which already places the analysis of this problem beyond what has been achieved using the detailed kernel methods familiar in geometric microlocal analysis. The natural boundary condition for the adjoint operator includes the condition: lim [∂x (xu(x, y, t)) − b(y)u(x, y, t)] = 0,

(1.28)

x→0+

allowing for solutions that behave like x b(y)−1, as x → 0+ . The solution operators for the 1-dimensional model problems are given by simple explicit formulæ. If b > 0, then the heat kernel is ktb (x, y)d y =

 y b t

where ψb (z) =

∞  j =0

If b = 0 then

x

kt0 (x, y) = e− t δ0 (y) +

e−

x+y t

ψb

 xy  dy t2

y

,

(1.29)

zj . j !( j + b)

x  t

e−

x+y t

ψ2

(1.30)  xy  dy t2

t

.

(1.31)

In either case ktb is smooth as x → 0+ and displays a y b−1 singularity as y → 0+ . It is notable that the character of the kernel changes dramatically as b → 0, nonetheless the regular solutions to these heat equations satisfy uniform estimates even as b → 0+ . This fact is quite essential for the success of our approach. The structure of these operators suggests that a natural functional analytic setting in which to do the analysis might be that provided by the anisotropic H¨older spaces defined by the singular, but incomplete metric on ⺢n+ × ⺢m : 2 dsW F =

n dx2  j j =1

xj

+

m 

d ym2 .

(1.32)

k=1

Similar spaces have been introduced by other authors for problems with similar degeneracies, see [11, 12, 28, 32, 23, 24]. In [22] Goulaouic and Shimakura proved a

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priori estimates in a H¨older space of this general sort in a case where the operator has this type of degeneracy, but the boundary is smooth. As was the case in these earlier works, we introduce two families of anisotropic H¨older spaces, which we denote by k,γ k,2+γ ᏯW F (P), and ᏯW F (P), for k ∈ ⺞0 , and 0 < γ < 1. In this context, heuristically k,γ k,2+γ an operator A is “elliptic of second order” if A−1 : ᏯW F (P) → ᏯW F (P). Note that k,2+γ k+1,γ k+2,γ k,2+γ ᏯW F (P) ⊆ ᏯW F (P), but ᏯW F (P)  ᏯW F (P), which explains the need for two families of spaces. In this monograph we consider the problem in (1.19) for f and g belonging to these H¨older spaces. The results obtained suffice to prove the existence of a semigroup on the space Ꮿ0 (P), but establishing the refined regularity properties of this semi-group and its adjoint require the usage of a priori estimates. These are of a rather different character from the analysis presented here; we will return to this question in a subsequent publication. In [23, 24], Graham studies model operators of the forms x(∂x2 + ⺢n ) + (1 − λ)∂x , and x∂x2 + ⺢n + (1 − λ)∂x , acting on functions defined on the half-space [0, ∞) × ⺢n . Using kernel methods, he proves sharp estimates for the solution of the inhomogeneous Dirichlet problem, in both isotropic and an-isotropic H¨older spaces. Graham works directly with the solution kernels for the elliptic problems, whereas we prove estimates for the parabolic problem and obtain the elliptic case via the Laplace transform. As manifolds with corners have non-smooth boundaries, and the Kimura diffusions are degenerate elliptic operators, the analysis of (1.19) can be expected to be rather challenging. We have already indicated a variety of problems that arise: 1. The principal part of L degenerates at the boundary. 2. The boundary of P is not smooth. 3. The “indicial roots” vary with the location of the point on b P. 4. The character of the solution operator is quite different at points where the vector field is tangent to b P. Along the boundary {x j = 0}, the first and second order terms in (1.20), b j ∂x j and x j ∂x2j , respectively, are of comparable “strength.” It is a notable and non-trivial fact that estimates for the solutions of these equations can be proved in these H¨older spaces, without regard for the value of b ≥ 0. As there is an explicit formula for the fundamental solution, the analysis of these model operators, while tedious and time consuming, is elementary. Indeed the solution of the homogeneous Cauchy problem, (∂t − L b,m )u = 0 in P × (0, ∞) with u( p, 0) = f ( p),

(1.33)

has an analytic extension to Re t > 0, which satisfies many useful estimates. To obtain a gain of derivatives where Re t > 0, in a manner that can be extended beyond the model problems, one must address the inhomogeneous problem, which has

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INTRODUCTION

somewhat simpler analytic properties. By this device, one can also estimate the Laplace transform of the heat semi-group, which is the resolvent operator: (μ − L b,m )

−1

∞ =

et L b,m e−μt dt.

(1.34)

0

The estimates for the inhomogeneous problem show that, in an appropriate sense, (μ− L b,m )−1 gains two derivatives and is analytic in the complement of (−∞, 0]. Finally one can re-synthesize the heat operator from the resolvent, via contour integration:  1 et L b,m = (μ − L b,m )−1 eμt dμ, (1.35) 2πi C π 2

where C is of the form | arg μ| = + α, for an 0 < α < positive real part, et L b,m also gains two derivatives.

π 2.

This shows that, for t with

Remark. In probability theory it is more common to consider the operators obtained from L b,m under the change of variables x j = operators become: L b,m

w2j 2 . In this coordinate system, the model

 n  m 2b j − 1 1 2 ∂w j + = ∂w j + ∂ y2k . 2 wj j =1

(1.36)

k=1

The processes these operators generate are called Bessel processes. For a recent paper on this subject see, for example, [5]. 1.3 PERTURBATION THEORY The next step is to use these estimates for the model problems in a perturbative argument to prove existence and regularity for a generalized Kimura diffusion operator L on a manifold with corners, P. The boundary of a manifold with corners is a stratified space, which produces a new set of difficulties. To overcome this we use an induction on the maximal codimension of the strata of b P. The induction starts with the simplest case where b P is just a manifold, and P is then a manifold with boundary. In this case, we can use the model operators to build a parametrix for the solution operator to the heat equation in a neighborhood of the

t . It is a classical fact that there is an exact solution operator, Q

t , defined boundary, Q b i in the complement of a neighborhood of the boundary, for, in any such subset of P, L is a non-degenerate elliptic operator. Using a partition of unity these operators can

t , for the solution operator. The Laplace be “glued together” to define a parametrix, Q transform ∞

t dt R(μ) = eμt Q (1.37) 0

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is then a right parametrix for (μ − L)−1 . Using the estimates and analyticity for the model problems, and the properties of the interior solution operator, we can show that

(μ − L) R(μ) = Id +E(μ),

(1.38)

where E(μ) is analytic in ⺓ \ (−∞, 0], and the Neumann series for (Id +E(μ))−1 converges in the operator norm topology for μ in sectors | arg μ| ≤ π − α, for any α > 0, if |μ| is sufficiently large. This allows us to show that

(μ − L)−1 = R(μ)(Id +E(μ))−1

(1.39)

is analytic and satisfies certain estimates in Tα,R = {μ : | arg μ| < π − α,

|μ| > R},

(1.40)

for any 0 < α, and R depending on α. For t in the right half plane we can now reconstruct the heat semi-group acting on the H¨older spaces:  1 et L = (μ − L)−1 eμt dμ (1.41) 2πi bTα,R

for an appropriate choice of α. This allows us to verify that et L has an analytic continuation to Re t > 0, which satisfies the desired estimates with respect to the anisotropic H¨older spaces defined above. The proof for the general case now proceeds by induction on the maximal codimension of the strata of b P. In all cases we use the model

t near the maximal codimensional part operators to construct a boundary parametrix Q b of b P. The induction hypothesis provides an exact solution operator in the “interior,”

t . A key step in

t , with certain properties, which we once again glue together to get Q Q i the argument is to verify that the heat operator we finally obtain satisfies the induction hypotheses. The representation of et L in (1.41) is a critical part of this argument. 1.4 MAIN RESULTS With these preliminaries we can state our main results. The sharp estimates for operators et L and (μ − L)−1 are phrased in terms of two families of H¨older spaces. For k,γ k,2+γ k ∈ ⺞0 and 0 < γ < 1, we define the spaces ᏯW F (P), ᏯW F (P), and their “heatk,γ k,2+γ space” analogues, ᏯW F (P × [0, T ]), ᏯW F (P × [0, T ]), see Chapter 5. For example: 0,γ in the 1-dimensional case f ∈ ᏯW F ([0, ∞)) if f is continuous and | f (x) − f (y)| sup √ √ γ < ∞; x = y | x − y| 0,2+γ

(1.42) 0,γ

it belongs to ᏯW F ([0, ∞)) if f, ∂x f, and x∂x2 f all belong to ᏯW F ([0, ∞)), with lim

x→0+ ,∞

x∂x2 f (x) = 0.

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INTRODUCTION k,γ

0,γ

For k ∈ ⺞, we say that f ∈ ᏯW F ([0, ∞)), if f ∈ Ꮿk ([0, ∞)), and ∂xk f ∈ ᏯW F ([0, ∞)). 0,γ A function g ∈ ᏯW F ([0, ∞) × [0, ∞)), if g ∈ Ꮿ0 ([0, ∞) × [0, ∞)), and |g(x, t) − g(y, s)| √ < ∞, √ γ (x,t ) =(y,s) [| x − y| + |t − s|] sup

√

(1.43)

etc. Much of this monograph is concerned with proving detailed estimates for the model problems with respect to these H¨older spaces and then using perturbative arguments to obtain analogous results for a general Kimura diffusion on an arbitrary compact manifold with corners. To describe the uniqueness properties for solutions to these equations, we need to consider the geometric structure of the boundary of P, and its relationship to L. As noted b P is a stratified space, with hypersurface boundary components {1, j : j = 1, . . . , N1 }. A boundary component of codimension n is a component of an intersection 1,i1 ∩ · · · ∩ 1,in ,

(1.44)

where 1 ≤ i 1 < · · · < i n ≤ N1 . A component of b P is minimal if it is an isolated point or a positive dimensional manifold without boundary. We denote the set of minimal components by b Pmin . Fix a generalized Kimura diffusion operator L. Let {ρ j : j = 1, . . . , N1 } be defining functions for the hypersurface boundary components. We say that L is tangent to 1, j if Lρ j 1, j = 0, and transverse if there is a c > 0 so that Lρ j 1, j > c.

(1.45)

D EFINITION 1.4.1. The terminal boundary of P relative to L, b Pter (L), consists of elements of b Pmin to which L is tangent, along with boundary strata,  of P to which L is tangent, and such that L  = L  is transverse to all components of b. For the model space we say that ˙ 1 (⺢n+ × ⺢m ) ∩ Ꮿ2 ((0, ∞)n × ⺢m ) f ∈ Ᏸ2W F (⺢n+ × ⺢m ) ⊂ Ꮿ

(1.46)

if the scaled second derivatives x i ∂x2i f (x; y), x i x j ∂x2i x j f (x, y), x i ∂x2i yl f (x, y), ∂ y2l yk f (x, y)

(1.47)

extend continuously to ⺢n+ × ⺢m . We also assume that x i x j ∂x2i x j f (x, y) tends to zero if either x i or x j goes to zero, and x i ∂x2i f (x; y) and x i ∂x2i yl f (x, y) go to zero as x i goes to zero. A function f ∈ Ꮿ1 (P) ∩ Ꮿ2 (int P) belongs to Ᏸ2W F (P) if it belongs to these local spaces in each local coordinate chart. Using a variant of the Hopf maximum principle, we can prove T HEOREM 1.4.2. Let P be a compact manifold with corners, and L a generalized Kimura diffusion defined on P. Suppose that L is either tangent or transverse to every hypersurface boundary component of b P, and let b Pter (L) denote the set of terminal components of the boundary stratification relative to L. The cardinality of the

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set b Pter (L) equals the dimension of the null-space of L acting on Ᏸ2W F (P), which ∗ is also the dimension of ker L . The null-space of L is represented by smooth non∗ negative functions; the null-space of L by non-negative measures supported on the components of b Pter (L). The existence and regularity results for the heat equation defined by a general Kimura diffusion, L, on a manifold with corners, P, are summarized in the next two results: T HEOREM 1.4.3. Let P be a compact manifold with corners, L a generalized k,γ Kimura diffusion on P, k ∈ ⺞0 and 0 < γ < 1. If f ∈ ᏯW F (P), then there is a unique solution k,γ v ∈ ᏯW F (P × [0, ∞)) ∩ Ꮿ∞ (P × (0, ∞)), to the initial value problem (∂t − L)v = 0 with v( p, 0) = f ( p).

(1.48)

This solution has an analytic continuation to t with Re t > 0. We have a similar result for the inhomogeneous problem: T HEOREM 1.4.4. Let P be a compact manifold with corners, L a generalized k,γ Kimura diffusion on P, k ∈ ⺞0 , 0 < γ < 1, and T > 0. If g ∈ ᏯW F (P × [0, T ]), then there is a unique solution k,2+γ u ∈ ᏯW F (P × [0, T ]) to (∂t − L)u = g with u( p, 0) = 0,

(1.49)

which satisfies estimates of the form uW F,k,2+γ ,T ≤ Mk,γ exp(Ck,γ T )gW F,k,γ ,T .

(1.50) k,2+γ

We also have a result for the resolvent of L acting on the spaces ᏯW F (P), showing that (μ − L)−1 is an elliptic operator with respect to our scales of Banach spaces. T HEOREM 1.4.5. Let P be a compact manifold with corners, L a generalized Kimura diffusion on P, k ∈ ⺞0 , 0 < γ < 1. The spectrum, E, of the unbounded, closed operator L, with domain k,2+γ

k,γ

ᏯW F (P) ⊂ ᏯW F (P),

is independent of k, and γ . It is a discrete set lying in a conic neighborhood of (−∞, 0]. The eigenfunctions belong to Ꮿ∞ (P). k,2+γ

k,γ

Remark. Note that ᏯW F (P) is not a dense subspace of ᏯW F (P).

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INTRODUCTION

1.5 APPLICATIONS IN PROBABILITY THEORY The principal sources for operators of the type studied here are infinite population limits of Markov chains in population genetics, and certain classes of “linear” models in mathematical finance. In this context the operator L, acting on a dense domain in Ꮿ0 (P) is called the backward Kolmogorov operator. Its formal adjoint, which acts on the dual space, ᏹ(P), of finite signed Borel measures on P, is the forward Kolmogorov operator. The standard way to address the adjoint operator is to study the martingale problem associated with L on Ꮿ0 ([0, ∞); P). Letting ω ∈ Ꮿ0 ([0, ∞); P), for each t ∈ [0, ∞), we define x(t) : Ꮿ0 ([0, ∞); P) → P, by x(t)[ω] = ω(t). We let Ᏺt denote the σ -field generated by {x(s) : 0 ≤ s ≤ t} and Ᏺ the σ -field generated by {x(s) : s ≥ 0}. For each q ∈ P, a probability measure ⺠q on (Ꮿ0 ([0, ∞); P), Ᏺ) is a solution of the martingale problem associated with L and

starting from q ∈ P at time t = 0, if

⺠q (x(0) = q) = 1 and

⎧ ⎨ ⎩

t L f (x(s))ds

f (x(t)) − 0

⎫ ⎬

(1.51)

⎭ t ≥0

is a ⺠q -martingale with respect to {Ᏺt }t ≥0, see [39]. The existence results in Theorems 1.4.3 or 1.4.5 suffice to prove that the associated martingale problem has a unique solution. A standard argument then shows that the paths for associated strong Markov process remain, almost surely, within P. From this we can deduce a wide variety of results about the forward Kolmogorov equation, and the solutions of the associated stochastic differential equation. The precise nature of these results depends on the behavior of the vector field V along b P. As this analysis requires techniques quite distinct from those employed here, we defer these questions to a future, joint publication with Daniel Stroock. Using the Lumer-Phillips theorem, these results also suffice to prove that the Ꮿ0 (P)graph closure, L, of L acting on Ꮿ3 (P) is the generator of a strongly continuous contraction semi-group. At present we have not succeeded in showing that the resolvent of L is compact, and will return to this question in a later publication. We have nonethe∗ less been able to characterize the null-space of adjoint operator L , under a natural clean intersection condition for the vector field V . This allows for an analysis of the asymptotic behavior of the solution to ∂t ν − L ∗ ν = 0, as t → ∞, see formula (12.63), and (12.62), for the asymptotics of et L f. In [1, 4] Bass, Perkins, et al. have employed methods, similar to our own, to study operators of the form LBP =

n  √ i, j =1

x i x j ai j (x)∂x2i x j +

n  i=1

bi (x)∂xi

(1.52)

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acting on functions in Ꮿ2b (⺢n+ ). Here bi (x) ≥ 0 along b⺢n+ . They have also considered other degenerate operators of this general type. Their main goal is to show the uniqueness of the solution to the martingale problem defined by L B P . To that end they introduce weighted H¨older spaces, which take the place of our anisotropic spaces. In the 1-dimensional case, the weighted γ -semi-norm is defined by  | f (x) − f (x + h)| γ 2 [| f ]| B P,γ = sup (1.53) x . hγ x∈⺢+ ; h>0 They prove estimates for the heat kernels of model operators, equivalent to L b,0 , with respect to these H¨older spaces. Under a smallness assumption on the off-diagonal elements of the coefficient matrix (ai j (x)), they are able to control the error terms introduced by replacing L B P by the model operator L B P,0 =

n 

[x i aii (0)∂x2i + bi (0)∂xi ]

(1.54)

i=1

well enough to construct a resolvent operator (L B P − μ)−1 for μ > 0. This suffices for their applications to the martingale problem defined by L B P . Notice that with this approach, only “pure corner” models are used, and no consideration is given to operators of the form L b,m with m > 0. For domains much more general than ⺢n+ it is difficult to see how to make such an approach viable. The operators we treat are somewhat more restricted, in that we take the coefficients of the off-diagonal terms to have the form x i x j ai j (x; y). Our method could equally well be applied to operators of the form considered by Bass and Perkins, i.e., √ with x i x j replaced by x i x j , if we were to append smallness hypotheses for the offdiagonal terms, similar to those they employ. We briefly consider this more general class of operators in Section 11.3. After slightly modifying the definitions of the higher order H¨older norms to include certain increasing weights, many of our results could be generalized to include certain non-compact cases. Our aims were of a more analytic character, and take us far beyond what is needed to show the uniqueness of the solution to the martingale problem. This leads us to consider such things as the higher order regularity of solutions with smoother initial data, the analytic extension of the semi-group in time, and the higher order mapping properties of the resolvent operator. We also show the ellipticity of the resolvent, with a gain of 2 derivatives with respect to the anisotropic H¨older norms. While this does not appear explicitly in [4], a similar result, with respect to the weighted H¨older norms, should follow from what they have proved. 1.6 ALTERNATE APPROACHES We wish to mention a few other approaches which have been successfully employed to study similar degenerate elliptic and parabolic equations. As outlined above, the approach here is a hybrid one: we use the explicit kernels for the solution operators of the model problems to derive a priori estimates in adapted H¨older spaces, and then,

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INTRODUCTION

using these a priori estimates, carry out a perturbation argument to pass from the model problem to the actual problem. A guiding principle throughout is to frame definitions and constructions based on the geometry of the underlying WF metrics. It is also viable to study such problems using only a priori estimates, or through a purely parametrix based approach. We have already mentioned the papers [11], [12] and [28]. Each of these treats very similar classes of degenerate operators on a manifold with boundary, with the defining function for the boundary multiplying either just the second normal derivative or else all second derivative terms, respectively. Motivation for these papers comes out of the study of the porous medium equation. The work of Daskalopoulos and Hamilton is based on a priori estimates derived through the maximum principle. Koch’s paper, by contrast, uses some powerful techniques involving singular integral operators to derive similar estimates. It is not immediately apparent, but the model operators L 1 := x∂x2 +

m 

∂ y2j + b∂x ,

j =1 m 

L 2 := x(∂x2 +

j =1

and

∂ y2j ) + b∂x

are essentially √ equivalent. Indeed, replacing L 1 by x L 1 and changing variables by setting ξ = x gives  1 1 2 2 1 ξ ∂ξ + (b − )ξ ∂ξ + ξ 2 ∂ y2j , 4 2 2 m

j =1

which has the same structure as x L 2 . In this representation the indicial roots of both x L 1 and x L 2 are α ∈ {0, 1 − b}. The functions ξ 2α (x α ) are solutions of the normal equation  1 1 2 2 1 ξ ∂ξ + (b − )ξ ∂ξ + λξ 2 ξ α = O(ξ α+1 ), (1.55) 4 2 2 which evidently do not depend on λ. The operators x L 1 and x L 2 are uniformly degenerate operators on a manifold with boundary. By definition, any such operator is one which can be written as sums of products of the basic vector fields x∂x and x∂ y j . Thus, a model prototype for second order “elliptic” differential operators has the form (x∂x )2 +

m 

(x∂ y j )2 + bx∂x .

(1.56)

j =1

In other words, the linear operators considered in [11], [12], [28] and [23, 24] are subsumed into the more general framework of uniformly degenerate operators.

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Another instance of a much-studied operator of this form is the Laplacian on hyperbolic space, ⺘n , which has the form (1.56) with b = 1 − n. A key difference between the usual theory developed for that operator, and indeed also for the linearizations of the porous medium equations in [11], [23, 24] and [28], and our study of the generalized Kimura diffusions is that the implicit boundary condition (1.23) used here is of Neumann type (at least when b < 1). In other words, for the hyperbolic Laplacian one typically picks out the solution determined by a simple growth condition, e.g., ⺘n u = f with u ∈ L 2 . Neumann conditions are well-known, even in the nondegenerate setting, to have a more global nature. On the other hand, a very important feature of our class of generalized Kimura operators L is the fact that the indicial root 0 is constant, and in particular is independent of both the first order coefficient b as well as the spectral parameter λ when we study the resolvent of L. This is in marked contrast with ⺘n − λ where the indicial roots depend on the spectral parameter. While this property of Kimura diffusions is not inconsistent with the general theory of elliptic uniformly degenerate operators, it indicates a special feature of these operators that makes tractable much of the analysis here. A comprehensive study of uniformly degenerate elliptic operators on manifolds with boundary, and of the slightly more general class of edge operators, was undertaken in [32]. The emphasis in that paper is on mapping properties on weighted Sobolev and H¨older spaces, and on the construction of parametrices using the methods of geometric microlocal analysis. The focus is on identifying the precise behavior of the Schwartz kernels of the various operators (the resolvent operator, heat kernel, etc.) along the boundaries and near to the intersection of the diagonal with the boundaries. These techniques give very detailed information, but have the disadvantage that they require an elaborate formalism. The geometric-microlocal methods and those employed in [11, 28] have been developed only for operators on manifolds with boundary, but not corners. It has proved challenging to generalize the microlocal approach to situations where the boundary is stratified, but see [33, 34]. It is likely that the amount of effort required to do this would be at least what has been done in the present monograph. Successful completion of such a program would give a precise description of the resolvent kernel itself, which would more than suffice to prove local regularity results. In this monograph we do not give precise descriptions of the resolvent or heat kernels, nor have we established local regularity results. 1.7 OUTLINE OF TEXT The book is divided in three parts: I. Wright-Fisher Geometry and the Maximum Principle: Chapters 2.1-3. Chapter 2.1 introduces the geometric preliminaries needed to analyze generalized Kimura diffusions. In Chapter 2.2 we show that coordinates (x 1 , . . . , x M ; y1 , . . . , y N−M )

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INTRODUCTION

can be introduced in the neighborhood of a boundary point of codimension M so that the boundary is locally given by {x 1 = · · · = x M = 0} and the second order purely normal part of L takes the form M  j =1

x i ∂x2i .

(1.57)

This generalizes a 1-dimensional result in [20]. In Chapter 3 we prove maximum principles for the parabolic and elliptic equations, (∂t − L)u = g and (μ − L)w = f, respectively,

(1.58)

from which the uniqueness results follow easily. Of particular note is an analogue of the Hopf boundary point maximum principle, which allows very detailed anal∗ yses of the ker L and ker L . II. Analysis of Model Problems: Chapters 4–9. In Chapter 4 we introduce the model problems and the solution operator for the associated heat equations. These operators, L b,m =

m  j =1

[x j ∂x2j + b j ∂x j ] +

m 

∂ y2l ,

(1.59)

l=1

act on functions defined on Sn,m = ⺢n+ × ⺢m , where n +m = N, and give a good approximation for the behavior of the heat kernel (∂t − L)−1 in neighborhoods of different types of boundary points. We state and prove elementary features of these operators, that generalize results proved in [15], and show that the model heat operators have an analytic continuation to the right half plane: H+ = {t : Re t > 0}.

(1.60)

In Chapter 5 we introduce the degenerate H¨older spaces on the spaces Sn,m , and their heat-space counterparts on Sn,m × [0, T ]. These are, in essence, H¨older spaces defined by the incomplete metric on Sn,m given by 2 dsW F =

n  dx2 i

j =1

xi

+

m 

d yl2 .

(1.61)

l=1

We also establish the basic properties of these spaces. Chapters 6–9 are devoted to analyzing the heat and resolvent operators for the model problems acting on the H¨older spaces defined in Chapter 5. This is a very long and tedious process because many cases need to be considered and, in each case, many estimates are required. Conceptually, however, these results are elementary. The estimates are pointwise estimates done in H¨older spaces, which

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means one can vary a single variable at a time. As the model heat kernels are products of 1-dimensional heat kernels, this reduces essentially every question one might want to answer to one of proving estimates for the 1-dimensional kernels. We call this the 1-variable-at-a-time method. In higher dimensions, the resolvent kernel, which is the Laplace transform of the heat kernel, is not a product of 1-dimensional kernels. This makes it far more difficult to deduce the mapping properties of the resolvent from its kernel, and explains why we use the representation as a Laplace transform. The proof of the estimates on the 1-dimensional heat kernels, defined by the operators x∂x2 + b∂x are given in Appendix A. Analogous results for the Euclidean heat kernel are stated in Chapter 8. The proofs of these lemmas, which are elementary, are left to the reader. A notable feature of the estimates for the degenerate model problem is the fact that the constants remain uniformly bounded as b → 0. This despite the fact that the character of the heat kernel changes quite dramatically at b = 0, see (1.29) and (1.31). This is also in sharp contrast to the analysis of similar problems in [11], where a positive lower bound is assumed for the coefficient of the analogous vector field. III. Analysis of Generalized Kimura Diffusions: Chapters 10–12. This part of the book represents the culmination of all the work done up to this point. We consider a generalized Kimura diffusion operator L defined on a compact manifold with corners P. In Chapter 10 we prove the existence of solutions to the heat equation (∂t − L)u = g in P × (0, T ] with u( p, 0) = f ( p), (1.62) k,γ

k,2+γ

with data in (g, f ) ∈ ᏯW F (P × [0, T ]) × ᏯW F (P). We show that the solution k,2+γ belongs to ᏯW F (P × [0, T ]) (Theorems 10.0.2 and 10.5.2). The case g = 0 provides a solution to the Cauchy problem, but it is not optimal as regards either the regularity of the solution, or the domain of the time variable, defects that are corrected in Chapter 11. The proof of these results is an intricate induction argument, where we induct over the maximal codimension of b P. This argument allows us to handle one stratum at a time. The underlying geometric fact is a “doubling theorem,” which shows that any neighborhood, complementary to the highest codimension stratum of b P, can be embedded into a manifold with  where the maximum codimension of b P  is one less than that of b P, corners P (Theorem 10.2.1). This explains why we need to consider domains well beyond those that can be easily embedded into Euclidean space. We first treat the lowest differentiability case (k = 0) and then use an extension of the contraction mapping theorem to towers of Banach spaces (Theorem 10.8.1), to obtain the mapping results for k > 0. These results (even in the k = 0 case) suffice to prove that the graph closure of L acting on Ꮿ3 (P) is the generator of strongly continuous semi-group in Ꮿ0 (P). 0,2+γ

We next consider the operators L γ , defined as L acting on the domain ᏯW F (P). 0,2+γ 0,γ As a map from ᏯW F (P) to ᏯW F (P), L γ is a Fredholm operator of index 0. In

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Chapter 11 we use essentially the same parametrix construction as used to prove Theorem 10.0.2 to prove the existence of the resolvent operator (μ − L γ )−1 : ᏯW F (P) −→ ᏯW F (P), k,γ

k,2+γ

for μ in the complement of discrete set lying in a conic neighborhood of (−∞, 0]. These are the expected “elliptic” estimates for operators with this type of degenk,γ eracy. In fact the spectrum of L acting on ᏯW F (P) does not depend on k or γ , as the resolvent is compact and all eigenfunctions belong to Ꮿ∞ (P). Using the analyticity properties of the resolvent, we give an alternate construction, using a 0,γ contour integral, for the semi-group, acting on ᏯW F (P) : e

tL

1 = 2πi



et μ (μ − L)−1 dμ,

(1.63)

α

where α bounds a region of the form | arg μ| > π − α, and |μ| > Rα , (Theorem 11.2.1). As we can take α to be any positive number, this shows that the semi-group is holomorphic in the right half plane. Finally in Chapter 12 we give a good description of the null-space of L γ , and show that the non-zero spectrum of L γ lies in a half plane Re μ < η < 0. We also deduce various properties of the semi-group, defined by the graph closure, ∗ L, of L, acting on Ꮿ0 (P). The adjoint operator, L , is defined on a domain ∗ Dom(L ) ⊂ ᏹ(P), which is not dense. We describe the subspace ᏹ (P), ∗ defined by Lumer and Phillips, on which L defines a C0 -semi-group. Although we have not yet proved the compactness of the resolvent of L, we obtain a rather ∗ complete description of the null-space of L . Using this we give the long time 0,γ asymptotics for et L f, assuming that f ∈ ᏯW F (P), for any 0 < γ , as well as ∗

those for et L ν, for ν in ᏹ (P). Finally we consider the problem of finding “irregular” solutions to the equation Lw = f, when the maximum principle precludes the existence of a regular solution.

IV. Proof of 1-dimensional Estimates: Appendix A. In the Appendix we give careful proofs of the estimates for the degenerate, 1-dimensional heat kernels used in the perturbation theory. These arguments are complicated by the fact that the heat kernel displays both the additive and multiplicative group structures on ⺢+ :  xy  dy  y b x+y ktb (x, y)d y = . (1.64) e− t ψb 2 t y t The arguments√involve Taylor’s theorem, the asymptotic expansion of the heat √ | x− y|

√ tends to infinity and Laplace’s method. We obtain mapkernel where t ping properties for b > 0, with uniform constants as b tends to zero. Using k,γ k, γ compactness of the embeddings, ᏯW F → ᏯW F , for  γ < γ , e.g., we then extend these results to b = 0.

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1.8 NOTATIONAL CONVENTIONS • We use ⺢+ = [0, ∞), hence Ꮿ∞ c (⺢+ ) consists of smooth functions on ⺢+ , supported in finite intervals [0, R]. • The right half plane is denoted H+ = {t ∈ ⺓ : Re t > 0}.

(1.65)

Sn,m = ⺢n+ × ⺢m .

(1.66)

• We let

• For b = (b1 , . . . , bn ) ∈ ⺢n+ , the model operator is: L b,m =

m  j =1

[x j ∂x2j + b j ∂x j ] +

m 

∂ y2l .

(1.67)

l=1

• If a(t) is an operator valued function, acting on functions in a space ᐄ, and g : [0, T ] → ᐄ is measurable, then the notation At g usually means t t

Ag=

a(t − s)g(s)ds.

(1.68)

0

• For φ ∈ [0, π2 ) we define the sector Sφ = {t ∈ ⺓ : | arg t|
0 such that the flow by the one-parameter family of diffeomorphisms t associated to the vector field V is defined on all of H for 0 ≤ t < . This gives a diffeomorphism between H × [0, ) and some neighborhood ᐁ of H . A rescaling in t gives a diffeomorphism from H × [0, 1).   L EMMA 2.1.4. Let K be a corner of codimension  in P. Let B+ = {x : |x| <   1} ∩ ⺢+ . Then there is a diffeomorphism between K × B+ and a neighborhood ᐁ of K in P.

P ROOF. Let H1, . . . , H be the boundary hypersurfaces which intersect along K . Let V j be an inward pointing vector fieldtransversal to H j , as constructed above. For  any x ∈ B+ with |x| < , let Vx = x j V j and x be the time one flow of the one-parameter family of diffeomorphisms associated to Vx . Then  K ×  B+  (q, x) −→ x (q)

is the desired mapping.



L EMMA 2.1.5. For each boundary face H of P there is a function ρ H which is everywhere positive in P \ H and which vanishes on H and has non-vanishing differential there. Any such function is called a boundary defining function for H . P ROOF. This may be constructed using a partition of unity exactly as in the first lemma. Alternately, if ᐁ is a neighborhood of H, which has been identified diffeomorphically with a product H × [0, 1), then we can set ρ H to equal the projection onto the second coordinate in this neighborhood. This function is then extended to the rest of P as a strictly positive function using a partition of unity.  There is one other construction which plays an important role below. We present it in a sequence of two lemmas.  L EMMA 2.1.6. For each boundary face H , there is a new manifold with corners P which is obtained by doubling P across H .  be the disjoint union of two copies of P identified by the identity P ROOF. Let P mapping along H . If we wish to work within the setting of oriented manifolds, then one copy should be P itself and the other −P, i.e., P with the opposite orientation. We give this space the structure of a manifold with corners by specifying a collection of coordinate charts. First, use all charts of P which are disjoint from H . Then, near any point p ∈ H , choose a chart ψ : ᐁ → (⺢+ ) × ⺢ N−−1 for H as a manifold with corners, and define the extended chart ψ˜ : ᐁ × (−, ) → (⺢+ ) × ⺢ N−−1 × ⺢ by  ˜ ψ(q, t) = (ψ(q), t). This provides a chart around all points of the image of H in P, and it is clear that the transition functions are smooth.  L EMMA 2.1.7. Let P be a compact manifold with corners. Let K be the corner of  which is a manifold with corners maximal codimension n. Then there is a new space P, only up to codimension n − 1, obtained by doubling P across K , such that P \ K is  identified with an open subset of P.

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Remark. In the proof of the main existence theorem we use a related though somewhat different doubling construction, see Theorem 10.2.1. P ROOF. Notice that K is a closed, and possibly disconnected, manifold of dimension N − n. For simplicity we assume that K is connected, but removing this only complicates the notation slightly. We first define the radial blowup of P along the submanifold K . Let B be the unit ball around 0 in ⺢n+ , which we describe via the polar n = {θ ∈ ⺢n+1 : |θ | = 1, θ ≥ 0 ∀ j }. coordinates (r, θ ) where 0 ≤ r < 1 and θ ∈ S+ j n This coordinate system is degenerate at r = 0 since the entire spherical orthant {0}× S+

is collapsed to a point. We define the radial blowup of B at 0, denoted B = [B, {0}], by n replacing the origin by a copy of S+ ; in other words,

B is simply a copy of the cylinder n [0, 1) × S+ .

= [P; K ]. In terms of the Next, define the radial blowup of P along K , denoted P identification of the tubular neighborhood ᐁ of K in P as K × B, we replace each B by

B. This space is still a manifold with corners up to codimension n; the codimension n with K . There is a new, possibly n corners are now the products of the vertices of S+ n disconnected, hypersurface boundary, H0 = K × S+ (at r = 0). 

across the face H0 in the sense The final step is to define P to be the double of P of the previous lemma. This space no longer has any corners of codimension n. From  as an open set. the construction it is clear that P \ K embeds in P  An important class of manifolds with corners is provided by the regular polyhedra P ⊂ ⺢ N . Recall first that a polyhedron is a domain in ⺢ N whose boundary lies in a union of hyperplanes and is a finite union of regions {Q i }, each itself a polyhedron in a hyperplane Hi ⊂ ⺢ N . Of particular interest to us here are the convex polyhedra, which are by definition determined by a finite number of affine inequalities: P = {z ∈ ⺢ N : z · ω j ≥ c j , j = 1, . . . , A}, where {ω1 , . . . , ω A } ⊂ ⺢ N is a finite set of vectors, and the c j are real numbers. The various faces and corners of P are the subsets determined by replacing any subcollection of these inequalities by the corresponding equalities. Amongst the convex polyhedra we distinguish the subclass of regular convex polyhedra P. By definition, P is a regular convex polyhedron if it is convex and if near any corner, P is the intersection of no more than N half-spaces with corresponding normal vectors ω j linearly independent. It is clear from these definitions that any regular convex polyhedron is a manifold with corners. Namely, if p ∈ P lies in a corner of codimension , hence is an element of  independent hyperplanes {z · ω j = c j }, then there is an affine change of variables which carries a neighborhood of p in P to a neighborhood of 0 in ⺢+ × ⺢ N− . A polyhedron that is not regular or non-convex, when endowed with the smooth structure given by the embedding into Euclidean space, does not satisfy one or more of the defining properties of manifolds with corners. It should be noted that unlike convex polyhedra, which are always contractible, manifolds with corners can have very complicated topologies.

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2.2 NORMAL FORMS AND WRIGHT-FISHER GEOMETRY We are now in a position to define the general class of elliptic Kimura operators on a manifold with corners P. These, and their associated heat operators, are our main objects of study. The definition we give is coordinate-dependent, but we indicate how to formulate this in a coordinate-independent way. Our goal in this chapter is to show that there is a local normal form for any operator L in this class which shows that it can be regarded as a perturbation in a small enough neighborhood of one of the model Kimura operators L b,m introduced in the Introduction. The reduction to this normal form is assisted by use of geometric constructions with respect to a singular Riemannian metric on P. This metric (or any one equivalent to it) is also instrumental in the formulation of the correct function spaces on which we let L act; this is the topic of Chapter 5. The normal form in this multidimensional setting generalizes the normal form, originally introduced by Feller, which is fundamental in the analysis of the 1-dimensional case in [15]. D EFINITION 2.2.1. Let P be a manifold with corners. A second order differential operator L defined on P is called a generalized Kimura diffusion operator if it satisfies the following set of conditions: i) L is elliptic in the interior of P. ii) If q is a boundary point of P which lies in the interior of a corner of codimension n, then there are local coordinates (x, y), x = (x 1 , . . . , x n ), y = (y1 , . . . , ym ) so that in the neighborhood ᐁ = {0 ≥ x j < 1 ∀ j ≤ n, |yk | < 1 ∀ k ≤ m}

the operator takes the form L=

n  j =1

aii x i ∂x2i +

 1≤i = j ≤n

x i x j ai j ∂x2i x j + m n   i=1 k=1

x i bik ∂x2i yk +

m 

ckl ∂ y2k yl + V .

(2.4)

k,l=1

For simplicity we assume that all coefficients lie in Ꮿ∞ (P). We also assume that (ai j ) and (ckl ) are symmetric matrices. iii) The vector field V is inward pointing at all boundaries and corners of P. iv) The matrices (aii ) and (ckl ) are strictly positive definite. The distinguishing features are the simple vanishing of the coefficients of the second order terms normal to the boundary and the ellipticity in all other directions. This

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leads to a coordinate-invariant definition. Recall that any second order operator L in the variables x, y has a principal symbol σ2 (L)(x, y, ξ , η) = n  i, j =1

aˆ i j (x, y)ξi ξ j +

m n  

bˆik (x, y)ξi ηk +

i=1 k=1

m 

cˆkl (x, y)ηk ηl .

k,l=1

Here ξ and η are the dual (cotangent) variables associated to x and y. We require then that σ2 (L)(x, y, ξ , η) is non-negative for all (ξ , η), and strictly positive when all x j > 0 and (ξ , η) = (0, 0). Furthermore its characteristic set Char(L) = {(x, y, ξ , η) : σ2 (L)(x, y, ξ , η) = 0} is equal to the set of all conormal vectors to b P, or more precisely to the set of all points (q, ν) where q ∈ b P and ν ∈ Tq∗ P vanishes on the tangent spaces of all boundary hypersurfaces which contain q. Finally, we require that σ2 (L) vanishes precisely to first order at this set. It is immediate to see that any operator which satisfies the first definition satisfies all the coordinate-invariant conditions above. For the converse, observe that in any local coordinate system, at a point q in the interior of the face where x j = 0, the conormal is spanned by the covector ν = d x j , i.e., ξ j = 0 and all other ξi and ηk vanish. Hence if we write σ2 (L)(x, y, ξ , η) as a quadratic form as above, then for this (q, ν), aˆ i j (q) = 0 for all i ≤ n, and this vanishing is simple. This gives aˆ j j (x, y) = x j a j j (x, y) and for i = j , aˆ i j (x, y) = x i x j ai j (x, y). The rest of the verification is straightforward. As noted in the Introduction, we could enlarge our family somewhat by allowing terms of the form √ √ x i x j ai j ∂x2i x j and x i bik ∂x2i yk , (2.5) if we append a smallness hypothesis on the coefficients {ai j (x; y), bik (x; y)}. Operators of this type were analyzed by Bass and Perkins. Indeed we can consider operators of this more general type where the coefficients are smooth functions of the variables √ √ ( x 1 , . . . , x n ; y1 , . . . , ym ). We discuss this in more detail in Section 11.3, but leave a more thorough discussion of these generalizations to the interested reader. Let H be a boundary hypersurface of P, with ρ a Ꮿ2 -function, vanishing on H. The first order part V of L is tangent to H if and only if Vρ ρ=0 = 0.

(2.6)

From the form of the operator it is easy to see that this is true if and only if Lρ ρ=0 = 0.

(2.7)

In this case we say that L is tangent to H. If L is tangent to H, then there is a naturally induced operator L H acting on Ꮿ∞ (H ).

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D EFINITION 2.2.2. If L is tangent to H, then the restriction of L to H, L H , is given by the prescription L H u := (L u )  H ∀ u ∈ Ꮿ∞ (H ). Here  u is any smooth extension of u to a neighborhood of H in P. The operator L H is a generalized Kimura diffusion operator on the manifold with corners H . As we have already mentioned, it is very helpful to consider a singular Riemannian metric on P such that the second order terms of L agree with those in the Laplacian for g. The change of variables which brings L into a normal form is then simply a Fermi coordinate system for this metric. To do this, we regard the principal symbol of L as a dual metric on the cotangent bundle; we write this as     X Ao X X B X Ad 0 g −1 = + . (2.8) 0 C Bt X 0 Here, B is an n × m matrix, C is a positive definite m × m matrix, (and has all diagonal entries equal to zero) and X and Ad are the given by ⎛ ⎛ ⎞ x1 0 . . . 0 a11 0 . . . ⎜ 0 x2 . . . 0 ⎟ ⎜ 0 a22 . . . ⎜ ⎜ ⎟ X =⎜ ⎟ , Ad = ⎜ .. .. ⎝ ⎝ ⎠ . . 0

0

...

xn

0

0

Ao is off-diagonal diagonal matrices

⎞ 0 0 ⎟ ⎟ ⎟. ⎠ . . . ann

(2.9)

We then compute that the inverse of this matrix, i.e., the metric tensor itself, takes the form:  −1  o  d + A X A B g= (2.10)  .  C Bt d positive definite and diagonal, A o The block submatrices here are all smooth, with A  positive definite. having vanishing diagonal entries and C The metric g is singular when any x j vanishes, but it can be desingularized by changing variables via dx j √ dζ j = √ , i.e., ζ j = 2 x j . (2.11) xj In these coordinates, the metric takes the form g=

n  i=1

 aii dζi2 +

n  i, j =1

 ai j ζi ζ j dζi dζ j +

m n   i=1 k=1

 bik ζi dζi d yk +

m 

 ckl d yk d yl . (2.12)

k,l=1

The coefficients a˜ i j , b˜ik and c˜kl are smooth functions of (ζ12 , . . . , ζn2 ; y1 , . . . , ym ). This implies that the metric g can be extended by even reflection across each hyperplane ζ j = 0 to define a smooth, non-degenerate metric on a full neighborhood ᐂ of the origin in ⺢n+m . This means that each boundary hypersurface Si = {ζi = 0} is the fixed point

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set of the locally defined isometry ζi → −ζi , and hence Si , and any intersection S J = ∩ j ∈ J Si j , is totally geodesic. Let S be the intersection of all the Si , i.e., the corner of maximal codimension which intersects ᐂ; for simplicity, assume that its codimension is n. Assuming that ᐂ is sufficiently small, then for each p ∈ ᐂ there is a unique closest point  J ( p) to p in S J ; if S J = S, the maximal codimension corner, then we write this projection as  S . The signed distance function ρi ( p) = s-distg ( p, Si ) to each hypersurface is smooth. Abusing notation slightly, let y = (y1 , . . . , ym ) be the composition of the original set of local coordinates restricted to S with the projection  S . Then (ρ1 , . . . , ρn , y1 , . . . , ym ) is a new set of smooth local coordinates in ᐂ. Let us compute the metric coefficients of g with respect to these coordinates. In fact, we first compute the coefficients for the corresponding co-metric, i.e., the entries of the matrix   dρi , dρ j  dρi , d yl  g −1 = . (2.13) d yk , dρ j  d yk , d yl  From general considerations, since each of the ρi are distance functions, we have dρi , dρi  = 1,

i = 1, . . . , n,

in the entire neighborhood ᐂ. Now, because of the reflectional symmetries, dρ j is clearly orthogonal to dρi when either ρi or ρ j equal 0, and similarly, dρi is orthogonal il such that to d yk when ρi = 0; this means that there are smooth functions  αi j and β k . αi j , and dρi , d yk  = ρi β dρi , dρ j  = ρi ρ j  Inserting these expressions into the matrix above and changing coordinates by set√ ting ρ j = 2 x j , we obtain the matrix of coefficients for this co-metric. This has the form (2.8) with Ad = Idn . Finally, taking the inverse of this matrix gives the normal form we are seeking. We summarize this in the P ROPOSITION 2.2.3. Let q ∈ b P lie in a boundary face of codimension n. Then there is a neighborhood ᐁ of q and smooth local coordinates (x, y) in this neighborhood, with q corresponding to (0; 0), in terms of which L takes the form L=

n  i=1

x i ∂x2i +



 ckl ∂ yk ∂ yl +

1≤k,l≤m

 1≤i = j ≤n

x i x j ai j ∂xi ∂x j +

m n  

x i bil ∂xi ∂ yl + V .

(2.14)

i=1 l=1

 (x, y)) is a smooth family of positive Here V is an inward pointing vector field, and (ckl definite matrices.

D EFINITION 2.2.4. The coordinates introduced in this proposition are called adapted local coordinates centered at q.

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We have written this expression in such a way as to emphasize that the first two sums on the right are in some sense the principal parts of L, and the other second order terms should be regarded as lower order perturbations. The body of our work below is devoted to showing that this is exactly the case. On the other hand, the first order term V (or at least the part of it which is not tangent to b P) is definitely not a lower order perturbation. The key point in this normal form is that all of the coefficients of the leading terms x i ∂x2i are simultaneously equal to 1. By a linear change of the y coordinates we can also  (0, 0) = δ . Hence restricting to an even smaller neighborhood, and writing make ckl kl  the normal part of V at (0, 0) as ni=1 bi ∂xi , then we see that L b,m :=

n   i=1

m   x i ∂x2i + bi ∂xi + ∂ y2k

(2.15)

k=1

should provide a good model for L. Note that since V is inward pointed, we have bi ≥ 0 for each i .

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Chapter Three Maximum Principles and Uniqueness Theorems One of the most important features associated with any scalar parabolic or elliptic problem is the use of the maximum principle. Although this seems to give only qualitative properties of solutions, it can actually be used to deduce many quantitative results, including even the parabolic Schauder estimates. On a more basic level, it is the key ingredient in proving uniqueness of solutions to such an equation. We now develop the maximum principle and its main consequences, both for the model operators ∂t − L b,m on an open orthant, and for the general Kimura diffusion operators ∂t − L on a compact manifold with corners, as well as their elliptic analogues. This generalizes the results for the 1-dimensional case in [15]. Of particular note in this regard is a generalization of the Hopf boundary point maximum principle, given in Lemma 3.2.6. This result allows us to precisely describe the null-space of L and its adjoint L ∗ . 3.1 MODEL PROBLEMS We begin with maximum principles for the model operators. P ROPOSITION 3.1.1. Suppose that u is a subsolution of the model Kimura diffusion equation ∂t u ≤ L b,m u on [0, T ] × Sn,m such that u ∈ Ꮿ0 ([0, T ] × Sn,m ) ∩ Ꮿ1 ((0, T ] × Sn,m ),

(3.1)

and u ∈ Ꮿ2 away from the boundaries of Sn,m for t > 0. Suppose also that x j ∂x2j u → 0 as x j → 0 for each j ≤ n. Suppose finally that 2)

|u(x, y, t)| ≤ Aea(|x|+| y|

(3.2)

for some a, A > 0. Then sup

[0,T ]×Sn,m

u(x, y, t) = sup u(x, y, 0).

(3.3)

Sn,m

P ROOF. We show that if u(x, y, 0) ≤ 0, then u(x, y, t) < 0 for all 0 < t ≤ T . It is straightforward to check that the function e x/(τ −t ) on [0, τ )t × ⺢+ is a solution of the equation (∂t − L 0 )u = 0. Using the relation ∂xk L 0 = L k ∂xk , we obtain that ∂xk e x/(τ −t ) =

1 e x/(τ −t ) (τ − t)k 34

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is a solution of (∂t − L k )u = 0, and hence also, if k is the multi-index (k1 , . . . , kn ) ∈ ⺞n , then n  U1,τ,k (x, t) = (τ − t)−k j −1 e x j /(τ −t ) j =1

solves (∂t − L k )U1,τ,k = 0. Suppose that k j > b j for each j . Then (∂t − L b,m )U1,τ,k =

n 

(k j + 1 − b j )x j ∂x j U1,τ,k > 0, 0 ≤ t < τ.

j =1

Next define U2,τ ( y, t) =

1 2 e|y| /2(τ −t ). (τ − t)m/2

Finally, given the subsolution u in the statement of the proposition, set v(x, y, t) = u(x, y, t) − 1 U1,τ,k (x, t) − 2 U2,τ ( y, t) + 3 We see that

1 . 1+t

(∂t − L b,m )v < 0,

so that v is a strict subsolution of this equation. Let D R = [0, τ  ] × B R (0) and D R = {0} × B R (0) ∪ [0, τ  ] × b B R (0), B R (0) is the ball of radius R around the origin in ⺢n+ × ⺢m and 0 < τ  < τ . We claim that the supremum of v on D R is attained on D R but not at any one of the boundaries where any x j = 0. The fact that this supremum must occur on D R follows from the standard maximum principle. If the supremum were to occur when x j = 0, then using that u ∈ Ꮿ1 up to this boundary, we see that the corresponding derivative ∂x j v ≥ 0 at this point, which is impossible. This proves the claim. Finally, since u grows no faster 2 than ea(|x|+|y| ) , we can choose 1/τ < a and R sufficiently large to ensure that v is as negative as we wish on the entire side boundary (0, τ  ] × b B R (0). We conclude that u(x, y, t) ≤ 1 U1,τ,k (x, 0) + 2U2,τ ( y, 0) − 3 , for any 1 , 2 , 3 > 0. Now let these parameters tend to zero to see that u ≤ 0 when t > 0, as desired.  3.2 KIMURA DIFFUSION OPERATORS ON MANIFOLDS WITH CORNERS The corresponding result for a general variable coefficient Kimura diffusion, L on a manifold with corners, P, requires slightly stronger hypotheses. We let Ᏸ2W F (P) denote a certain subspace of Ꮿ1 (P) ∩ Ꮿ2 (int P) adapted to the degeneracies of L. In a

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neighborhood, U, of a boundary point p0 of codimension M we introduce local coordinates (x 1 , . . . , x M , y1 , . . . , y N−M ), so that the stratum of the boundary through p0 is locally given by  ∩ U = {(0, y) : | y| < 1}.

(3.4)

A function f ∈ Ꮿ1 (U ) ∩ Ꮿ2 (U ) belongs to Ᏸ2W F (U ) if, for 1 ≤ i, j ≤ M, and 1 ≤ l, m ≤ N − M the functions x i ∂x2i f, x i x j ∂xi ∂x j f, x i ∂xi ∂ yl f, ∂ yl ∂ ym f

(3.5)

extend continuously to b P ∩ U, with the first three types of expressions vanishing whenever x i or x j vanishes. These conditions are clearly coordinate invariant. D EFINITION 3.2.1. A function in Ꮿ1 (P) ∩ Ꮿ2 (int P) belongs to Ᏸ2W F (P) if its restrictions to neighborhoods of boundary points belong to each of these local Ᏸ2W F spaces. Our first result shows that on Ᏸ2W F (P), a Kimura diffusion is a dissipative operator. L EMMA 3.2.2. Let w ∈ Ᏸ2W F (P), and suppose that w assumes a local maximum at p0 ∈ P, then Lw( p0 ) ≤ 0. P ROOF. If p0 ∈ int P, then this is obvious, as L is strongly elliptic in the interior, and annihilates the constant function. Suppose that p0 is a boundary point of codimension n, and (x 1 , . . . , x n , y1 , . . . , ym ) are adapted local coordinates. We normalize so that p0 corresponds to 0; the stratum b P through p0 is locally given by  = {x 1 = · · · = x n = 0}. The regularity assumptions show that w restricted to  is locally Ꮿ2 and the local form for L given in (2.14) shows that   ckl ∂ yk ∂ yl w( p0 ) + V w( p0 ). (3.6) Lw( p0 ) = k,l

Since V ( p0) is inward pointing and p0 is a local maximum, it is clear that V w( p0 ) ≤ 0.

(3.7)

Since w  is locally Ꮿ2 , the second order part of Lw at p0 is also non-positive, thus proving the lemma.  In order to refine this result, we must describe in more detail the structure of b P, and the relationship of L to the various components of the stratification of b P. First, let {1, j : j = 1, . . . , N1 } denote the connected hypersurface boundary components of P and {ρ j } their respective defining functions. If  is a component of b P of codimension n, then there are n hypersurface boundary components {1, ji : i = 1, . . . , n} so that  is a connected component of the intersection 1, j1 ∩ · · · ∩ 1, jn .

(3.8)

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The first order part V is tangent to  near p0 , if and only if Vρ ji = 0 for i = 1, . . . , n; this is evidently equivalent to the condition Lρ ji = 0 for this collection of indices. If this holds at all points of , then we say that L is tangent to . If, on the other hand, there is a c > 0 so that Lρ ji  > c, for i = 1, . . . , n,

(3.9)

then we say that L is transverse to . These conditions are independent of the choice of defining function. We write b P T (L) for the union of boundary components to which L is tangent, and b P (L) for the union of boundary components to which L is transverse. The following non-degeneracy assumption about L simplifies many of the global results. D EFINITION 3.2.3. We say that L meets b P cleanly, if for each 1 ≤ j ≤ N1 , either Lρ j {ρ j =0} ≡ 0, or there exists a c j > 0, so that Lρ j {ρ j =0} > c j .

(3.10)

Briefly, for each j , the vector field is either tangent or transverse to 1, j , so cleanness prevents such behavior as V lying tangent to some 1, j only along a proper closed subset (possibly in b1, j ). A boundary component belongs to b P T (L) if and only if it is a component of the intersection of a collection of boundary faces to which L is tangent. A boundary component belongs to b P (L) if and only if it is a component of the intersection of a collection of boundary faces to which L is transverse. There may be boundary components that belong to neither of these extremes. Any boundary component  is itself a manifold with corners. If L is tangent to , then we write L  for the generalized Kimura diffusion defined by restriction of L to . It is clear that if U is any neighborhood of a point p0 in the interior of , if w ∈ Ᏸ2W F (U ), and if L is tangent to , then [Lw] ∩U = L  [w ∩U ].

(3.11)

The first basic result is the following. L EMMA 3.2.4. Let w ∈ Ᏸ2W F (P), and suppose that w is a subsolution of L, i.e., Lw ≥ 0. Then w cannot assume a local maximum in the interior of P, or in the interior of any component  ∈ b P T (L), unless w is constant on P, or on that component, respectively. P ROOF. Since L is a non-degenerate elliptic operator in int P, it follows from the standard strong maximum principle that w does not assume a local maximum in int P unless w is constant. The regularity hypothesis, and the assumption that L is tangent to , show that w  ∈ Ᏸ2W F (), and L  [w  ] ≥ 0.

(3.12)

Hence the first part of this proof applies to show that w cannot assume its maximum in int , unless w  is constant. 

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In order to apply this result to determine the null-space of L, we need to discuss two further types of boundary components. First, amongst the collection of all components of the stratification of b P, certain ones are minimal in the sense that they themselves have no boundary; these components are either points or closed manifolds. We denote by b Pmin the union of all such components; the different components of this set may have different dimensions. These minimal components are the minimal elements in the maximal well-ordered chains of boundary components, where the ordering is given by containment in the closure. L EMMA 3.2.5. Every component of b P is either minimal or else contains elements of b Pmin in its closure. P ROOF. This follows directly by induction on the maximal codimension of corners in P. If  ∈ b Pmin , then  is a manifold with corners with maximal codimension no more than one less than that of P. Hence there is some boundary component   of  which is minimal. Clearly   is also a boundary component of P, and since it has no boundary, it must be minimal for P as well.  T (L) = b P T (L) ∩ b P Note that if  ∈ b Pmin min (L), then either  is a point and all coefficients of L vanish at , or else dim  > 0 and L  is a non-degenerate elliptic operator. It follows immediately that if Lw ≥ 0 as above, and if w  attains a local T (L), then w  is constant. maximum on  ∈ b Pmin  Finally, the terminal boundary of P relative to L, denoted b Pter (L), consists of the union of boundary components  ⊂ b P T (L) such that L  is transverse to all components of b (i.e., b ∈ b (L  )). In particular, if L is transverse to all components T (L) have empty boundary, it of b P itself, then b Pter (L) = P. As elements of b Pmin T (L) ⊂ b P (L). follows that b Pmin ter There is a version of the Hopf boundary point lemma, adapted to this setting.

L EMMA 3.2.6. Let P be a compact, connected manifold with corners, and L a generalized Kimura diffusion operator that meets b P cleanly. Suppose that w ∈ Ᏸ2W F is a subsolution of L, Lw ≥ 0, in a neighborhood, U, of a point p0 ∈ b P which lies in the interior of a boundary component  ∈ b P (L). If w attains a local maximum at p0, then w is constant on U . This has an immediate and important consequence. L EMMA 3.2.7. Let P be a compact manifold with corners, and L a generalized Kimura diffusion on P, which meets b P cleanly. Suppose that L is transverse to every face of b P, and w ∈ Ᏸ2W F (P) is a subsolution of L. Then w is constant. This follows directly from Lemma 3.2.6. We defer the proof of Lemma 3.2.6 momentarily and derive its main consequence. The following result shows that, at least when L meets b P cleanly, the null-space of L on Ᏸ2W F (P) is finite dimensional. P ROPOSITION 3.2.8. Let P be a compact, connected manifold with corners and L a generalized Kimura diffusion which meets b P cleanly. Let w ∈ Ᏸ2W F (P) be a

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solution to Lw = 0. Then w is determined by its (constant) values on the components of b Pter (L). P ROOF. We prove this by induction on the dimension of P. If the dimension of P is 1, then P is an interval. The statement of the proposition in this case was established in [15]. Now suppose the result has been proved for all compact manifolds with corners P  of dimension N − 1, and all general Kimura diffusion operators L  on them. Assume that dim P = N, and that w  = 0 for all  ∈ b Pter (L). Obviously, if P has no boundary faces to which L is tangent, then P is itself a terminal boundary and we already have that w ≡ 0. We henceforth assume that b P T (L) = ∅. If w does not vanish identically, then we can assume that it is positive somewhere, and therefore attains a positive maximum somewhere in P. The induction hypothesis shows that for any boundary component  ∈ b P T (L), we have a solution w  , which vanishes on bter (L  ). This follows because the terminal components of b P relative to L, which are contained in the closure of , are the same as the terminal components of b relative to L  . Indeed, if L is tangent to some component 0 of b, then L  is also tangent to 0 . Furthermore, the restriction of L  to 0 is the same as L 0 , the restriction of L to 0 . Thus the condition that L  be transverse to all components of b is the same, whether restricting from P or . This means that w 0 vanishes on all 0 ∈ bter (L  ), and hence by induction, w  = 0. Note that w vanishes on every hypersurface boundary component to which L is tangent. Lemma 3.2.6 shows that w cannot attain a positive maximum on a boundary component to which L is transverse. Thus w must attain its maximum at a point p0 lying in a boundary component, 1 to which L is neither tangent, nor transverse. That is, 1 lies in the intersection of boundary faces some of which belong to b P T (L) and some of which L is transverse to. This implies that there is a boundary hypersurface 0 to which L is tangent, and such that 1 ⊂ b0 . As w = 0 on 0 and 1 ⊂ 0 , this contradiction establishes that w ≡ 0 on P.  Remark. This theorem is proved in [38] in the special case of the classical Kimura diffusion, without selection, acting on the N-simplex. Lemma 3.2.7 is a special case of this proposition. Two other corollaries are: C OROLLARY 3.2.9. If L is a generalized Kimura diffusion on the compact manifold with corners and L is everywhere tangent to b P, then any solution w ∈ Ᏸ2W F (P) to T . Lw = 0 is determined by its constant values on b Pmin = b Pmin C OROLLARY 3.2.10. If P is a compact manifold with corners, and L a generalized Kimura diffusion on P meeting b P cleanly, then the dimension of the null-space of L acting on Ᏸ2W F (P) is bounded above by the cardinality of the set b Pter (L). Remark. These results give powerful support for our assertion that the regularity condition w ∈ Ᏸ2W F (P) is a reasonable replacement for a local boundary condition, at

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least when L meets b P cleanly. In applications to probability, one often considers solutions of equations of the form Lw = g, where w satisfies a Dirichlet condition on b P. Frequently g is non-negative, and our uniqueness results easily imply that there cannot be a regular solution. The simplest example arises in the 1-dimensional case, with L = x(1 − x)∂x2 . The solution, w to the equation Lw = −1 with w(0) = w(1) = 0,

(3.13)

gives the expected time to arrive at {0, 1}, for a path of the process starting at 0 < x < 1. There cannot be a regular solution as x(1 − x)∂x2w(x) = −1 cannot converge to 0 as x → 0+ , 1− . The solution, given by w(x) = −[x log x + (1 − x) log(1 − x)], is plainly not regular. In applications to probability this situation often pertains. The fact that Lw ≥ 0 has no non-trivial regular solutions shows that the required solutions cannot be regular, and therefore involve the non-zero indicial root. We now turn to the proof of the “Hopf lemma”: P ROOF OF L EMMA 3.2.6. The proof of this lemma relies on the construction of barrier functions and a simple scaling argument. To motivate the argument we first give the proof for a model operator and a boundary point of codimension 1. Let (x, y) denote normalized local coordinates so that L = x∂x2 + b∂x +

m 

∂ y2l , with b > 0,

(3.14)

l=1

and w assumes a local max at the (0, 0). We assume that w is not constant. The strong maximum principle implies that there is a neighborhood U of 0 so that if (x, y) ∈ U and x > 0, then w(x, y) < w(0, 0). (3.15) For R, r positive numbers and

1 2

≤ α < 1, we define anisotropic balls in ⺢n+ × ⺢m :

+ (n, m) = {(x, y) ∈ ⺢n+ × ⺢m : |x α − r α |2 + | y|2 ≤ n R 2α } ∩ {x ≥ 0}. (3.16) B R,r,α

Here r = (r, . . . , r ), and x α = (x 1α , . . . , x nα ), etc. We now construct a non-negative local barrier, v λ,α,r that satisfies + + Lv λ,r > 0 in Br,r,α (1, m) \ B +r ,r,α (1, m), and v λ,r b Br,r,α (1,m) = 0, 2β

where β =

1 . 2α

(3.17)

Note that B +r ,r,α (1, m) is a compact subset of P lying a positive distance from b P. 2β

Figure 3.1 shows the set B +

1,1, 21

(2, 0) \ B + 1

1 4 ,1, 2

(2, 0) ⊂ ⺢2+ .

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1.5

1

0.5

0 0

Figure 3.1: The set B +

1,1, 21

between the black curves.

0.5

(2, 0) \ B + 1

1

1 4 ,1, 2

1.5

(2, 0) ⊂ ⺢2+ lies in the positive quadrant

We first define barrier functions in the 1-codimensional case, letting wλ,α,r = e−λ[(x

α −r α )2 +|y|2 ]

,

(3.18)

then

 Lwλ,r = 4λ2 [α 2 x 2α−1 (r α − x α )2 + |y|2]+   2λ α[b − (1 − α)]x α−1 (r α − x α ) − α 2 x 2α−1 − m wλ,r .

(3.19)

If 1 − b < α < 1, then we see that Lwλ,r (x, y) tend to +∞ as x tends to zero. As [α 2 x 2α−1 (r α − x α ) + |y|2 ] only vanishes at (r, 0) and (0, 0) (if α = 12 ) it is not difficult to see that for large enough λ we have that + Lwλ,α,r > 0 in Br,r,α (1, m) \ B +r ,r,α (1, m).

(3.20)

v λ,α,r = wλ,α,r − wλ,α,r (0, 0),

(3.21)

2β

Let + (1, m). b Br,r,α

so that the barrier vanishes on + (1, m) ⊂ U and a λ and α as above. The hypothesis that We fix an r so that Br,r,α w is non-constant shows that there is an  > 0 so that (w + v λ,α,r ) b B +r

2β

,r,α

(1,m)
0. (3.24) Under this change of variables, the model operator L becomes  1 [X ∂ X2 + b∂ X + ∂Y2l ]. μ m

(3.25)

l=1

That is, up to a positive factor, this operator is invariant under these changes of coordinate. √ If we let W (X, Y ) = w(μX, μY ), then evidently LW ≥ 0, and W attains a local maximum at (0, 0). The ball (X α − R α )2 + |Y |2 ≤ R 2α where R =

r μ

(3.26)

is contained in this coordinate chart. In the original coordinates we have L = x∂x2 + b∂x + +

m 

k=1 m 

∂ y2k +

m 

xal (x, y)∂x ∂ yl +

l=1

ckl (x, y)∂ yk ∂ yl +

k,l=1

n 

 b(x, y)∂x +

i=1

m 

dk (x, y)∂ yk , (3.27)

k=1

where  b(0, 0) = ckl (0, 0) = 0. Letting x = μX and y =

√ μY , we obtain:

 m m  1 √  √ X∂ X2 + b∂ X + Lμ = ∂Y2k + μ Xal (μX, μY )∂ X ∂Yl + μ k=1

m 

ckl (μX,

l=1

 m √ √ √  √ μY )∂Yk ∂Yl +  b(μX, μY )∂ X + μ dk (μX, μY )∂Yk .

k,l=1

k=1

(3.28) Even though we may let μ get very small, we can fix a positive R. It is then not hard to see that, with a possibly larger α < 1, by taking μ small enough we can arrange for + L μ wλ,α,R > 0 in B R,R,α (1, m) \ B +R

2β

,R,α

(1, m).

(3.29)

From this point the argument proceeds as before showing that if Lw ≥ 0 and w attains a maximum at p0 , then w is constant.

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Suppose that p0 is a point on a stratum of codimension n, we can choose (x, y) adapted local coordinates, with p0 corresponding to (0, 0), so that the operator takes the form: L=

n 

[x i ∂x2i + bi ∂xi ] +

i=1 n 

m 

∂ y2k +

k=1

x i x j ai j (x, y)∂xi ∂x j +

i = j =1

m n  

m 

x i ail (x, y)∂xi ∂ yl +

i=1 l=1 n 

k,l=1 m 

 bi (x, y)∂xi +

i=1

ckl (x, y)∂ yk ∂ yl + dk (x, y)∂ yk ,

(3.30)

k=1

where ckl (0, 0) =  bi (0, 0) = 0.

(3.31)

Let b = (b1 , . . . , bn ) > 0. For the model operator L b,m we consider barrier functions of the form   wλ,α,r = exp −λ(|x α − r α |2 + | y|2 ) . (3.32) Applying the model operator we see that ⎡ 2⎣ 2

L b,m wλ,α,r = 4λ

α

n 

⎤ x 2α−1 (x αj j

α 2

− r ) + | y|

2⎦

wλ,α,r +

j =1

⎤ n    [b j − (1 − α)]x α−1 − m ⎦ wλ,α,r . (r α − x αj ) − αx 2α−1 2λ ⎣α j j ⎡

(3.33)

j =1

In order for wλ,α,r to be a subsolution, we need to choose

1 2

≤ α < 1, so that

1 − min{b1 , . . . , bn } < α.

(3.34)

With such a choice of α, we see that L b,m wλ,α,r tends to +∞ as any x j tends to zero. We can therefore find λ0 so that for λ0 < λ we have + L b,m wλ,α,r > 0 in Br,r,α (n, m) \ B + r

(2n)β

where, as before, β =

1 2α .

,r,α

(n, m),

(3.35)

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As before we can scale the variables x = μX and y = Lμ =

√ μY to obtain

 n m n   1  √ [X i ∂ X2 i + bi ∂ X i ] + ∂Y2k + μ X i X j ai j (μX, μY )∂ X i ∂ X j + μ √ μ

i=1 m n  

k=1

X i ail (μX,

i=1 l=1

√ μY )∂ X i ∂Yl +

i = j =1 m 

ckl (μX,

√ μY )∂Yk ∂Yl +

k,l=1 n 

√ √  bi (μX, μY )∂ X i + μ

i=1

m 

dk (μX,

 √ μY )∂Yk .

(3.36)

k=1

A calculation shows that   n 1 4λ2 L μ wλ,α,r (X, Y ) = α 2 X i2α−1 (X iα − r α )2 + |Y |2 + μ i=1  √ 2 α ai j (μX, μY )X i X j (X iα − r α )(X αj − r α )+ i = j

√

μ

m n  

m  √ √ α α α αail (μX, μY )X i (X i − r )Yl + ckl (μX, μY )Yk Yl +

i=1 l=1

k,l=1

 n   $ % √ 2λ bi +  bi (μX, μY ) − (1 − α) X iα−1 (r α − X iα ) − α 2 X i2α−1 − i=1

 m  √ √ √ [1 + cll (μX, μY ) + μdl (μX, μY )Yl ] .

(3.37)

l=1

If we take r and μ sufficiently small, then the O(λ2 )-term is bounded below by a positive multiple of n  α 2 X i2α−1 (X iα − r α )2 + |Y |2 . (3.38) i=1

By taking α < 1 a little larger and possibly reducing μ, we can assure that √ + bi (μX, μY ) − (1 − α) : (X, Y ) ∈ Br,r,α (n, m); i = 1, . . . , n} min{bi + 

(3.39)

is strictly positive. With these choices, there is a λ0 so that if λ > λ0 , then + (n, m) \ B + r L μ wλ,α,r > 0 in Br,r,α

(2n)β

,r,α

(n, m).

(3.40)

Note also that ∂ X i wλ,α,r (X, Y ) tends to +∞ as X i → 0+ . Finally we set v λ,α,r = wλ,α,r − wλ,α,r (0, 0).

(3.41)

The argument then proceeds as before. We are assuming that w is a non-constant solution to Lw ≥ 0, which assumes a local maximum at (0, 0). This implies that

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w(X, Y ) < w(0, 0) for (X, Y ) ∈ int P. Thus we can choose an  > 0 so that for (X, Y ) ∈ b B + r ,r,α (n, m) we have the estimate (2n)β

(w + v λ,α,r )(X, Y ) < (w + v λ,α,r )(0, 0).

(3.42)

+ (n, m), we see that (w + v Since v λ,α,r vanishes on b Br,r,α λ,α,r ) must assume its max-

+ (n, m). As before this implies that the derivatives imum at a point p0 on b P ∩ Br,r,α ∂ X j w( p) tend to −∞ as p approaches p0, contradicting our assumptions about the smoothness of w. This completes the proof of the lemma. 

Remark. Because the derivatives of the barrier functions blowup at a rate dictated by the choice of α, the regularity hypotheses on w can be considerably weakened. These results do not address the case when L fails to meet b P cleanly. While a different argument is needed, it seems likely that a result like that in Proposition 3.2.8 remains true. In particular, the dimension of the null-space of L acting on Ᏸ2W F (P) should be finite dimensional. 3.3 MAXIMUM PRINCIPLES FOR THE HEAT EQUATION We now turn to maximum principles for the heat equation: P ROPOSITION 3.3.1. Let u be a subsolution of the Kimura diffusion equation ∂t u ≤ Lu on [0, T ] × P, where P is a compact manifold with corners, such that u ∈ Ꮿ0 ([0, T ] × P) ∩ Ꮿ1 ((0, T ] × P), and u(t, ·) ∈ Ᏸ2W F (P) for t > 0, then sup u( p, t) = sup u( p, 0).

[0,T ]×P

P

P ROOF. This is proved in almost exactly the same way as Proposition 3.1.1. Because P is compact and u(·, t) is continuous up to b P, there is no need to assume a growth condition on u. The hypotheses are such that we can verify as before that no local maximum occurs along b P × (0, T ], and by the usual maximum principle, there is also no local maximum in the interior of P when 0 < t ≤ T .  C OROLLARY 3.3.2. Let u 1 and u 2 be two solutions of (∂t −L b,m )u = g, u(x, y, 0) = 2 f (x, y) in ⺢+ × Sn,m such that |u j | ≤ C T ea(|x|+| y| ) for some C > 0 uniformly in any [0, T ] × Sn,m , satisfying the regularity hypotheses of Proposition 3.1.1. Then u 1 ≡ u 2 . Similarly, if u 1 and u 2 are two solutions of (∂t − L)u = f , u(·, 0) = f (·) in ⺢+ × P, satisfying the regularity hypotheses of Proposition 3.3.1, where P is a compact manifold with corners, then u 1 ≡ u 2 .

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Remark. The regularity assumption up to the boundaries where x j = 0 are fundamental. For example, if 0 < b < 1, then x 1−b is a stationary solution of (∂t − L b )u = 0 on ⺢+ × ⺢+ which is certainly subexponential as x → ∞. However, from results of [15] it follows that there is another solution w(x, t) to this equation with initial data w(x, 0) = x 1−b . This solution is smooth up to x = 0 for t > 0, so that w(x, t) = x 1−b for t > 0. Then x 1−b − w(x, t) is a homogeneous solution with zero Cauchy data at t = 0 and which has subexponential growth. It is neither Ꮿ1 up to x = 0, nor does it satisfy limx→0+ x∂x2 u(x, t) = 0. We record one other easy extension of these results. P ROPOSITION 3.3.3. Let L be a general Kimura operator on a compact manifold with corners P, and suppose that c ∈ Ꮿ0 (P). Suppose that u is a subsolution of the diffusion equation associated to L + c, i.e., ∂t u ≤ (L + c)u, such that u ∈ Ꮿ0 ([0, T ] × P) ∩ Ꮿ1 ((0, T ] × P), and u(·, t) ∈ Ᏸ2W F (P) for t > 0, then sup u( p, t) ≤ eαt sup u( p, 0),

[0,T ]×P

P

where α = ||c||∞ . Consequently, if u 1 and u 2 are two solutions of ∂t u = (L + c)u which satisfy the regularity conditions above and which have the same initial condition at t = 0, then u1 ≡ u2. Finally, we also state the corresponding maximum principle and uniqueness result for Kimura equations. P ROPOSITION 3.3.4. Let L be a general Kimura operator on a compact manifold with corners P and that c ∈ Ꮿ0 (P) is a non-positive function. Let f satisfy 0 ≤ (L + c) f and f ∈ Ᏸ2W F (P), then sup f ( p) = sup f ( p). P

bP

If f 1 and f 2 are any two solutions of (L + c) f = 0 which satisfy all the regularity assumptions above and which agree on b P, then f 1 ≡ f 2 . There is an important special case, where a much sharper result is true. P ROPOSITION 3.3.5. Let L be a general Kimura operator on a compact manifold with corners P and that c ∈ Ꮿ0 (P) is a non-positive function, with c b P strictly negative. Let f satisfy (L + c) f = 0, and also suppose that f ∈ Ᏸ2W F (P), then f ≡ 0.

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MAXIMUM PRINCIPLES AND UNIQUENESS THEOREMS

P ROOF. We show that f can neither attain a positive maximum nor a negative minimum, and hence is identically zero. Since we are considering the equation (L + c) f = 0, it suffices to show that f cannot attain a negative minimum. We suppose that f does attain a negative minimum. The regularity assumptions and the compactness of P show that f attains its minimum at some point p0 ∈ P. The strict maximum principle implies that p0 ∈ / int P. Suppose that p0 belongs to a point of the boundary of codimension M, so that in local coordinates Lf =

M 

[x i ai (x,

y)∂x2i

f + bi (x, y)∂xi f ] +

x i x j ai j (x, y)∂xi ∂x j f +

i, j =1

i=1 M n−M  

M 

x i cil (x, y)∂xi ∂ yl f +

i=1 l=1

n−M 

clm (x, y)∂ ym ∂ yl f +

l,m=1

n−M 

dm (x, y)∂ yl f.

(3.43)

l=1

At p0 = (0, y0 ) we see that the regularity assumptions show that L f ( p0 ) =

M  i=1

bi (0, y0 )∂xi f +

n−M 

clm (0, y0 )∂ ym ∂ yl f.

(3.44)

l,m=1

As p0 is a local minimum, the second order part is non-negative; since the vector field is inward pointing, so is the first order part. We therefore conclude that at p0 we have L f ( p0 ) + c( p0) f ( p0 ) > 0, contradicting our assumption that f is in the null-space of L + c. 

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Part II

Analysis of Model Problems

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Chapter Four The Model Solution Operators t In this chapter we introduce the model heat kernels k b,m , i.e., the solution operators for the model problems ∂t − L b,m , and then prove a sequence of basic estimates for these operators which are direct generalizations of the estimates for the 1-dimensional version of this problem considered in [15]. We recall and slightly extend the Ꮿ0 and Ꮿk theory in the 1-dimensional case proved in [15] and then derive the straightforward extensions of these results to higher dimensions. This sets the stage for the more difficult H¨older estimates for solutions, which is carried out in the next several chapters, and which forms the technical heart of this monograph. We also define the resolvent families (L b,m − μ)−1 , describe their holomorphic behavior as functions of μ and relate this to the analytic semi-group theory for the model parabolic problems. At the end of the chapter we describe why the estimates we prove here are not adequate for the perturbation theoretic arguments needed to construct the solution operator for general Kimura diffusions ∂t − L.

4.1 THE MODEL PROBLEM IN 1-DIMENSION First recall the 1-dimensional model operator, L b = x∂x2 + b∂x

on ⺢+ × ⺢+ ,

(4.1)

where b is any non-negative constant, and the general inhomogeneous Cauchy problem & ∂t w − L b w = g on ⺢+ × (0, T ] (4.2) w(x, 0) = f (x), x ∈ ⺢+ . So long as g and f have moderate growth, then Corollary 3.3.2 guarantees that there is a unique solution with moderate growth and satisfying certain regularity hypotheses at x = 0; it is given by the integral formula  t ∞

∞ ktb−s (x, x)g( ˜ x, ˜ s) d xds ˜

w(x, t) = 0

0

ktb (x, x) ˜ f (x) ˜ d x. ˜

+

(4.3)

0

The precise form of the heat kernel for this problem was derived in [15]: for any b > 0,   x˜ b−1 − x+x˜ x x˜ b t , (4.4) ˜ = b e ψb kt (x, x) t t2 51

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CHAPTER 4

where ψb (z) =

∞  j =0

zj . j !( j + b)

(4.5)

When b = 0, the Schwartz kernel takes a somewhat different form:    x  x+x˜ x x x˜ 0 − t kt (x, x) + e− t δ0 (y). ˜ = 2 e ψ2 t t2

(4.6)

The defining equation for this kernel is that (∂t − L b )ktb = 0 for t > 0 (along with the initial condition that limt →0+ k b (x, x) ˜ = δ(x − x)). ˜ However, it can be checked directly from the explicit expression that (∂t − L tb,x˜ )ktb (x, x) ˜ = 0, where L tb,x˜ = ∂x˜ (∂x˜ x˜ − b)

(4.7)

is the formal adjoint operator. Note too that we can verify directly from (4.4) that ˜ = 0 and ktb (x, x) ˜ = ᏻ(x˜ b−1 ) lim (∂x˜ x˜ − b)ktb (x, x)

+ x→0 ˜

(4.8)

when t > 0 and b > 0. We write the solution as a sum w = v + u where v is a solution to the problem with g = 0 and u is a solution to the problem with f = 0. We often call the first of these the homogeneous Cauchy problem and the second the inhomogeneous problem. In the following, we describe the various estimates for solutions on integer order spaces. More specifically, we use the standard spaces of -times continuously differ entiable functions Ꮿ (⺢+ ),  ∈ ⺞, and their parabolic analogues, Ꮿ, 2 (⺢2+ ), which are 2 the closures of Ꮿ∞ c (⺢+ ) with respect to the norms g,  = 2



q p

∂t ∂x g(x, t)∞ .

(4.9)

0≤ p+2q≤

Remark. To make the notation less cumbersome, in the context of these parabolic spaces, we always take 2 to mean the greatest integer in /2. ˙  (⺢+ ), which is To keep track of behavior of solutions as x → ∞, we also use Ꮿ ∞ the closure of Ꮿc (⺢+ ) with respect to the norm  f  =

 

p

∂x f (x)∞ .

(4.10)

p=0

We first discuss solutions of the homogeneous problem and after that solutions of the inhomogeneous problem. The first result is a slight improvement of a theorem from [15].

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THE MODEL SOLUTION OPERATORS

˙  (⺢+ ), then v ∈ Ꮿ, 2 (⺢+ × ⺢+ ). MoreL EMMA 4.1.1. For each  ∈ ⺞, if f ∈ Ꮿ over if p + 2q ≤ , then q p ∂t ∂x v(x, t)

∞ =

b+ p

kt

q

p

(x, x)L ˜ b+ p ∂x˜ f (x) ˜ d x. ˜

(4.11)

0

P ROOF. Let v be defined by (4.3) with g = 0. It is shown in [15] that v ∈ Ꮿ0 ([0, ∞)t ; Ꮿb ([0, ∞)x )) ∩ Ꮿ∞ ((0, ∞)t × [0, ∞)x ), and (4.11) is established for any 0 ≤ p ≤ , but only for q = 0. We establish the general case as follows. For t > 0, differentiate the representation formula for v (i.e., (4.3) with g = 0) and use (4.7) to obtain ∞

∞ ∂t ktb (x, x) ˜ f (x) ˜ d x˜

∂t v(x, t) =

L tb,x˜ ktb (x, x) ˜ f (x) ˜ d x. ˜

= lim

→0+

0

(4.12)



If  ≥ 2 and b > 0, we can integrate by parts twice in x˜ and let  → 0 to obtain that ∞ ktb (x, x)L ˜ b f (x) ˜ d x, ˜

∂t v(x, t) =

(4.13)

0

and hence ∂t v ∈ Ꮿ0 (⺢+ × ⺢+ ). In particular, if f ∈ Ꮿ2 (⺢+ ), then v ∈ Ꮿ2,1 (⺢+ × ⺢+ ). Using (4.11) and (4.13) inductively shows that v has the stated regularity and also gives (4.11) for all p, q with p+2q ≤ . The result for b = 0 follows from the formula (4.11) above when b > 0 and Proposition 7.8 in [15].  Now turn to the inhomogeneous equation, (∂t − L b )u = g, u(·, 0) = 0. The solution is given by the Duhamel formula  t ∞ ktb−s (x, x)g( ˜ x, ˜ s) d xds. ˜

u(x, t) = 0

(4.14)

0

Since ktb (x, x) ˜ → δ(x − x) ˜ as t → 0+ , we understand this to mean that t −∞ ktb−s (x, x)g( ˜ x, ˜ s) d xds. ˜

u(x, t) = lim

→0+

0

(4.15)

0

Denote this Volterra operator by K tb . Lemma 4.1.1 implies the basic regularity result.

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CHAPTER 4 



L EMMA 4.1.2. If g ∈ Ꮿ, 2 (⺢+ × [0, T ]), then u = K tb g ∈ Ꮿ, 2 (⺢+ × [0, T ]); furthermore, for any p, q with p + 2q ≤ , q p

b+ p

∂t ∂x u = K t

q

p

q−1 

q−l−1 p ∂x g(x, t),

L lb+ p ∂t

(4.16)

kτb (x, x)g( ˜ x, ˜ s) d x; ˜

(4.17)

L b+ p ∂x˜ g +

l=0

where the sum is absent when q = 0. P ROOF. Let

∞ G(τ, s, x) = 0

Lemma 4.1.1 implies that it follows easily that

j ∂xi ∂t ∂τk G

t u(x, t) =

∈ Ꮿ0 (⺢3+ ), provided i + 2( j + k) ≤ . From this



G(t − s, s, x)g(x, ˜ s) ds ∈ Ꮿ, 2 (⺢+ × ⺢+ ).

(4.18)

0

Equation (4.16) with q = 0 follows directly from Lemma 4.1.1. Without loss of generality, we can therefore let p = 0, and prove the remainder of the formula by induction. The case q = 1 is simply the equation ∂t u = L b u + g.

(4.19)

Assume that the formula holds for some q with 2q ≤  − 2. The results in [15] show that we can differentiate the equation in (4.16) to obtain that q+1

∂t

q

u = ∂t K tb L b g +

q−1 

q−l

L lb ∂t

g.

(4.20)

l=0 q

q

The first term on the right equals L b K tb L b g + L b g. This completes the proof since q q+1  L b K tb L b g = K tb L b g. 4.2 THE MODEL PROBLEM IN HIGHER DIMENSIONS We now generalize these results and formulæ to the higher dimensional model operators L b,m . Using the multiplicative nature of heat kernels, we can immediately write the model heat kernels in terms of the 1-dimensional ones, ktb,m (x, y, x, y) = ktb (x, x)kteuc,m ( y, y), where x) ktb (x,

=

n ' i=1

ktbi (x i , x˜i ),

kteuc,m ( y, y) =

1 (4πt)

m 2

(4.21)

e−

| y− y|2 4t

.

(4.22)

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THE MODEL SOLUTION OPERATORS

Note that this makes sense, even when bi = 0 for some indices. This is because there is, at most, one δ-factor in each coordinate. The general problem is & ∂t w − L b,m w = g on Sn,m × (0, T ] (4.23) w(x, y, 0) = f (x, y), (x, y) ∈ Sn,m . Uniqueness of moderate growth solutions with appropriate regularity and moderate growth data is then given by Corollary 3.3.2, and this solution has the integral representation  w(x, y, t) =

t Sn,m

0

ktb,m x, y)g( x, y, s) d xd yds −s (x, y, (4.24)



+

Sn,m

ktb,m (x, y, x, y) f ( x, y) d xd y.

As before we discuss the homogeneous (g = 0) and inhomogeneous ( f = 0) problems  separately, analyzing regularity in the elementary spaces Ꮿ (Sn,m ) and Ꮿ, 2 (Sn,m × ∞ [0, T ]), which are defined as the completions of the spaces Ꮿ∞ c (Sn,m ) and Ꮿc (Sn,m × [0, T ]) with respect to the norms || f || =

β

max ∂ xα ∂ y f  L ∞ (Sn,m )

|α|+|β|≤

||g||,  = 2

and max

2 j +|α|+|β|≤

β

∂t ∂ xα ∂ y g L ∞ (Sn,m ×[0,T ]) , j

(4.25)

respectively. The following result will be helpful. P ROPOSITION 4.2.1. Fix b¯ ∈ ⺢n+ and f ∈ Ꮿ0 (Sn,m ). For any b ∈ ⺢n+ , let v b be the unique moderate growth solution of (4.23) with Cauchy data f (and with g = 0). ¯ Then limb→b¯ v b = v b in Ꮿ0 (Sn,m × [0, T ]), for every T > 0. P ROOF. Note that the result is trivial if all entries of b¯ are strictly positive since varies smoothly with b ∈ (0, ∞)n and is uniformly integrable, so we assume that some entries of b¯ vanish. This result is the multidimensional generalization of [15, Prop. 7.8], and we review the proof of this 1-dimensional case because we wish to use this same argument inductively. That proof both starts the induction and provides the inductive step. So, first let f ∈ Ꮿ0 (⺢+ ). If f (0) = 0, then we can approximate f uniformly by f  ∈ Ꮿ0c ((0, ∞)). Since ktb converges smoothly to kt0 away from x = x˜ = 0, it is clear that ktb f → kt0 f . Now estimate ktb,m

|ktb f − kt0 f | ≤ |ktb ( f − f )| + |kt0 ( f − f  )| + |(ktb − kt0 ) f  |. Given f , choose  so that sup | f − f  | < ; by the maximum principle, the first and second terms are each less than . Then, for this , choose b sufficiently small so that the third term is less than  too.

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For arbitrary f ∈ Ꮿ0 (⺢+ ), choose a smooth cutoff χ(x) which equals 1 for x ≤ 1 and vanishes for x ≥ 2, and write f (x) = f (0) + χ(x)( f (x) − f (0)) + (1 − χ(x))( f (x) − f (0)). Applying ktb to this sum, then ktb f (0) = f (0) for all b, and the other two terms vanish at zero, so we may apply the previous reasoning to each of them. This proves the 1-dimensional case. Now consider the higher dimensional case. For simplicity, assume that bn = 0;  write b = (b1 , . . . , bn−1 ) and b¯ = (b¯1 , . . . , b¯n−1 ), and also set x  = (x 1 , . . . , x n−1 ). 0 Suppose that f ∈ Ꮿ (Sn,m ). Decompose f (x, y) as in the 1-dimensional case, as f (x  , 0, y) + χ(x n )( f (x, y) − f (x  , 0, y)) + (1 − χ(x n ))( f (x, y) − f (x  , 0, y)). Since the first term is independent of x n , we have  Sn,m

ktb,m (x, y, x, y) f ( x  , 0, y) d xd y=  Sn−1,m



ktb ,m (x  , y , x  , y ) f ( x  , 0, y) d x  d y, 

so we may apply the inductive hypothesis to see that this is continuous in b up to b¯ . The third term is supported away from x n = 0 already, so the result is clear for this term. Finally, for the second term, which we denote by f 2 , we can argue exactly as in the 1-dimensional case, choosing a continuous function h 2 supported away from x n = 0 ¯ and such that sup | f 2 − h 2 | < . Then |ktb,m ( f 2 − h 2 )| <  and |ktb,m ( f2 − h 2 )| < , ¯ and we may choose b sufficiently close to b¯ so that sup |(ktb,m − ktb,m )h 2 | <  too.  Remark. In cases where some of the {b j } vanish, the solution kernel is quite a bit more complicated than when all the b j > 0. Suppose k < n and that b1 = · · · = bk = 0, but b j > 0 for k = k + 1, . . . , n. The heat kernel for L b,0 takes the form x) ktb (x,

=

k '

[kt0,D (x j , x˜ j ) + e− j =1

xj t

δ0 (x˜ j )]

n '

b

kt j (x j , x˜ j ).

(4.26)

j =k+1

If H j = {x˜ j = 0}, then this kernel has a δ-distribution on the incoming boundary strata {Hi1 ∩ · · · ∩ Hil : ∀1 ≤ l ≤ k, and sequences: 1 ≤ i 1 < · · · < i l ≤ k}. A similar result is given in [37, 38] for the case of the Fleming-Viot operator defined on the simplex. The analogue of Lemma 4.1.1 is

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THE MODEL SOLUTION OPERATORS 

P ROPOSITION 4.2.2. For  ∈ ⺞, if f ∈ Ꮿ (Sn,m ), then v ∈ Ꮿ, 2 (Sn,m × [0, ∞)). For j ∈ ⺞0 and any multi-index of non-negative integers α and β, if 2 j +|α|+|β| ≤ , then  j β j β ∂t ∂ xα ∂ y v = ktb+α,m (x, y, x, y)L b+α,m ∂xα ∂y f ( x, y) d xd y. (4.27) Sn,m

P ROOF. We assume that all entries bi > 0, since if some bi = 0 then we can prove ( j) the result for an approximating sequence b( j ) → b with all bi > 0 and then apply the previous proposition. For t > 0 (and all bi > 0) the kernel is smooth in (t, x, y) and we can differentiate under the integral sign. Using (4.8) and [15, Cor. 7.4], we obtain (4.27) with j = 0. The argument needed to handle the ∂t derivatives is slightly more delicate. We claim that ∂t ktb (x, x) ˜ = (x∂x2 + b∂x )ktb (x, x) ˜ = (∂x2˜ x˜ − b∂x˜ )ktb (x, x), ˜

(4.28)

˜ tb and ∂x˜ ktb are not integrable at 0 when b ≤ 1. but some care is needed because ∂x2˜ xk We overcome this issue with the following lemma. L EMMA 4.2.3. If f ∈ Ꮿ2 (Sn,m ), then  L bi ,xi

ktb,m (x, y, x, y) f ( x, y) d xd y=

Sn,m

 Sn,m

ktb,m (x, y, x, y)L bi ,x˜i f ( x, y) d xd y.

(4.29)

m P ROOF. To simplify notation, relabel so that i = 1, and write Sn−1,m = ⺢n−1 + ×⺢ . ∞ First assume that b1 > 0. If f ∈ L (⺢+ ), then

∞

L b1 ,x1 ktb1 (x 1 , x˜1 ) f (x˜1 ) d x˜1 and

0

∞

∂t ktb1 (x 1 , x˜1 ) f (x˜1 ) d y1

(4.30)

0

are both absolutely integrable when t > 0. Thus using (4.28), we re-express 

ktb,m (x, y, x, y) f ( x, y) d xd y=

L bi ,xi Sn,m

∞ →0+

b



L tb1 ,x˜1 kt 1 (x 1 , x˜1 )

lim



Sn−1,m

n '

b

kt i (x i , x˜i )kteuc,m ( y, y) f ( x, y) d xd y.

(4.31)

i=2

Since f ∈ Ꮿ2 (Sn,m ) and limx˜1 →0+ (∂x˜1 x˜1 − b1 )ktb1 (x 1 , x˜1 ) = 0, we can integrate by parts in the x˜1 -integral and let  → 0 to obtain (4.29). The case b1 = 0 follows by applying Proposition 4.2.1. 

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The assertion that  ktb,m (x, y, x, y) f ( x, y) d xd y= ∂t Sn,m

 Sn,m

ktb,m (x, y, x, y)L b,m f ( x, y) d xd y (4.32)

follows directly from this lemma.  The fact that v ∈ Ꮿ, 2 if f ∈ Ꮿ then follows from these formulæ and the more elementary fact that if f ∈ Ꮿ0 (Sn,m ), then v ∈ Ꮿ0 (Sn,m × [0, ∞)).  Finally consider the inhomogeneous problem (∂t − L b,m )u = g,

u(x, y, 0) = 0,

(4.33)

with solution given by (4.23). This is treated just as before. Let  kτb,m (x, y, x, y)g( x, y, s) d xd y, G(τ, s, x, y) =

(4.34)

Sn,m

and observe that

t − u(x, y, t) = lim G(t − s, s, x, y) ds. →0+

(4.35)

0

The following result is an immediate consequence of Proposition 4.2.2 and (4.35): 

P ROPOSITION 4.2.4. If g ∈ Ꮿ, 2 (Sn,m × [0, T ]) for some  ∈ ⺞, then the unique  bounded solution u to (4.33) is given by (4.35) and lies in Ꮿ, 2 (Sn,m × [0, T ]). If 2 j + |α| + |β| ≤ l, then  j α β β ∂t ∂ x ∂ y u = ktb+α,m (x, y, x, y)(L b+α,m ) j ∂xα ∂y g( x, y, s) d xd y Sn,m

+

j −1 

j −r−1

∂t

β

(L b+α,m )r ∂ y ∂ xα g. (4.36)

r=0

We prove one final result, concerning the behavior at spatial infinity of solutions corresponding to compactly supported data f, g. P ROPOSITION 4.2.5. Let w be given by (4.23) with data f ∈ Ꮿ0c (Sn,m ) and g ∈ × [0, T ]). For any sets of non-negative integers α, β and N,

Ꮿ0c (Sn,m

β

lim

(1 + |(x, y)|) N |∂ xα ∂ y v(x, y, t)| = 0

lim

(1 + |(x, y)|) N |∂ xα ∂ y u(x, y, t)| = 0

|(x, y)|→∞ |(x, y)|→∞

for each t > 0.

β

(4.37)

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THE MODEL SOLUTION OPERATORS

P ROOF. This follows easily from the fact that the singularities of these kernels are located on the diagonal at t = 0, and from their exponential rates of decay at spatial infinity. If the incoming variables  x, y are confined to a compact set, this decay is uniform for t ∈ [0, T ] for any T < ∞.  4.3 HOLOMORPHIC EXTENSION The kernel functions ktb (x, y) extend to be analytic for t lying in the right half plane Re t > 0. By the permanence of functional relations, the functional equation ∞ ktb (x, y)ksb (y, z)d y = ktb+s (x, y)

(4.38)

0

holds for s and t in this half plane. Therefore the solution to the homogeneous Cauchy problem, ∞ v(x, t) = ktb (x, y) f (y)d y, (4.39) 0

is analytic in Re t > 0, and satisfies ∂t v − L b v = 0 where Re t > 0, where ∂t is the complex derivative. If we let t = τ eiθ , where τ ∈ (0, ∞) and |θ | < kτbeiθ (x, y)

π 2,

(4.40)

then

y b−1 e−ibθ − (x+y)e−iθ τ = e ψb τb

(

x ye−2iθ τ2

) .

(4.41)

For any α > 0, the asymptotic expansion 1

b

z 4 − 2 e2 ψb (z) ∼ √ 4π

√ z

⎛ ⎝1 +

∞  cb, j j =1

z

j 2

⎞ ⎠

(4.42)

holds uniformly for | arg z| ≤ π − α. This shows that the kernel has an asymptotic expansion: ⎛ ⎞ √ √ ∞ − iθ2   b − 1 j ei j θ  ( x− y)2 e−iθ e τ y 2 4 ⎝1 + ⎠. τ e− cb, j (4.43) kτbeiθ (x, y) ∼ b √ j τ y x (x y) 2 j =1

This explicit expression shows that the qualitative behavior of this kernel, as τ → 0, is uniform in sectors π (4.44) Sφ = {t : | arg t| < − φ}, 2

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CHAPTER 4

for any φ > 0. Moreover, if f ∈ Ꮿ0b ([0, ∞)), then lim v(x, t) = f (x)

(4.45)

t→0 t∈Sφ

uniformly in the Ꮿ0 -topology. In the sequel we shall also be analyzing the resolvent R(μ) = (L b − μ)−1 . If μ ∈ (0, ∞) and f ∈ Ꮿ0b ([0, ∞)), then R(μ) f is defined by expression 1

 R(μ) f = lim

→0+



⎞ ⎛∞  ⎝ ktb (x, y) f (y) d y ⎠ e−μt dt.

(4.46)

0

Using the asymptotics of ktb , we can apply Morera’s theorem to show that for each fixed x ∈ ⺢+ , R(μ) f (x) is an analytic function of μ in the right half plane. Applying L b to the integral on the right and integrating by parts, and using estimates proved below, we show that if f is H¨older continuous and bounded on [0, ∞), then (μ − L b )R(μ) f = f.

(4.47)

Using Cauchy’s theorem and the asymptotics of the heat kernel, the contour for the t-integral can be deformed to show that for μ ∈ (0, ∞) and | arg θ | < π2 we also have 1

 R(μ) f = lim

→0+



⎞ ⎛∞  ⎝ k b iθ (x, y) f (y) d y ⎠ e−μτ eiθ eiθ dτ. τe

(4.48)

0

This expression is analytic in μ in the region where Re(μeiθ ) > 0, and hence R(μ) f extends analytically to ⺓ \ (−∞, 0], and this extension satisfies (4.47). For f ∈ Ꮿ0 , we show that the following limits exist locally uniformly for x ∈ (0, ∞) : 1

  ∞ lim

→0+



iθ

∂x kτbeiθ (x, y) f (y)d ye−μτ e eiθ dτ

0

(4.49)

1 

 ∞ lim

→0+



iθ

x∂x2 kτbeiθ (x, y) f (y)d ye−μτ e eiθ dτ.

0

This demonstrates the R(μ) f is twice differentiable in (0, ∞). If f is only in Ꮿ0 ([0, ∞)), then the limits limx→0+ ∂x R(μ) f (x), and limx→0+ x∂x2 R(μ) f (x) = 0 may not exist. For |θ | < π2 and  > 0, we let θ, = {teiθ : t ∈ [,  −1 ]}.

(4.50)

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61

THE MODEL SOLUTION OPERATORS

For t ∈ S0 the kernel function satisfies the equation ∂t ktb (x, y) = L b ktb (x, y),

(4.51)

where ∂t is the complex derivative. For  > 0 we see that  ∞ Lb

ktb (x, y) f (y)d ye−μt d ydt

 ∞ =

θ, 0

∂t ktb (x, y) f (y)d ye−μt d ydt.

(4.52)

θ, 0

For  > 0 we can interchange the order of the integrations and integrate by parts to obtain:  ∞ Lb

ktb (x, y) f (y)d ye−μt d ydt

 ∞ =μ

θ, 0

θ, 0

∞

∞

−1 iθ kb−1 eiθ (x, y) f (y)d ye−μ e d y

−

0

ktb (x, y) f (y)d ye−μt d ydt+

iθ

b −μe ke d y. (4.53) iθ (x, y) f (y)d ye

0

Provided that Re(eiθ μ) > 0, and f ∈ Ꮿ0 (⺢+ ), we can let  → 0+ to obtain that (μ − L b )R(μ) f = f.

(4.54)

We summarize these observations as a proposition: P ROPOSITION 4.3.1. The solution to the homogeneous Cauchy problem (∂t − L b )v = 0 with v(x, 0) = f (x) ∈ Ꮿ0b ([0, ∞)) extends to an analytic function of t with Re t > 0. The resolvent operator R(μ) is analytic in the complement of (−∞, 0], and is given by the integral in (4.48) provided that Re(μeiθ ) > 0. Moreover, (μ − L b )R(μ) f = f, for μ ∈ ⺓ \ (−∞, 0]. From the corresponding fact for its 1-dimensional factors, it is obvious that the kerx, y) extends analytically to Re t > 0 and hence the solution v b (x, y, t) nel ktb,m (x, y, does as well. Indeed, if f ∈ Ꮿ0b (Sn,m ) then t → v b (·, ·, t) is an analytic function 0 from the right half plane with values in Ꮿ∞ b (int Sn,m ) ∩ Ꮿb (Sn,m ). The asymptotic formula (4.43) and the standard asymptotics for the Euclidean heat kernel then give that for any φ > 0, lim v b (·, ·, t) = f, (4.55) t →0 in Sφ

in the uniform topology. For f ∈ Ꮿ0b (Sn,m ) and Re μ > 0, the Laplace transform is defined by the limit 1

 R(μ) f (x, y) = lim

→0+



v b (x, y, t)e−μt dt.

(4.56)

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62

CHAPTER 4

Assuming that f ∈ Ꮿ0W F (Sn,m ) then again using the analyticity and asymptotic behavior of the kernel, we can use Cauchy’s theorem to deform the contour of integration in (4.56). For |θ | < π2 and μ ∈ (0, ∞), we have that 1

 R(μ) f (x, y) = lim

→0+

iθ

v b (x, y, τ eiθ )e−μτ e eiθ dτ.

(4.57)



The expression in (4.57) defines an analytic function of μ where Re μeiθ > 0. This in turn shows that R(μ) f has an analytic continuation to ⺓ \ (−∞, 0]. In order to establish the identity (μ − L b,m )R(μ) f = f

(4.58)

in the higher dimensional case, it is simpler to assume that f is H¨older continuous. 0,γ Specifically, in the next chapter we shall define H¨older spaces ᏯW F (Sn,m ) which are specially adapted to this problem. For such data one can show that the individual terms in L b,m R(μ) f are continuous on Sn,m , and satisfy (4.58). If f is only continuous, then arguing as before, one can show that the limit, as  → 0+ of 1

 L b,m

iθ

v b (x, y, τ eiθ )e−μτ e eiθ dτ,

(4.59)



exists in Ꮿ0 (Sn,m ). It satisfies the identity (μ − L b,m )R(μ) f = f in the Ꮿ0 -graph closure sense. Generally the individual terms of L b,m R(μ) f are not defined as (x, y) approaches the boundary of Sn,m . We summarize these results in a proposition. P ROPOSITION 4.3.2. The solution v to the homogeneous Cauchy problem, with initial data v(x, y, 0) = f (x, y) ∈ Ꮿ0b (Sn,m ), extends to an analytic function of t with Re t > 0. The resolvent operator R(μ) is analytic in the complement of (−∞, 0], and is given by the integral in (4.57) provided that Re(μeiθ ) > 0. If f ∈ Ꮿ0W F (Sn,m ), then (μ − L b,m )R(μ) f = f, for μ ∈ ⺓ \ (−∞, 0].

4.4 FIRST STEPS TOWARD PERTURBATION THEORY Our primary goal in this monograph is to construct the solution operator ᏽt for a general Kimura operator ∂t − L and to use it to study properties of the associated semi-group on various function spaces. This is done by perturbing an approximate solution obtained by patching together the solution operators ᏽtb,m for the models ∂t − L b,m associated to the normal forms of L in various coordinate charts. This strategy works very well in the 1-dimensional problem considered in [15], but turns out to be substantially more complicated in higher dimensions. We explain this now.

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THE MODEL SOLUTION OPERATORS

63

The relative simplicity of this method for operators in 1-dimension is not hard to explain. As already pointed out in Chapter 2.2, the normal form for the second order part of a 1-dimensional Kimura operator is exactly x∂x2 in a full neighborhood of a boundary point. Thus we can choose coordinates and a constant b ≥ 0 so that L = L b + W , where W is a vector field which vanishes at the boundary point. Hence the error term incurred by using ᏽtb as an approximate solution operator is W ◦ ᏽtb . In an appropriate sense, this operator is smoothing of order 1, and restricted to data on suitably small time intervals [0, T ], it also has small norm acting on continuous functions. Hence it is easy to solve away this error term using a convergent Neumann series. When carrying out the same procedure in higher dimensions, the difference between L and any one of the models L b,m is unavoidably second order. Hence the error term E incurred by applying ∂t − L to a parametrix formed by patching together these model heat kernels is no longer smoothing, since it is the result of applying a differential operator of order 2 to an operator which is smoothing of order 2. Even worse, this error is not bounded on Ꮿ0 . This is a well-known fact in classical potential theory, that Ꮿ0 and higher Ꮿk spaces are ill-suited for the study of regularity properties of elliptic and parabolic problems in higher dimensions, and that one should use H¨older spaces instead. The applications of these Kimura diffusions in probability and biology demand that we study the semi-group for L on Ꮿ0 . This leaves us with a slightly unsatisfactory state of affairs. We are only able to construct the solution operator for ∂t − L on a suitable scale of H¨older spaces. We can still prove the existence and many properties of the semi-group on Ꮿ0 , but this must be done in an indirect fashion.

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Chapter Five ¨ Degenerate Holder Spaces The starting point to implement this perturbation theory is a description of the various function spaces we shall be using. As described above, we seek function spaces on the domain P for which the diffusion associated to a general Kimura diffusion operator L is well posed. More pragmatically, we wish to define spaces on which one can prove analogues of the standard parabolic Schauder estimates, so that we can pass from the model to more general operators. This chapter is devoted to a description of the various spaces on which this is possible, and to an explanation of the relationships between them. Two familiar guiding principles when choosing the right function spaces for a problem are that one should choose spaces which respect the natural scaling properties of the operator, and in addition, that these spaces should be based on the geometry of an associated metric. In the classical setting, the operator ∂t − ∂ y2j on ⺢+ × ⺢m is homogeneous with respect to the parabolic dilations (t, y) → (λ2 t, λ y), and is naturally associated to the Euclidean metric. The first of these principles indicates that t and y derivatives should be weighted differently; the second suggests that we formulate mapping properties in terms of H¨older spaces with semi-norms defined using the Euclidean distance function. This is indeed the case, and we review the definitions of the standard parabolic H¨older spaces below. Other examples where these principles are applied include [11, 12, 28, 23, 24] and [32]. To apply the same two principles in the present setting, we observe that ∂t − L b,m is homogeneous with respect to the slightly different scaling, √ (t, x, y) → (λt, λx, λ y), which indicates that derivatives with respect to t and the x i should be twice as strong as derivatives with respect to the y j . On the other hand, when all x j are strictly positive, we must revert to the standard scaling corresponding to the interior problem. In other words, whatever function spaces we use must incorporate both types of homogeneity. The metric naturally associated to L b,m is ds 2 =

n  dx2 i

i=1

xi

+

m 

d y 2j ;

j =1

note that this metric is homogeneous with respect to (x, y) → (λx, associated Laplacian is simply L b,m with b = ( 12 , . . . , 12 ). 64

(5.1) √

λ y), and that the

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¨ DEGENERATE HOLDER SPACES

65

Before embarking on the many definitions below, we make two remarks. First, the basic definition of a H¨older semi-norm with respect to a given metric g is [|u|]g;0,γ := sup

z =z 

|u(z) − u(z  )| , dg (z, z  )γ

(5.2)

where dg is the Riemannian distance between the two points. It is very useful to observe that instead of taking the supremum over all distinct pairs z, z  , it suffices to take the supremum only over pairs with z = z  and dg (z, z  ) ≤ 1. This is simply because if dg (z, z  ) > 1, then the quotient on the right, evaluated at z, z  , is bounded by 2 sup |u|. For this reason, we introduce the notation 1

sup ≡

z =z 

sup

,

{z =z  : d(z,z  ) 0, define W j,δ = {(x, y) : δ ≤ x j } ⊂ Sn,m . From the hypotheses again, it is clear that if δ > 0, then √ lim  x j ∂x2j y p ( f − f i )∞,W j,δ = 0, i→∞

(5.49)

√ so we must only show that | x j ∂x2j y p ( f − f i )(x, y)| is uniformly small when i is large and x j is small. We have √ | x j ∂x2j y p ( f − f i )(x, y)|  √ ≤ | x j ∂x2j y p f (x, y) − ( x j + ηi )∂x2j y p f (x + ηi , y)| √ √ | x j − x j + ηi |  + |( x j + ηi )∂x2j y p f (x + ηi 1, y)|. √ x j + ηi

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74

CHAPTER 5 0,2+γ

By definition of the ᏯW F -norm again, and using (5.46), this gives:   γ √ γ /2 | x j ∂x2j y p ( f − f i )(x, y)| ≤  f W F,0,2+γ n γ ηi + (x j + ηi ) 2 .

(5.50)

Together with (5.49), this implies that √ lim  x j ∂x j y p ( f − f i )∞ = 0.

(5.51)

i→∞

√ Finally, we must consider terms of the form x j x k |∂x2j xk ( f − f i )|. Once again, for any δ > 0, √ lim  x j x k ∂x2j xk ( f − f i )∞,W j,δ ∩Wk,δ = 0. (5.52) i→∞

Near the boundary, we have √ | x j x k ∂x2j xk ( f − fi )(x, y)| ≤  √ | x j x k ∂x2j xk f (x, y) − (x j + ηi )(x k + ηi ) ∂x2j xk fi (x + ηi 1, y)|+  √ | (x j + ηi )(x k + ηi ) − x j x k |   (x j + ηi )(x k + ηi )|∂x2j xk f i (x + ηi 1, y)|, (5.53) (x j + ηi )(x k + ηi ) whence, by (5.46), √ | x j x k ∂x2j xk ( f − fi )(x, y)| ≤

 . -+ +γ γ γ /2  f W F,0,2+γ n γ ηi min +x j + ηi + 2 , |x k + ηi | 2 .

This implies that

√ lim  x j x k ∂x2j xk ( f − f i )∞ = 0,

(5.54) (5.55)

i→∞



and proves the lemma. ˙ k (Sn,m ) belongs to Ꮿ (Sn,m ) if A function f ∈ Ꮿ WF k,γ

 f W F,k,γ =  f Ꮿk + β

β

|∂ xα ∂ y f (x, y) − ∂ xα ∂ y f (x  , y )| [ρs (x, x  ) + ρe ( y, y )]γ {|α|+|β|=k} (x, y) =(x  , y ) sup

1

sup

(5.56)

˙ k,2 (Sn,m ) is the closure of Ꮿk+2 is finite. Similarly, Ꮿ c (Sn,m ) with respect to the norm WF  f W F,k,2 =  f Ꮿk−1 +

sup

β

{|α|+|β|=k}

∂ xα ∂ y f W F,2 ,

(5.57)

˙ k,2 (Sn,m ) belongs to Ꮿ and a function f ∈ Ꮿ W F (Sn,m ) if WF k,2+γ

 f W F,k,γ =  f Ꮿk +

sup

{|α|+|β|=k}

β

∂ xα ∂ y f W F,0,2+γ < ∞.

(5.58)

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75

The analogue of Lemma 5.2.5 is straightforward and shows that these are Banach spaces. ˙ 0 (Sn,m × The parabolic H¨older spaces are defined similarly. A function g ∈ Ꮿ 0,γ [0, T ]) belongs to ᏯW F (Sn,m × [0, T ]) provided gW F,0,γ =g∞ + |g(x, y, t) − g(x  , y , t  )| √ < ∞.    γ (x, y,t ) =(x  , y  ,t  ) [ρs (x, x ) + ρe ( y, y ) + |t − t |] 1

sup

(5.59)

The semi-norm |[g|]W F,0,γ is the second term on the right. When it is important to emphasize the maximum time T, we use the notation |[g|]W F,0,γ ,T for this semi-norm. ˙ k (Sn,m × [0, T ]) belongs to Ꮿk,γ (Sn,m × [0, T ]) if A function g ∈ Ꮿ WF

gW F,k,γ = gᏯk + sup

j

1

{|α|+|β|+2 j =k}

sup

(x, y,t)

=(x  , y ,t  )

β

j

β

|∂t ∂ xα ∂ y g(x, y, t) − ∂t ∂ xα ∂ y g(x  , y , t  )| √ < ∞. [ρs (x, x  ) + ρe ( y, y ) + |t − t  |]γ

(5.60)

0,γ

We now list several basic estimates and facts. First, for functions f ∈ ᏯW F (Sn,m ) 0,γ and g ∈ ᏯW F (Sn,m × [0, T ]), we have | f (x, y) − f (x  , y )| ≤ 2 f W F,0,γ dW F ((x, y), (x  , y ))γ

(5.61)

|g(x, y, t) − g(x  , y , t  )| ≤ 2gW F,0,γ dW F ((x, y, t), (x  , y , t  ))γ . 0,γ

Furthermore, there are Leibniz formulæ for these semi-norms: if f, g ∈ ᏯW F (Sn,m ), 0,γ or f, g ∈ ᏯW F (Sn,m × [0, T ]), then [| f g|]W F,0,γ ≤ [| f ]|W F,0,γ g L ∞ + [|g|]W F,0,γ  f  L ∞ .

(5.62) 0,γ

L EMMA 5.2.6. Let 0 < γ  < γ < 1 and suppose that f ∈ ᏯW F (Sn,m ) and 0,γ g ∈ ᏯW F (Sn,m × [0, T ]). Then [| f ]|W F,0,γ  ≤

γ γ



1− γ 2 |[ f ]|W F,0,γ  f ∞ γ γ

(5.63)

.

(5.64)



1− γγ

[|g|]W F,0,γ  ≤ 2 |[g|]Wγ F,0,γ g∞

,

P ROOF. These follow directly from the identity |h(x, y, t) − h(x  , y , t  )|  = dW F ((x, y, t), (x  , y , t  ))γ  γ % γ |h(x, y, t) − h(x  , y , t  )|  γ $    1− γ |h(x, y, t) − h(x , y , t )| dW F ((x, y, t), (x  , y , t  ))γ

(5.65)

where h is defined on Sn,m × [0, T ], or the analogous identity for functions defined on Sn,m . 

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76

CHAPTER 5

˙ 2,1 (Sn,m × [0, T ]) is the closure of Ꮿ2,1 The space Ꮿ c (Sn,m × [0, T ]) with respect to WF the norm √ β gW F,2,1 = g∞ + ∇ x, y,t g∞ + sup ( x∂ x )α ∂ y g∞ , (5.66) |α|+|β|=2

and

0,2+γ ᏯW F (Sn,m

× [0, T ]) is the subspace on which

gW F,0,2+γ = gW F,2,1 + ∇ x, y,t gW F,0,γ +

sup

|α|+|β|=2

√ β ( x∂ x )α ∂ y gW F,0,γ < ∞.

(5.67)

The basic lemma now reads: n m L EMMA 5.2.7. Let g ∈ Ꮿ1 (Sn,m × [0, T ]) ∩ Ꮿ2,1 W F ((0, ∞) × ⺢ × [0, T ]) satisfy √ √ lim x j x k ∂x2j xk g(x, y, t) = 0 and lim x j ∂x2j y p g(x, y, t) = 0 x j or x k →0+

x j →0+

for j, k ≤ n, p ≤ m, and / lim

(x, y)→∞

|g(x, y, t)| + |∇ x, y,t g(x, y, t)| +

sup

|α|+|β|=2

0 √ α β |( x∂ x ) ∂ y g(x, y, t)| = 0.

0,2+γ

If gW F,0,2+γ < ∞, then g ∈ ᏯW F (Sn,m × [0, T ]). The proof is nearly identical to the one for Lemma 5.2.5, and this implies as before that 0,2+γ

ᏯW F (Sn,m × [0, T ]) is a Banach space.

We finally define the higher parabolic H¨older spaces in the expected way. Namely,

k ˙ k+2, 2 +1 (Sn,m Ꮿ WF

× [0, T ]) is the closure of Ꮿ∞ c (Sn,m × [0, T ]) with respect to the norm

||g||W F,k+2,k/2+1

=

g

k

k, Ꮿ 2

+

sup

|α|+|β|+2 j =k

β

||∂ xα ∂ y ∂t g||W F,2,1. j

(5.68)

k

k,2+γ ˙ k+2, 2 +1 (Sn,m × [0, T ]) on We define ᏯW F (Sn,m × [0, T ]) to be the subspace of Ꮿ WF which + + + α β j + ||g||W F,k,2+γ = ||g|| k, k + sup < ∞. +∂ x ∂ y ∂t g + Ꮿ 2

W F,0,2+γ

|α|+|β|+2 j =k

As before, if the upper limit T, for the time variable, is important we sometimes denote these norms by ||g||W F,k,γ ,T , and ||g||W F,k,2+γ ,T , respectively. These various spaces satisfy some obvious inclusions: if k  > k, or k  = k and 1 ≥ γ  > γ > 0, then k  ,γ 

k,γ

k  ,2+γ 

ᏯW F (Sn,m ) ⊂ ᏯW F (Sn,m ), ᏯW F

k,2+γ

(Sn,m ) ⊂ ᏯW F (Sn,m )

(5.69)

and k  ,γ 

k,γ

ᏯW F (Sn,m × [0, T ]) ⊂ ᏯW F (Sn,m × [0, T ]) k  ,2+γ 

ᏯW F

k,2+γ

(Sn,m × [0, T ]) ⊂ ᏯW F (Sn,m × [0, T ]).

(5.70)

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77

P ROPOSITION 5.2.8. If k < k  or k  = k and γ < γ  , then the restrictions of the inclusions in (5.69) and (5.70) to subspaces of functions which are supported in a ball of finite radius in Sn,m or Sn,m × [0, T ] are compact. P ROOF. These facts can all be deduced in a fairly straightforward manner from the Arzela-Ascoli theorem. We illustrate this by considering the inclusion 0,γ 

0,γ

{u ∈ ᏯW F (Sn,m ) : u = 0 for |(x, y)| > R} → ᏯW F (Sn,m ). If {u j } is a sequence in the space on the left with uniformly bounded norm, then by (5.61), this sequence is uniformly bounded and equicontinuous, hence some subsequence converges in Ꮿ0 to a limit function u. Now apply (5.63) to see that this subse0,γ  quence is Cauchy in ᏯW F .  As described in remark 5.2.3 in the 1-dimensional case, the higher order estimates in the general case are deduced by using formulæ (4.27) and (4.36). Again this suggests that the higher order norms should include weighted derivatives. As noted above, for our applications to Kimura operators on compact manifolds with corners we only need these results for data with fixed bounded support. To somewhat shorten this already long text, we have omitted these terms from the definitions of the higher order norms. Using the Leibniz formula we easily deduce the following estimates: P ROPOSITION 5.2.9. Fix an R > 0, a k ∈ ⺞, a non-negative vector 0 ≤ b, and a 0 < γ < 1. There is a constant C R so that k,γ

1. If f ∈ ᏯW F (Sn,m ) has support in the set {(x; y) : x ≤ R}, then if 2q + |α| + |β| ≤ k, we have the estimate β

L b,m ∂ xα ∂ y f W F,0,γ ≤ C R  f W F,k,γ . q

(5.71)

k,2+γ

2. If f ∈ ᏯW F (Sn,m ) has support in the set {(x; y) : x ≤ R}, then if 2q + |α| + |β| ≤ k, we have the estimate β

L b,m ∂ xα ∂ y f W F,0,2+γ ≤ C R  f W F,k,2+γ . q

(5.72)

k,γ

3. If g ∈ ᏯW F (Sn,m × [0, T ]) has support in the set {(x; y, t) : x ≤ R}, then if 2 p + 2q + |α| + |β| ≤ k, we have the estimate β

∂t L b,m ∂ xα ∂ y gW F,0,γ ,T ≤ C R gW F,k,γ ,T . p

q

(5.73)

k,2+γ

4. If g ∈ ᏯW F (Sn,m × [0, T ]) has support in the set {(x; y, t) : x ≤ R}, then if 2 p + 2q + |α| + |β| ≤ k, we have the estimate β

∂t L b,m ∂ xα ∂ y gW F,0,2+γ ,T ≤ C R gW F,k,2+γ ,T . p

q

(5.74)

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Chapter Six ¨ Holder Estimates for the 1-dimensional Model Problems In this and the following three chapters we establish H¨older estimates for the solutions of the model problems, i.e., w such that (∂t − L b,m )w = g with w( p, 0) = f ( p),

(6.1)

where f and g belong to the anisotropic H¨older spaces introduced in Chapter 5. It may appear that we are taking a circuitous path, by first considering the 1-dimensional case, then pure corner models, ⺢n+ , followed by Euclidean models (⺢m ) before finally treating the general case, ⺢n+ × ⺢m . In fact, all cases need to be treated, and in the end nothing is really wasted. We give a detailed treatment of the 1-dimensional case, both because it establishes a pattern that will be followed in the subsequent cases, and because all of the higher dimensional estimates are reduced to estimates on heat kernels for the 1-dimensional model problems. The derivation of parabolic Schauder estimates is now an old subject, and there are many possible approaches to follow. Our proof of these estimates for the model operator ∂t − L b is elementary. It uses the explicit formula for the heat kernel, (1.29), along with standard tools of analysis, like Taylor’s formula and Laplace’s method. The paper [11] considers a similar degenerate diffusion operator in 2 + 1-dimensions, and contains proofs of parabolic Schauder estimates for that problem. We present different arguments to derive the analogous estimates here. This allows us to handle the case b = 0, which is somewhat different than the situation in [11]. It is straightforward from the definitions that for any k ∈ ⺞0 and 0 < γ < 1, k,2+γ

k,γ

∂t − L b : {u ∈ ᏯW F (⺢+ × [0, T ]) : u(x, 0) = 0} −→ ᏯW F (⺢+ × [0, T ]). Our goal is to prove the converse, and of course also to study the regularity effects of nontrivial initial data. We shall prove the following two results: P ROPOSITION 6.0.10. Fix k ∈ ⺞0 , γ ∈ (0, 1), 0 < R and b ≥ 0. Suppose that k,γ f ∈ ᏯW F (⺢+ ), and let v be the unique solution to (4.2), with g = 0. If k > 0, then k,γ also assume that f has support in [0, R]. Then v ∈ ᏯW F (⺢+ × [0, T ]) for any T > 0 and there is a constant Ck,γ ,b,R so that If 0 < γ  < γ , then

vW F,k,γ ≤ Ck,γ ,b,R  f W F,k,γ .

(6.2)

lim v(·, t) − f W F,k,γ  = 0.

(6.3)

t →0+

78

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79

k,2+γ

If f ∈ ᏯW F (⺢+ ), then vW F,k,2+γ ≤ Ck,γ ,b,R  f W F,k,2+γ .

(6.4)

The constants Ck,γ ,b,R are uniformly bounded on any finite interval 0 ≤ b ≤ B. If 0 < γ  < γ , then lim v(·, t) − f W F,k,2+γ  = 0. (6.5) t →0+

If k = 0, then the constants in these estimates do not depend on R. P ROPOSITION 6.0.11. Fix k ∈ ⺞0 , γ ∈ (0, 1), 0 < R, and b ≥ 0. Let u be the k,γ unique solution to (4.2), with f = 0 and g ∈ ᏯW F (⺢+ × [0, T ]). If k > 0 we assume k,2+γ that g(x, t) is supported in {(x, t) : x ≤ R}. Then u ∈ ᏯW F (⺢+ × [0, T ]) and there is a constant Ck,γ ,b,R so that uW F,k,2+γ ,T ≤ Ck,γ ,b,R (1 + T )gW F,k,γ ,T .

(6.6)

The constants Ck,γ ,b,R are uniformly bounded for 0 ≤ b ≤ B. For any 0 < γ  < γ , the k,2+γ 

solution u(·, t) tends to zero in ᏯW F of R.

(⺢+ ). If k = 0, then the constant is independent

The assertions about the behavior of solutions as t → 0+ follow easily from Proposition 5.2.8, the following lemma, and the obvious facts that v(·, t) tends to f and u(·, t) tends to zero in Ꮿ0 (⺢+ ). L EMMA 6.0.12. Let X 2 ⊂ X 1 ⊂ X 0 be Banach spaces with the first inclusion precompact, and the second bounded. If for some M, the family v(t) ∈ X 2 satisfies: sup v(t) X 2 ≤ M and lim v(t) X 0 = 0,

(6.7)

lim v(t) X 1 = 0.

(6.8)

t →0+

t ∈[0,1]

then

t →0+

P ROOF. If limt →0+ v(t) X 1 = 0, then, by compactness, we can choose a sequence < tn >, tending to zero so that < v(tn ) > converges, in X 1 , to v ∗ = 0. The boundedness of the inclusion X 1 ⊂ X 0 , implies that < v(tn ) > must also converge, in X 0 , to v ∗ , but then v ∗ must equal 0.  Our final results concern the resolvent operator defined, for μ ∈ (0, ∞), by 1

 R(μ) f = lim

→0+

e−μt v(x, t)dt.

(6.9)



As noted in Proposition 4.3.2, R(μ) f extends to define an analytic function for μ ∈ ⺓ \ (−∞, 0]. Our final proposition gives a more refined statement of the mapping properties of R(μ) for the 1-dimensional model problem.

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P ROPOSITION 6.0.13. The resolvent operator R(μ) is analytic in the complement of (−∞, 0], and is given by the integral in (4.48) provided that Re(μeiθ ) > 0. For α ∈ (0, π], there are constants Cb,α so that if α − π ≤ arg μ ≤ π − α,

(6.10)

then for f ∈ Ꮿ0b (⺢+ ) we have: R(μ) f  L ∞ ≤

Cb,α  f L ∞ , |μ|

(6.11)

with Cb,π = 1. Moreover, for 0 < γ < 1, there is a constant Cb,α,γ so that if f ∈ 0,γ ᏯW F (⺢+ ), then Cb,α,γ  f W F,0,γ . R(μ) f W F,0,γ ≤ (6.12) |μ| k,γ

k,2+γ

If for a k ∈ ⺞0 , and 0 < γ < 1, f ∈ ᏯW F (⺢+ ), then R(μ) f ∈ ᏯW F (⺢+ ), and, we have (μ − L b )R(μ) f = f. (6.13) 0,2+γ

If f ∈ ᏯW F (⺢+ ), then

R(μ)(μ − L b ) f = f.

There are constants Cb,k,α so that, for μ satisfying (6.10), we have  1  f W F,k,γ . R(μ) f W F,k,2+γ ≤ Cb,k,α 1 + |μ|

(6.14)

(6.15)

For any B > 0, these constants are uniformly bounded for 0 ≤ b ≤ B. Remark. Unlike the results for the heat equations, the higher order estimates for the resolvent do not require an assumption about the support of the data. This is because the estimates for this operator only involve spatial derivatives; it is the time derivatives that lead to the x j -weights.

6.1 KERNEL ESTIMATES FOR DEGENERATE MODEL PROBLEMS The proofs of the estimates in one and higher dimensions rely upon estimates for the kernel functions ktb (x, y) and their derivatives. These kernels are analytic in the right half plane Re t > 0, and many of these estimates are stated and proved for this analytic continuation. Since we often need to refer to these results, we first list these estimates as a series of lemmas. Most of the proofs are given in Appendix A. Throughout this book we let C, Cb or Cb,∗ (where ∗ are other parameters) denote positive constants that are uniformly bounded for 0 ≤ b ≤ B, and a fixed value of γ . We often make use of the following elementary inequalities.

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L EMMA 6.1.1. For each k ∈ ⺞, and 0 < γ < 1, there is a constant Ck,γ such that for non-negative numbers {a1 , . . . , ak } we have ⎡ ⎤γ k k k    γ γ −1 Ck,γ aj < ⎣ a j ⎦ < Ck,γ aj . (6.16) j =1

j =1

j =1

P ROOF. As everything is homogeneous of degree γ it suffices to consider nonnegative k-tuples, (a1 , . . . , ak ), with k 

a j = 1,

(6.17)

j =1



for which the statement is obvious.

L EMMA 6.1.2. For 0 < γ < 1, there is a constant m γ so that, if x and y are non-negative, then |x γ − y γ | ≤ m γ |x − y|γ . (6.18) P ROOF. We can assume that x > y, and therefore the inequality is equivalent to the assertion that, for 1 < x, we have |x γ − 1| ≤ m γ |x − 1|γ .

(6.19)

The existence of m γ follows easily from the observation that x γ − 1 = γ (x − 1) + O((x − 1)2 ).

(6.20) 

The remaining lemmas are divided according to the order of the derivative being estimated. Proofs are given in Appendix A. The reader can skip the rest of this subsection and refer to it later, as needed. Recall that, for 0 < b, ktb (x, y) =

y b−1 − (x+y)  x y  e t ψb 2 , tb t

where ψb (z) =

∞  j =0

zj . j !( j + b)

(6.21)

(6.22)

This heat kernel is a smooth function in [0, ∞)x × (0, ∞) y × (0, ∞)t , which has an analytic extension in the t variable to the right half plane S0 , where the sectors Sφ are defined in (4.44). The asymptotic expansion (4.43) is valid in any sector Sφ , with φ > 0.

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6.1.1 Basic Kernel Estimates Recall that as b → 0+ , the kernels ktb converge, in the sense of distributions, to x

kt0 (x, y) = kt0,D (x, y) + e− t δ0 (y), where

kt0,D (x, y) =

x t2

e−

x+y t

ψ2

(6.23)

xy 

(6.24)

t2

is the solution operator for the equation ∂t v = x∂x2 v with v(0, t) = 0. As we will see, the solutions to the equations ∂t u − L b u = g, and their higher dimensional analogues satisfy H¨older estimates with constants uniformly bounded as b → 0+ . The kernel estimates are therefore proved for 0 < b, and the properties of solutions to the PDE with b = 0 are obtained by taking limits of solutions. A trivial but crucial fact is the following: L EMMA 6.1.3. For t ∈ S0 and b > 0 we have: ∞ ktb (x, y)d y = 1.

(6.25)

0

There is a constant Cφ so that, for t ∈ Sφ , ∞ |ktb (x, y)|d y ≤ Cφ .

(6.26)

0

P ROOF. The integral is absolutely convergent for any t ∈ S0 , and clearly defines an analytic function of t. For t ∈ (0, ∞), the integral equals 1, which proves the first assertion of the lemma. For the second, suppose that t = τ eiθ , and change variables, setting w = y/τ and λ = x/τ, to obtain: ∞

∞ |ktb (x, y)|d y

=

0

wb e− cos θ(w+λ) |ψb (wλe−2iθ )|

0

dw . w

(6.27)

We split the integral into the part from 0 to 1/λ and the rest. In the compact part we use the estimate   1 |ψb (wλe−2iθ )| ≤ + Cb |wλ| . (6.28) (b) Inserting this into the integral from 0 to 1/λ, it is clear that this term is uniformly bounded. In the non-compact part we use the asymptotic expansion for ψb to see that this term is bounded by I+ = C b

∞  1 λ

w  b2 − 14 − cos θ(√w−√λ)2 dw e √ . λ w

(6.29)

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This integral is O(e− cos θ/(2λ)) as λ → 0. As λ → ∞, we let z = that b− 1 ∞  2 z 2 I+ = C b √ +1 e− cos θ z dz. λ √ √1 − λ

83 √ √ w − λ, to obtain (6.30)

λ

It is elementary to see that this integral is bounded by a constant depending only on θ.  Remark. The proofs of the remaining estimates are in Appendix A. L EMMA 6.1.4. For b > 0, 0 < γ < 1, and 0 < φ < uniformly bounded with b, so that for t ∈ Sφ ∞

γ

π 2,

there are constants Cb,φ

γ

|ktb (x, y) − ktb (0, y)|y 2 d y ≤ Cb,φ x 2 .

(6.31)

0

L EMMA 6.1.5. For b > 0 there is a constant Cb,φ so that for t ∈ Sφ ∞ |ktb (x, y) − ktb (0, y)|d y ≤ Cb,φ 0

x/|t| . 1 + x/|t|

(6.32)

For 0 < c < 1 there is a constant Cb,c,φ so that, if cx 2 < x 1 < x 2 , and t ∈ Sφ , then ⎛ √ x −√ x ⎞ 2 1 ∞ √ |t | b b ⎝ √ ⎠. √ |kt (x 2 , y) − kt (x 1 , y)|d y ≤ Cb,c,φ (6.33) x2 − x1 1+ √ |t | 0

L EMMA 6.1.6. For b > 0, 0 < γ < 1 and t ∈ Sφ , 0 < φ < that ∞ √ γ √ |ktb (x, y)|| x − y|γ d y ≤ Cb,φ |t| 2 .

π 2,

there is a Cb,φ so (6.34)

0

For fixed 0 < φ, and B, these constants are uniformly bounded for 0 < b < B. For several estimates we need to split ⺢+ into a collection of subintervals. We let J = [α, β], where   √ √ √ √  √ 3 x2 − x1 3 x1 − x2 , 0 and β = . (6.35) α = max 2 2 L EMMA 6.1.7. We assume that x 1 /x 2 > 1/9 and J = [α, β], as defined in (6.35). For b > 0, 0 < γ < 1 and 0 < φ < π2 , there is a Cb,φ so that if t ∈ Sφ , then  √ √ √ √ |ktb (x 2 , y) − ktb (x 1 , y)|| y − x 1 |γ d y ≤ Cb,φ | x 2 − x 1 |γ . (6.36) Jc

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L EMMA 6.1.8. For b > 0, 0 < γ < 1 and c < 1 there is a Cb such that if c < s/t < 1, then ∞ + + √ γ √ + b + +kt (x, y) − ksb (x, y)+ | x − y|γ d y ≤ Cb |t − s| 2 .

(6.37)

0

We also have the simpler result, which holds without restriction on s < t, and when γ = 0. L EMMA 6.1.9. For b > 0 there is a Cb such that if s < t, then  ∞ + + + b + +kt (x, y) − ksb (x, y)+ d y ≤ Cb 0

 t/s − 1 . 1 + [t/s − 1]

(6.38)

6.1.2 First Derivative Estimates The following lemma is central to many of the results in this paper. L EMMA 6.1.10. For b > 0, 0 ≤ γ < 1, and 0 < φ < for t ∈ Sφ we have ∞ 0

π 2,

there is a Cb,φ so that

γ

√ |t| 2 −1 √ |∂x ktb (x, y)|| y − x|γ d y ≤ Cb,φ , 1 1 + λ2

(6.39)

where λ = x/|t|. The case γ = 0 is Lemma 8.1 in [15]. L EMMA 6.1.11. For b > 0, 0 < γ < 1, 0 < φ < constant Cb,c,φ so that for cx 2 < x 1 < x 2 , t ∈ Sφ , ∞

π 2,

and 0 < c < 1, there is a

√ √ √ √ | x 1 ∂x ktb (x 1 , y) − x 2 ∂x ktb (x 2 , y)|| x 1 − y|γ d y ≤

0



Cb,c,φ |t|

γ −1 2

√ √  | x2 − x1 | √ |t |  √ √ . | x2 − x1 | √ 1+ |t |

(6.40)

L EMMA 6.1.12. For b > 0, 0 < γ < 1, there is a constant Cb so that for t1 < t2 < 2t1 , we have: t1 ∞ t2 −t1 0

√ γ √ γ |∂x ktb2 −t1 +s (x, y) − ∂x ksb (x, y)|| x − y| 2 d yds < Cb |t2 − t1 | 2 .

(6.41)

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This result follows from the more basic: L EMMA 6.1.13. For b > 0, 0 ≤ γ < 1, and 0 < t1 < t2 < 2t1 , we have for s ∈ [t2 − t1 , t1 ] that there is a constant C so that ∞

√ |∂x ktb2 −t1 +s (x, y) − ∂x ksb (x, y)|| x

−

√

0

γ

(t2 − t1 )s 2 −1 . y| d y < C √ (t2 − t1 + s)(1 + x/s) γ

(6.42)

6.1.3 Second Derivative Estimates L EMMA 6.1.14. For b > 0, 0 < γ < 1, and 0 < φ < for t = |t|eiθ with |θ | < π2 − φ, |t | ∞ 0 0 |t | ∞

0

π 2,

there is a Cb,φ so that

√ γ γ √ b |x∂x2kse x| d yds ≤ Cb,φ x 2 and iθ (x, y)|| y − (6.43) √ b |x∂x2kse iθ (x, y)|| y −

√

γ 2

x|γ d yds ≤ Cb,φ |t| .

0

This follows from the more basic result: L EMMA 6.1.15. For b > 0, 0 ≤ γ < 1, 0 < φ < π2 , there is a Cb,φ so that if t ∈ Sφ , then γ ∞ √ λ|t| 2 −1 √ γ 2 b , (6.44) |x∂x kt (x, y)|| x − y| d y ≤ Cb,φ 1+λ 0

where λ = x/|t|. L EMMA 6.1.16. For b > 0, 0 < γ < 1, 0 < φ < there is a constant Cb,φ so that, for t ∈ Sφ , we have

π 2,

and 0 < x 2 /3 < x 1 < x 2 ,

|t | + + + + b b +(∂ y y − b)kse iθ (x 2 , α) − (∂ y y − b)k seiθ (x 2 , β)+ ds ≤ C b,φ ,

(6.45)

0

where α and β are defined in (6.35). L EMMA 6.1.17. For b > 0, 0 < γ < 1, φ < π2 , and 0 < x 2 /3 < x 1 < x 2 , if J = [α, β], with the endpoints given by (6.35), there is a constant Cb,φ so that if

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CHAPTER 6 π 2

|θ |
0, 0 < γ < 1, 0 < φ < π2 , and 0 < x 2 /3 < x 1 < x 2 , if J = [α, β], with the endpoints given by (6.35), there is a constant Cb,φ so that if |θ | < π2 − φ, then |t | 

√ √ γ √ √ γ b b |L b kse iθ (x 2 , y)−L b k seiθ (x 1 , y)|| y− x 1 | d yds ≤ C b,φ | x 2 − x 1 | . (6.47)

0 Jc

L EMMA 6.1.19. For b > 0, 0 < γ < 1, and t1 < t2 < 2t1 there is a constant Cb so that t1 ∞

√ γ √ |L b ktb2 −t1 +s (x, y) − L b ksb (x, y)|| x − y|γ d yds ≤ Cb |t2 − t1 | 2 .

(6.48)

t2 −t1 0

This lemma follows from the more basic result: L EMMA 6.1.20. For b > 0, 0 < γ < 1, and t1 < t2 < 2t1 and s > t2 − t1 , there is a constant Cb so that ∞

γ √ √ |L b ktb2 −t1 +s (x, y) − L b ksb (x, y)|| x − y|γ d y ≤ Cb (t2 − t1 )s 2 −2 .

(6.49)

0

6.1.4 Large t Behavior To study the resolvent kernel of L b , which is formally given by (μ − L b )

−1

∞ =

e−μt et L b dt,

(6.50)

0

and the off-diagonal behavior of the heat kernel in many variables, it is useful to have estimates for ∞ ∞ j j j b |∂x kt (x, y)|d x, and |x 2 ∂x ktb (x, y)|d y (6.51) 0

0

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87

valid for 0 < t. In the previous section we gave such results, but these were intended to study the behavior of these kernels as t → 0+ , and assumed the H¨older continuity of the data. To study the resolvent we also need estimates as t → ∞, valid for bounded, continuous data. L EMMA 6.1.21. For 0 < b < B, 0 < φ < π2 , and j ∈ ⺞ there is a constant C j,B,φ so that if t ∈ Sφ , then ∞ C j,B,φ j |∂x ktb (x, y)|d y ≤ , (6.52) |t| j 0

and

∞

j

j

|x 2 ∂x ktb (x, y)|d y ≤ 0

C j,B j

|t| 2

.

(6.53)

6.1.5 The Structure of the Proofs of the Lemmas We close this subsection by considering the structure of the proofs of these estimates. Recall that xy  1  y b − x+y ktb (x, y) = (6.54) e t ψb 2 . y t t In most of the estimates that follow we set w = y/|t|, λ = x/|t|, and eθ = e−iθ ; in these variables ktb (x, y)d y = (eθ w)b−1 e−(w+λ)eθ ψb (wλe2θ )dw.

(6.55)

Using Taylor’s theorem when wλ < 1, and the asymptotic expansions for the functions, ψb , ψb when wλ ≥ 1, we repeatedly reduce our considerations to the estimation of a small collection of types of integrals. Most of these are integrals that extend from 0 to 1/λ, or from 1/λ to ∞. We need to consider what happens as λ itself varies from 0 to ∞. The following results are used repeatedly in the proofs of the foregoing lemmas.  L EMMA 6.1.22. For γ > 0, 0 < φ < π2 , there are constants Cb,φ , Cb,φ uniformly π bounded for 0 < b < B, so that for 0 < λ < ∞, |θ | ≤ 2 − φ, we have 1

λ w 0

b−1 − cos θw

e

√ √ | w − λ|γ dw ≤

&C

γ

λ 2 −b as λ → ∞ C b,φ γ +b  2 + Cb,φ as λ → 0. b λ b,φ

b

(6.56)

P ROOF. The proof of this estimate follows easily from the change of variables w = λσ.  L EMMA 6.1.23. For γ ≥ 0, 0 < φ < π2 , and ν ∈ ⺢, there are constants Cν,γ ,φ and aν,γ , uniformly bounded for |ν| < B, so that for 0 < λ < ∞, |θ | < π2 − φ, we

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have ∞

ν

w 2 e− cos θ(

√ √ w− λ)2

1 λ

√ √ dw | w − λ|γ √ ≤ w & cos θ Cν,γ ,φ λaν,γ e− λ ν Cν,γ ,φ λ 2

P ROOF. Setting z =



I =

w λ

λ

as λ → 0+ as λ → ∞.

(6.57)

− 1, the integral in (6.57) becomes: ∞

ν+γ +1 2

2

2

(1 + z)ν e− cos θλz |z|γ dz.

(6.58)

1 λ −1

The estimate as λ → 0 follows easily from this and Lemma 6.1.24, proved below. To prove the result as λ → ∞, we need to split the integral into the part from λ1 − 1 to − 12 , and the rest. A simple application of Laplace’s method shows that the unbounded ν part is estimated by Cν,γ ,φ λ 2 . We can estimate the compact part by e

− cos θ λ4

λ

1

− 2

ν+γ +1 2

2 λ 4

e− cos θ λ 2

ν+γ +1 2

⎧ ⎨

1 ν+1

⎩ log

  ν+1

$λ%

1 2

−

 ν+1

(1 + z)ν =

1 λ −1

(6.59)



1 λ

if ν = −1 if ν = −1.

2

λ

In all cases this quantity is bounded by Cν,γ ,φ e− cos θ 8 , completing the proof of the lemma.  The following lemma is used to prove these estimates: L EMMA 6.1.24. Letting μ ∈ ⺢ and a > 0, we define ∞ G μ (λ, a) =

2

e−λz z μ dz.

(6.60)

a

There are constants Cμ so that 2

G μ (λ, a) ≤ Cμ

e−λa λa 1−μ

For μ > −1, G μ (λ, a) ≤ Cμ

1 λ

1+μ 2

√ 1 for a λ > . 2

(6.61)

√ 1 for a λ ≤ , 2

(6.62)

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if μ = −1, then √ G μ (λ, a) ≤ C−1 | log(a λ)| if μ < −1, then

√ 1 for a λ ≤ , 2

√ 1 for a λ ≤ . 2

G μ (λ, a) ≤ Cμ a 1+μ

(6.63)

(6.64)

P ROOF. The proofs are elementary. A simple change of variables shows that G μ (λ, a) =

1 λ

1+μ 2

√ G μ (1, a λ).

(6.65)

The second estimate is immediate from this formula and the fact that G μ (1, 0) is finite, for μ > −1. To prove the first relation we integrate by parts to obtain that ∞

2

e−z dz = −

w

This easily implies that

∞ −z 2 2 +∞ e−z ++ e + . 2z +w 2z 2

(6.66)

w

∞ w

2

2

e−z dz ≤

e−w , w

(6.67)

which implies the first estimate. The final two estimates follow from the fact that √ √ G μ (1, a λ) diverges as a λ → 0 at a rate determined by μ ≤ −1.  ¨ 6.2 HOLDER ESTIMATES FOR THE 1-DIMENSIONAL MODEL PROBLEMS With these rather extensive preliminaries out of the way, we now give the proofs for the H¨older estimates on solutions stated above. P ROOF OF P ROPOSITION 6.0.10. We first assume that b > 0, and begin with (6.2) for the case k = 0. Using Proposition 5.2.8, Lemma 4.1.1, the b = 0 case follows from the b > 0 case. To prove the higher order estimates we need to assume that the data has support in [0, R], then these results follow easily from the k = 0 case by using Propositions 5.2.8 and 5.2.9. From the maximum principle it is immediate that the supnorm of v is bounded by  f W F,0,γ . In light of Lemma 6.1.1 it suffices to separately prove that √ √ |v(x, t) − v(y, t)| ≤ C f W F,0,γ | x − y|γ and γ

|v(x, t) − v(x, s)| ≤ C f W F,0,γ |t − s| 2 .

(6.68)

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We start the spatial estimate, by estimating |v(x, t) − v(0, t)|. Because for every x, and t > 0, (6.25) holds, we use the formula for ktb , to deduce that v(x, t) − v(0, t) =

∞ 

 ktb (x, y) − ktb (0, y) ( f (y) − f (0))d y.

(6.69)

0

The basic estimate (5.21) shows that ∞ + + γ + b + +kt (x, y) − ktb (0, y)+ y 2 d y;

|v(x, t) − v(0, t)| = 2 f W F,0,γ

(6.70)

0

we apply Lemma 6.1.4 to see that γ

|v(x, t) − v(0, t)| ≤ Cb x 2  f W F,0,γ ,

(6.71)

for all t > 0, and that, for any B, the {Cb } are uniformly bounded for 0 < b < B. This is a very useful estimate, for observe that if M > 1, then γ

y2 ≤

M −1 γ x2 M γ

Thus (6.71) implies that if y 2 ≤ M, so that v satisfies

γ

γ

γ

⇒ x 2 ≤ M(x 2 − y 2 ).

M−1 γ2 M x ,

(6.72)

then there is a constant Cγ ,b , depending on

|v(x, t) − v(y, t)| ≤ |v(x, t) − v(0, t)| + |v(0, t) − v(y, t)| γ

≤ 2Cγ ,b  f W F,0,γ x 2 ≤ Cγ ,b  f W F,0,γ (x

γ 2

(6.73) γ 2

− y ).

Applying Lemma 6.1.2 we see that (6.73) implies that √ √ |v(x, t) − v(y, t)| ≤ Cγ ,b  f W F,0,γ | x − y|γ . as

(6.74)

To complete the spatial part of the estimate we just need to show that (6.74) holds, → ∞, for pairs (x, y) so that

x t

c≤

y < 1, x

(6.75)

with c a positive number less than 1. To that end we introduce a device, familiar from the Euclidean case, that will allow us to obtain the needed estimate. For pairs of points 0 ≤ x 1 < x 2 we define J = [α, β] where α and β are defined by   √ √ √ √  √ 3 x2 − x1 3 x1 − x2 , 0 and β = . (6.76) α = max 2 2 As noted above, this is the WF ball centered on the WF midpoint of [x 1 , x 2 ], with radius equal to d W F (x 1 , x 2 ).

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Using the fact that ktb (x, y) has y-integral 1, we easily deduce that v(x 2 , t) − v(x 1 , t) = ( f (x 2 ) − f (x 1 ))+   b kt (x 2 , y)( f (y) − f (x 2 ))d y − ktb (x 1 , y)( f (y) − f (x 1 ))d y+ 

J



ktb (x 2 , y)( f (x 1 ) − f (x 2 ))d y + Jc

J

(ktb (x 2 , y) − ktb (x 1 , y))( f (y) − f (x 1 ))d y. Jc

(6.77) It is a simple matter to see that the first four terms are estimated by √ √ C f W F,0 γ | x 2 − x 1 |γ ,

(6.78)

leaving just the second integral over J c . Terms of this type are estimated, for c > 1/9, 0,γ in Lemma 6.1.7. Thus for f ∈ ᏯW F (⺢+ ) Lemma 6.1.7 shows that there is a constant C independent of b ≤ B so that v satisfies (6.74). We now turn to the time estimate. We begin by estimating |v(x, t) − v(x, 0)|. Arguing as above we see that we have the estimate: +∞ + + + + + |v(x, t) − v(x, 0)| = ++ ktb (x, y)( f (y) − f (x))d y ++ + + 0 (6.79) ∞ √ √ ≤ 2 f W F,0,γ ktb (x, y)| x − y|γ d y. 0

Integrals of this type are estimated in Lemma 6.1.6, which shows that γ

|v(x, t) − v(x, 0)| ≤ Ct 2  f W F,0,γ .

(6.80)

γ 2

γ 2

Using the estimate in (6.72), we see that for M > 1, if Ms ≤ (M − 1)t , then (6.80) implies that γ γ |v(x, t) − v(x, s)| ≤ 2MC f W F,0,γ |t 2 − s 2 |. (6.81) Using Lemma 6.1.2 this estimate gives γ

|v(x, t) − v(x, s)| ≤ Cγ ,b  f W F,0,γ |t − s| 2 ,

(6.82)

for a constant Cγ ,b uniformly bounded for b ≤ B. This leaves only the case of of pairs s, t with c < s/t < 1, for a c < 1. To complete the last case, we write ∞ + + + + |v(x, t) − v(x, s)| ≤ +ktb (x, y) − ksb (x, y)+ | f (y) − f (x)|d y 0

≤ 2 f W F,0,γ

∞ + + √ √ + b + +kt (x, y) − ksb (x, y)+ | x − y|γ d y. 0

(6.83)

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This case follows from Lemma 6.1.8. Using this lemma we easily complete the proof 0,γ of the Proposition 6.0.10 for the case k = 0. The assertion that v ∈ ᏯW F (⺢+ × ⺢+ ) follows easily from these estimates. Notice that Lemma 6.1.15 applies to show that 0,γ even if f is only in ᏯW F (⺢+ ) then lim x∂x2 v(x, t) = 0 for any t > 0.

(6.84)

lim v(x, t) = 0 for any t > 0,

(6.85)

x→0

To show that

x→∞

we fix an R >> 0 and write R

∞ ktb (x, y) f (y)d y +

v(x, t) =

ktb (x, y) f (y)d y.

(6.86)

R

0

Proposition 4.2.5 shows that for any fixed R the first term tends uniformly to zero as 0,γ x → ∞. As f ∈ ᏯW F (⺢+ ) it follows that limx→∞ f (x) = 0. Hence given  > 0 we can choose R0 so that | f (x)| <  for x > R0 . For this choice of R0 , the second integral is at most , for all x, and the first tends to zero as x → ∞. Thus lim sup |v(x, t)| ≤ , x→∞

(6.87)

which proves (6.85). The estimates for the (2 + γ )-spaces follow easily from what we have just proved ˙ m ([0, ∞)), and 2 j + k ≤ m, then and Lemma 4.1.1. This shows that if f ∈ Ꮿ b j ∂t ∂xk v(x, t)

∞

j

ktb+k (x, y)L b+k ∂ yk f (y)d y.

=

(6.88)

0

In particular, the relations ∞ ktb+1 (x, y)∂ y f (y)d y,

∂x v(x, t) = 0

∞ ktb (x, y)L b f (y)d y and

∂t v(x, t) =

(6.89)

0

x∂x2 v(x, t)

= (∂t − b∂x )v,

and the γ -case, show that vW F,0,2+γ is bounded by  f W F,0,2+γ . Using these identities along with (6.84) and (6.85) allows us to conclude that lim [|v(x, t)| + |∂x v(x, t)| + |x∂x2 v(x, t)|] = 0.

x→∞

(6.90)

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0,2+γ

We can therefore apply Lemma 5.2.3 to see that v ∈ ᏯW F (⺢+ × ⺢+ ). For the k > 0 cases, we need to assume that f is supported in [0, R]. Now, using (6.88) and Proposition 5.2.9 we easily derive (6.2), and (6.4) for any k ∈ ⺞, and k,2+γ k,2+γ can again conclude that v ∈ ᏯW F (⺢+ × ⺢+ ), provided that f ∈ ᏯW F (⺢+ ), and supp f ⊂ [0, R]. Finally we consider what happens as b → 0+ . We begin with the k = 0 case; Proposition 7.8 in [15] shows that the solutions to the Cauchy problem for b > 0 converge uniformly to the solution with b = 0 in sets of the form ⺢+ × [0, T ]. If v b (x, t) denotes these solutions, then we have established the existence of constants Cγ so that for 0 < b < 1, x = y and t = s, the following estimates hold: √ γ √ |v b (x, t) − v b (y, s)| ≤ Cγ  f W F,0,γ [| x − y|γ + |t − s| 2 ].

(6.91)

As the constants Cγ are independent of b, we can let b tend to zero, and apply Proposition 7.8 of [15] to conclude that this estimate continues to hold for b = 0. Using (6.88) k,γ as above we can extend all the remaining estimates for the ᏯW F -spaces to the b = 0 case as well. k,2+γ k,2+γ To treat the ᏯW F -spaces, we use Proposition 5.2.8. If f ∈ ᏯW F (⺢+ ), then the b b solutions v to (4.2), with g = 0, v (x, 0) = f (x) and 0 < b < 1, are a bounded k,2+γ family in ᏯW F (⺢+ × ⺢+ ). Thus for any 0 < γ  < γ there is a subsequence < v bn > k,2+γ 

with bn → 0, which converges to v ∗ ∈ ᏯW F

(⺢+ × ⺢+ ). Evidently v ∗ satisfies

(∂t − L 0 )v ∗ = 0 with v ∗ (x, 0) = f (x).

(6.92)

The uniqueness theorem implies that v ∗ = v 0 , and therefore the family < v b > k,2+γ  converges in ᏯW F (⺢+ × ⺢+ ) to v 0 . Since each element of {v b : 0 < b < 1} satisfies the estimates in (6.4), with uniformly bounded constants, we conclude that k,2+γ v 0 ∈ ᏯW F (⺢+ × ⺢+ ), and v 0 also satisfies the estimate in (6.4).  We now turn to the proof of Proposition 6.0.11. Many parts of the foregoing argument can be recycled. P ROOF OF P ROPOSITION 6.0.11. We begin by studying the operator:  t ∞ K tb g(x)

ktb−s (x, y)g(y, s)d yds,

= 0

(6.93)

0

0,γ

assuming that g ∈ ᏯW F (⺢+ × [0, T ]). We want to show that 0,γ

0,2+γ

K tb : ᏯW F (⺢+ × [0, T ]) → ᏯW F (⺢+ × [0, T ]) is bounded. This entails differentiating under the integral defining K tb g, which is somewhat subtle near to s = t. If g ∈ Ꮿ∞ , then we can apply Corollary 7.6 of [15] to

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conclude that j ∂t ∂xk L lb u(x, t)

=

j lim ∂t ∂xk L lb →0+

t −∞ ktb−s (x, y)g(y, s)d yds. 0

(6.94)

0

In the arguments that follow we show that if g is sufficiently smooth, then the derivaj tives, ∂t ∂xk L lb u, exist and can be defined by this limit. Once cancellations are taken into account, the limits are, in fact, absolutely convergent. Provided that g is sufficiently smooth, we may use Lemma 4.1.2 to bring derivatives past the kernel onto g. 0,γ

Of special import is the case g ∈ ᏯW F (⺢+ × [0, T ]). For 0 <  < t, we let: t −∞ ktb−s (x, y)g(y, s)d yds.

u  (x, t) = 0

(6.95)

0

It follows easily that u  converges uniformly to u in ⺢+ × [0, T ]. Using the standard estimate on the difference √ √ |g(x, t) − g(y, t)| ≤ gW F,0,γ ,T | x − y|γ

(6.96)

and the facts that t −∞ ∂x ktb−s (x, y)[g(y, s) − g(x, s)]d yds

∂x u  (x, t) = 0

0

(6.97)

t −∞ x∂x2 u  (x, t) =

x∂x2 ktb−s (x, y)[g(y, s) − g(x, s)]d yds, 0

0

we can apply Lemmas 6.1.10 and 6.1.15 to establish the uniform convergence of ∂x u  and x∂x2 u  on ⺢+ × [0, T ]. This establishes the continuous differentiability of u in x on [0, ∞) × [0, T ], and the twice continuous differentiability of u in x on (0, ∞) × [0, T ]. We can differentiate u  in t to obtain that ∞ kb (x, y)g(y, t − ) + [x∂x2 + b∂x ]u  (x, t).

∂t u  (x, t) =

(6.98)

0

The right-hand side converges uniformly to g(x, t) + (x∂x2 + b∂x )u(x, t), thereby establishing the continuous differentiability of u in t and the fact that [∂t − (x∂x2 + b∂x )]u = g for (x, t) ∈ ⺢+ × [0, T ].

(6.99)

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This argument, or a variant thereof, is used repeatedly to establish the differentiability of u, the formulæ for its derivatives:  t ∞ ∂x ktb−s (x, y)[g(y, s) − g(x, s)]d yds

∂x u(x, t) = 0

0

(6.100)

 t ∞ x∂x2 u(x, t) =

x∂x2 ktb−s (x, y)[g(y, s) − g(x, s)]d yds, 0

0 0,γ

along with the fact that, for g ∈ ᏯW F (⺢+ × [0, T ]), these are absolutely convergent integrals. We let u(x, t) = K tb g(x). From the maximum principle it is evident that |u(x, t)| ≤ tg L ∞ .

(6.101)

The estimate in (6.74) can be integrated to prove that √ √ |u(x, t) − u(y, t)| ≤ Cb tgW F,0,γ | x − y|γ .

(6.102)

Using (6.105) and (6.110), proved below, and the equation ∂t u = L b u + g, we see that, for s < t, γ

|u(x, t) − u(x, s)| ≤ C(1 + t 2 )gW F,0,γ |t − s|+ γ

γ

γ

Ct 1− 2 (1 + t 2 )gW F,0,γ |t − s| 2 .

(6.103)

Note that (6.101), (6.102) and (6.103) show that there is a constant C so that γ

γ

uW F,0,γ ,T ≤ C T 1− 2 (1 + T 2 )gW F,0,γ ,T .

(6.104)

Below we show that there is a constant Cb so that γ

γ

|x∂x2u(x, t)| ≤ Cb min{x 2 , t 2 }gW F,0,γ ;

(6.105)

dividing by x and integrating gives the H¨older estimate for the first spatial derivative: γ

γ

|∂x u(x, t) − ∂x u(y, t)| ≤ Cb gW F,0,γ |x 2 − y 2 |.

(6.106)

Lemma 6.1.2 then implies that √ √ |∂x u(x, t) − ∂x u(y, t)| ≤ Cb gW F,0,γ | x − y|γ .

(6.107)

To complete the analysis of ∂x u we need to show that there is a constant Cb so that γ

|∂x u(x, t) − ∂x u(x, s)| ≤ Cb gW F,0,γ |t − s| 2 .

(6.108)

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In Lemma 6.1.10 it is shown that there are constants Cb , uniformly bounded for b < B, so that, with λ = x/t, we have γ

|∂x v(x, t)| ≤ C f W F,0,γ

t 2 −1 1

1 + λ2

= Cx

γ 2 −1

γ

 f W F,0,γ

λ1− 2 1

1 + λ2

.

(6.109)

It follows by integrating that γ

|∂x u(x, t)| ≤ Cb gW F,0,γ t 2 ,

(6.110)

and therefore, for any c < 1, there is a C so that if s < ct, then (6.108) holds with Cb = C. We are left to consider ct2 < t1 < t2 , for any c < 1. For 12 t2 < t1 < t2 we have: t 2 −t1∞

∂x ksb (x, y)[g(y, t2 − s) − g(y, t1 − s)]d yds+

∂x u(x, t2 ) − ∂x u(x, t1 ) = 0

0

2t1 −t2∞

[∂x ktb2 −s (x, y) − ∂x ktb1 −s (x, y)](g(y, s) − g(x, s))d yds+ 0

0

t1 ∞ ∂x ktb2 −s (x, y)(g(y, s) − g(x, s))d yds. (6.111) 2t1 −t2 0

To handle the first term, we observe that, for j = 1, 2 we have t 2 −t1∞

∂x ksb (x, y)g(y, t j − s)d yds = 0

0 t 2 −t1∞

∂x ksb (x, y)[g(y, t j − s) − g(x, t j − s)]d yds, (6.112) 0

0

which can be estimated by t 2 −t1∞

gW F,0,γ 0

√ √ γ |∂x ksb (x, y)|| x − y| 2 d yds.

(6.113)

0

Using Lemma 6.1.10, we see that these terms are bounded by the right-hand side of (6.108). In the third integral in (6.111) we use (5.32) to estimate (g(y, s) − g(x, s)), and again apply Lemma 6.1.10 to see that this term is also bounded by the right-hand side of (6.108). This leaves only the second integral in (6.111). To estimate this term we use Lemma 6.1.12.

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We now establish (6.105), and then the H¨older continuity of x∂x2 u(x, t). Because integrates to 1 w.r.t. y, for any x, and t > 1, it follows from (6.94) that

ktb (x, y)

 t ∞ x∂x2 u(x, t)

x∂x2 ksb (x, y)[g(y, t − s) − g(x, t − s)]d yds.

= 0

(6.114)

0

Using the estimate √ √ |g(y, t − s) − g(x, t − s)| ≤ 2gW F,0,γ | x − y|γ

(6.115)

and Lemma 6.1.14 gives:  t ∞ |x∂x2u(x, t)|

≤ gW F,0,γ 0

√ √ |x∂x2 ksb (x, y)|| y − x|γ d yds

0 γ 2

(6.116)

γ 2

≤ Cb gW F,0,γ min{t , x }. This completes the proof of (6.105), and therefore the proof of the spatial H¨older continuity of ∂x u. This argument also establishes that lim x∂x2 u(x, t) = 0.

(6.117)

x→0+

To verify the hypotheses of Lemma 5.2.3 we also need to show that lim [|u(x, t)| + |∂x u(x, t)| + |x∂x2u(x, t)|] = 0.

x→∞

(6.118)

The claim for |u(x, t)| follows as in the proof of (6.85). To estimate the derivatives we need to split the integral defining u into a compact and non-compact part, though more carefully than before. Let ϕ ∈ Ꮿ∞ satisfy ϕ(x) = 1 for x ≤ 0, For R, m ∈ (0, ∞) we let

ϕ(x) = 0 for x > 1. 

ϕ R,m (x) = ϕ

x−R m

(6.119)

 .

(6.120)

Using the mean value theorem we can easily show that there is a constant, C, independent of γ , R, m so that √ *+ ,+ C R+m +ϕ R,m + . (6.121) ≤ W F,0,γ m ,+ *+ We let m = R, so that lim R→∞ +ϕ R,m + W F,0,γ = 0.

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Define T ∞ u 0R (x, t)

ksb (x, y)ϕ R,R (y)g(y, t − s)d yds

= 0

u∞ R (x, t) =

0

(6.122)

T ∞ ksb (x, y)(1 − ϕ R,R (y))g(y, t − s)d yds. 0

0

For any fixed R, it follows from Proposition 4.2.5 that lim [|∂x u 0R (x, t)| + |x∂x2 u 0R (x, t)|] = 0.

x→∞

(6.123)

Given  > 0 we can choose R so that (1 − ϕ R,R (y))g L ∞ < .

(6.124)

Fix a 0 < γ  < γ . Applying (5.31) with (6.121) and (6.124) along with Lemma 5.2.6 we see that lim (1 − ϕ R,R (y))gW F,0,γ  = 0.

R→∞

(6.125)

It now follows from (6.105) and (6.110) that, for a possibly larger R, we have the estimate: 2 ∞ |∂x u ∞ R (x, t)| + |x∂ x u R (x, t)| ≤ .

(6.126)

Combining this with (6.123) we easily complete the proof of (6.118). To finish the spatial estimate, we need only show that x∂x2 u is H¨older continuous. As before, the estimate (6.105) implies that for any c < 1, there is a constant C so that √ √ x 1 < cx 2 ⇒ |x 2 ∂x2 u(x 2 , t) − x 1 ∂x2 u(x 1 , t)| ≤ CgW F,0,γ | x 2 − x 1 |γ . (6.127) We are left to consider pairs x 1 , x 2 with cx 2 < x 1 < x 2 .

(6.128)

Since we have already established the H¨older continuity of the first derivative, it suffices to show that L b u(x, t) is H¨older continuous, which technically, is a little easier. This is a rather delicate estimate; we need to decompose the integral expression for L b u(x 1 , t) − L b u(x 2 , t) as in (6.77). We use the notation introduced there, with

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J = [α, β], etc. L b u(x 2 , t) − L b u(x 1 , t) = t   L b ksb (x 2 , y)(g(y, t − s) − g(x 2, t − s))d y− 0



J

L b ksb (x 1 , y)(g(y, t − s) − g(x 1 , t − s))d y− J



L b ksb (x 2 , y)(g(x 2, t − s) − g(x 1, t − s))d y+ 

Jc

(L b ksb (x 2 , y) − L b ksb (x 1 , y))(g(y, t − s) − g(x 1 , t − s))d y ds

Jc

= I1 + I2 + I3 + I4 .

(6.129)

In this formula the operator L b acts in the x-variable. The justification for this formula is essentially identical to that given for (6.114). We begin by estimating I3 . For this purpose we observe that, for t > 0, we have: L b,x ktb (x, y) = ∂t ktb (x, y) = L tb,y ktb (x, y).

(6.130)

The operator L tb,y = ∂ y (∂ y y − b), so we can perform the y-integral to obtain that t   I3 = (∂ y y − b)ksb (x 2 , α) − (∂ y y − b)ksb (x 2 , β) (g(x 2 , t − s) − g(x 1 , t − s))ds. 0

(6.131)

As usual we use the estimate √ √ |g(x 2 , t − s) − g(x 1 , t − s)| ≤ 2| x 2 − x 1 |γ .

(6.132)

Lemma 6.1.16 therefore completes this step; it shows that there is a constant Cb so that √ √ (6.133) |I3 | ≤ Cb gW F,0,γ | x 1 − x 2 |γ . We now turn to the compactly supported terms I1 and I2 . These terms are estimated by  t β gW F,0,γ

√ √ |L b ksb (x 2 , y)|| y − x 2 |γ d yds

0 α

 t β gW F,0,γ 0 α

(6.134) √ √ |L b ksb (x 2 , y)|| y − x 1 |γ d yds.

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The needed bounds are given in Lemma 6.1.17. This lemma shows that the terms I1 and I2 are estimated by √ √ (6.135) Cb gW F,0,γ | x 2 − x 1 |γ . This leaves only the non-compact term, I4 . Recall that t  (L b ksb (x 2 , y) − L b ksb (x 1 , y))(g(y, t − s) − g(x 1 , t − s))d yds,

I4 =

(6.136)

0 Jc

and that J c = [0, α) ∪ (β, ∞). We use (5.32) to estimate the difference |g(y, t − s) − g(x 1, t − s)|, hence Lemma 6.1.18 completes this case. Using Lemma 6.1.18 we see that I4 also satisfies the bound in (6.135), which therefore completes the proof of the spatial part of the H¨older estimate. To complete the k = 0 case all that remains is to estimate |L b u(x, t1 ) − L b u(x, t2 )|. The time estimate begins very much as the estimate for |v(x, t2 )−v(x, t1 )|; we first show that γ |L b u(x, t)| ≤ Cb gW F,0,γ t 2 . (6.137) This implies that for any M > 1, there is a C M,b so that if t2 > Mt1 , then γ

|L b u(x, t2 ) − L b u(x, t1 )| ≤ C M,b gW F,0,γ |t2 − t1 | 2 ,

(6.138)

which leaves only the case that 1 < t2 /t1 < M. To prove (6.137) we use Lemma 6.1.10 and Lemma 6.1.15. The estimate in (6.110) shows that to prove (6.137) it suffices to show that γ

|x∂x2 u(x, t)| ≤ Cb gW F,0,γ t 2 .

(6.139)

x∂x2 ks (x, y)[g(y, t − s) − g(x, t − s)]d yds,

(6.140)

To prove this we write  t ∞ x∂x2 u(x, t)

= 0

0

which implies that  t ∞ |x∂x2 u(x, t)|

≤ 2gW F,0,γ 0

0

t ≤ CgW F,0,γ

s 0

√ √ |x∂x2 ksb (x, y)|| x − y|γ d yds

γ 2 −1



x/s 1 + x/s

(6.141)

 ds.

The second line follows from Lemma 6.1.15; an elementary argument shows that the γ last integral is bounded by a constant times t 2 , completing the proof of (6.137).

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To complete the time estimate we need to show that, for M > 1, there is a constant C M,b , so that x 1 < x 2 < M x 1 implies that γ

|L b u(x, t2 ) − L b u(x, t1 )| ≤ C M,b gW F,0,γ |t2 − t1 | 2 .

(6.142)

The proof of (6.142), for the remaining cases, is broken into several parts, where we observe that, for t1 < t2 < 2t1 we have: t 2 −t1∞

L b ksb (x, y)[g(y, t2 − s) − g(y, t1 − s)]d yds+

L b u(x, t2 ) − L b u(x, t1 ) = 0

0

2t1 −t2∞

[L b ktb2 −s (x, y) − L b ktb1 −s (x, y)](g(y, s) − g(x, s))d yds+ 0

0

t1 ∞ L b ktb2 −s (x, y)(g(y, s) − g(x, s))d yds. (6.143) 2t1 −t2 0

We denote these terms by I1 , I2 and I3 . We start by estimating I1 , which we split into two parts, I11 − I12 ; each part we rewrite as t 2 −t1∞

L b ksb (x, y)[g(x, t j − s) − g(y, t j − s)]d yds

I1 j = 0

j = 1, 2.

(6.144)

0

Indeed this really explains the meaning of this term as a convergent integral. These are estimated, using the same argument, after we employ the estimate: √ √ |(g(y, t j − s) − g(x, t j − s))| ≤ 2gW F,0,γ || x − y|γ .

(6.145)

This shows, using Lemma 6.1.15, that t 2 −t1∞

|I1 j | ≤ 2gW F,0,γ | 0 0 t 2 −t1

γ

√ √ |L b ksb (x, y)|| x − y|γ d yds

s 2 −1

≤ CgW F,0,γ | 0



x/s 1 + x/s

(6.146)

 ds.

An elementary argument now applies to show that this is bounded by γ

CgW F,0,γ |t2 − t1 | 2 .

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Essentially the same argument works to estimate I3 , which, using (6.145), satisfies: t1 ∞ |I3 | ≤ 2gW F,0,γ

√ √ |L b ktb2 −s (x, y)|| x − y|γ d yds

2t1 −t2 0

(6.147)

2(t2 −t1 )∞

√

|L b ksb (x, y)|| x −

= 2gW F,0,γ t2 −t1

√

y|γ d yds.

0

The last line is again estimated using Lemma 6.1.15. To complete the proof in the k = 0 case all that remains is to estimate I2 . This term is bounded by applying Lemma 6.1.19. This completes the proof of the H¨older estimates for L b u(x, t) in the k = 0 case. As ∂t u = L b u + g,

(6.148)

the estimates on ∂t u are now an immediate consequence. Using equations (6.117) 0,2+γ and (6.118) we apply Lemma 5.2.3 to conclude that u ∈ ᏯW F (⺢+ × [0, T ]), which completes the proof of (6.6) in the k = 0 case. For the k > 0 cases, we need to add the assumption that supp g ⊂ {(x, t) : x ∈ [0, R]}. To prove the higher order estimates we use Lemma 4.1.2, which is an extension of Corollaries 7.6 and 7.7 of [15], and Proposition 5.2.9 to reduce the higher order k,γ estimates to the k = 0 case and elementary estimates for functions in ᏯW F (⺢ ×[0, T ]). All that remains is to consider what happens as b → 0. The estimates proved above hold uniformly for b > 0, with uniform bounds on the constants Cb for b in bounded 0,γ subsets of [0, ∞). If g ∈ ᏯW F (⺢+ × [0, T ]), then the solutions u b are uniformly 0,2+γ bounded in ᏯW F (⺢+ × [0, T ]). Proposition 5.2.8 implies that if 0 < γ  < γ , then there is a subsequence < u bn >, with bn → 0, that converges to some function u ∗ in 0,2+γ  ᏯW F (⺢+ × [0, T ]). Since u ∗ solves (∂t − L 0 )u ∗ = g and u ∗ (x, 0) = 0,

(6.149)

the uniqueness of the solution implies that in fact u ∗ = u 0 , and that 0,2+γ 

lim u b = u 0 in ᏯW F

b→0+

(⺢+ × [0, T ]).

(6.150)

We can therefore take limits in the estimates satisfied by u b for b > 0, to conclude that, 0,2+γ in fact, u 0 ∈ ᏯW F (⺢+ × [0, T ]), and satisfies (6.6), with k = 0. The higher order estimates for the b = 0 case follow from this argument and (4.16). The final claims in Proposition 6.0.11, asserting the convergence of u(x, t) to 0,  as t → 0+ , with respect to the Ꮿk,2+γ -norms, follow easily from these estimates, Proposition 5.2.8, and Lemma 6.0.12. This completes the proof of Proposition 6.0.11. 

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6.3 PROPERTIES OF THE RESOLVENT OPERATOR We conclude this section by proving the estimates for R(μ) f stated in Proposition 6.0.13. P ROOF OF P ROPOSITION 6.0.13. As in the proofs of the previous results, we begin by establishing these results for the k = 0 case, and arbitrary 0 < b. The cases of arbitrary k ∈ ⺞ are obtained using Lemma 4.1.1. As noted earlier, no assumption about the support of the data is needed for the resolvent operator, since we do not have to estimate time derivatives. Hence we only require (6.88) with q = 0. We fix an angle 0 < φ ≤ π2 . 0,γ We begin by showing that if f ∈ ᏯW F (⺢+ ), then R(μ) f ∈ Ꮿ2W F (⺢+ ). First we see that Lemma 6.1.3 implies that R(μ) f  L ∞ ≤ Cb,φ

 f L ∞ . |μ|

(6.151)

The argument in the proof of Proposition 6.0.10 between (6.69) and (6.78) applies mutatis mutandis to show that there is a constant Cb,φ , so that if t ∈ Sφ , then √ √ (6.152) |v(x, t) − v(y, t)| ≤ Cb,φ  f W F,0,γ | x − y|γ . Integrating this shows that there is a constant Cb,α,γ so that ∞

[|v(·, t)|]W F,0,γ |e−μt dt| ≤ Cb,α,γ

0

 f W F,0,γ , |μ|

(6.153)

completing the proof of (6.12). Next observe that, for t ∈ S0 , we have the formulæ: ∞ ∂x ktb (x, y)( f (y) − f (x))d y

∂x v(x, t) = 0

(6.154)

∞ x∂x2 v(x, t) =

x∂x2 ktb (x, y)( f (y) − f (x))d y. 0

Using the first formula and the estimate in Lemma 6.1.10 we see that, for t ∈ Sφ γ

|t| 2 −1 . |∂x v(x, t)| ≤ Cb,φ  f W F,0,γ √ 1 + x/|t|

(6.155)

If | arg μ| < π2 + φ, then, by choosing an appropriate ray in the right half plane we see that there is a positive constant m φ , so that ∞ |∂x R(μ) f (x)| ≤ Cb,φ  f W F,0,γ 0  ≤ Cb,φ

 f W F,0,γ γ

(m φ |μ|) 2

.

γ

e−m φ |μ|s s 2 −1 ds √ 1 + x/s

(6.156)

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Using the estimate in Lemma 6.1.15 we can show that ∞ |x∂x2 R(μ)

f (x)| ≤ Cb,φ  f W F,0,γ 0

≤

  f W F,0,γ Cb,φ γ (m φ |μ|) 2

γ

e−m φ |μ|s xs 2 −1 ds x +s

(6.157)

.

It is useful to note that by splitting this integral into a part from 0 to x and the rest, we can also show that γ

 |x∂x2 R(μ) f (x)| ≤ Cb,φ  f W F,0,γ x 2 .

(6.158)

These estimates show that R(μ) f ∈ Ꮿ2W F (⺢+ ) and, by integrating (6.158), establish the H¨older estimate on the first derivative: [|∂x R(μ) f ]|W F,0,γ ≤ Cb,φ  f W F,0,γ .

(6.159)

With these estimates in hand, we can integrate by parts to establish that (μ − L b )R(μ) f = f for μ ∈ ⺓ \ (−∞, 0].

(6.160)

0,2+γ

Below we show that R(μ) f ∈ ᏯW F (⺢+ ). By the open mapping theorem, to show that R(μ) is also a left inverse for (μ − L b ) it suffices to show that the null-space of μ − L b is trivial. For μ ∈ S0 , this follows immediately from the estimate in (6.2), and the uniqueness of the solution to the Cauchy problem. If, for some μ ∈ S0 , there 0,2+γ were a solution f ∈ ᏯW F (⺢+ ) to L b f = μ f, then the solution to the Cauchy problem with this initial data would be v(x, t) = eμt f. This solution grows exponentially, 0,2+γ contradicting (6.2). Thus for μ ∈ S0 , and f ∈ ᏯW F (⺢+ ) we also have the identity R(μ)(μ − L b ) f = f.

(6.161)

The permanence of functional relations implies that this holds for μ ∈ ⺓ \ (−∞, 0]. We can also apply the observation in (6.72), along with the estimate in (6.158), to see that if any 0 < c < 1 is fixed, then there is a Cb,c,φ so that if y < cx, then √ √ |x∂x2 R(μ) f (x) − y∂x2 R(μ) f (y)| ≤ Cb,c,φ  f W F,0,γ | x − y|γ . 0,2+γ

(6.162)

To complete the proof that R(μ) f ∈ ᏯW F (⺢+ ) we only need to show that there is a 0 < c < 1, so that a similar estimate holds for cx < y < x. This is accomplished exactly as in the proof of Proposition 6.0.11. It suffices to

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estimate |[L b R(μ) f ]|W F,0,γ , and use a decomposition like that given in (6.129): L b R(μ) f (x 2 ) − L b R(μ) f (x 1 ) =  ∞ μseiθ b e L b kse iθ (x 2 , y)( f (y) − f (x 2 ))d y− 0



J b L b kse iθ (x 1 , y)( f (y) − f (x 1 ))d y−

J



b L b kse iθ (x 2 , y)( f (x 2 ) − f (x 1 ))d y+



Jc b [L b kse iθ (x 2 , y)

−

b L b kse iθ (x 2 , y)]( f (y) −

f (x 1 ))d y eiθ ds

Jc

= I1 + I2 + I3 + I4 . (6.163) Here we select θ ∈ (− π2 , π2 ), so that π . (6.164) 2 Fix a positive constant 1/3 < c < 1, and assume that cx 2 < x 1 < x 2 , so that we can apply Lemmas 6.1.16–6.1.18 as in the earlier argument. We use the fact that L b,x ktb (x, y) = L tb,y ktb (x, y), to perform the y-integral in I3 . As the estimate in Lemma 6.1.16 holds uniformly for all |t|, it applies to show that √ √ |I3 | ≤ Cb,φ  f W F,0,γ | x 2 − x 1 |γ . (6.165) | arg μeiθ |
0. The case of b = 0 is obtained by using the fact that the constants in the estimates are uniformly bounded for 0 < b < 1, 0,γ and Proposition 5.2.8. This shows that if we let f ∈ ᏯW F (⺢+ ) and set wb = (μ − 0,2+γ L b )−1 f for 0 < b, then {wb : 0 < b < 1} are uniformly bounded in ᏯW F (⺢+ ). Proposition 5.2.8 shows that for any 0 <  γ < γ , this sequence has a subsequence that 0,2+ γ 0,2+γ converges in ᏯW F (⺢+ ). Any such limit w0 ∈ ᏯW F (⺢+ ) and satisfies (μ − L 0 )w0 = f.

(6.167)

The uniqueness result, Proposition 3.3.4, shows that w0 is uniquely determined, which 0,2+ γ implies that {wb } itself converges in ᏯW F (⺢+ ) to w0 , and that w0 therefore satisfies the estimates in the statement of the proposition. Finally we use Lemma 4.1.1 to commute the x-derivatives past ktb (x, y) and follow the argument above to establish this theorem for arbitrary k ∈ ⺞. 

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Remark. The solution to the Cauchy problem can be expressed as contour integral involving R(μ) :  1 v(x, t) = eμt R(μ) f dμ, (6.168) 2πi α,R

where

α,R := −b{μ : | arg μ| < π − α and |μ| > R}.

(6.169)

Here the − sign indicates that α is taken with the opposite orientation to that it inherits as the boundary of the region on the right-hand side in (6.169). Using this formula we easily establish the analytic continuation of v to t ∈ H+ as well as estimates of the form v(·, t)W F,k,2+γ ≤ C(t) f W F,k,γ . (6.170) The constant C(t) tends to infinity as t → 0 at a rate that depends on γ . As eμt is exj ponentially decreasing along α,R , we can also estimate the time derivatives ∂t v(·, t), for t > 0.

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Chapter Seven ¨ Holder Estimates for Higher Dimensional Corner Models The estimates proved in the previous chapter form a solid foundation for proving analogous results in higher dimensions for model operators of the form L b,m =

n  j =1

[x j ∂x2j + b j ∂x j ] +

m 

∂ y2k ,

(7.1)

k=1

n

here b ∈ ⺢+ . In this context we exploit the fact that the solution operator for L b,m is a product of solution operators for 1-dimensional problems. In 2-dimensions we can write u(x 1 , x 2 , t) − u(y1 , y2 , t) = [u(x 1 , x 2 , t) − u(x 1 , y2 , t)] + [u(x 1, y2 , y) − u(y1, y2 , t)], (7.2) and in n > 2 dimensions we rewrite u(x, t) − u( y, t) as u(x, t) − u( y, t) =

n−1  [u(x j , x j +1 , yj , t) − u(x j , y j +1 , yj , t)],

(7.3)

j =0

where: x j = (x 1 , . . . , x j ) if 1 ≤ j and ∅ if j ≤ 0, x j = (x j +2 , . . . , x n ) if j < n − 1 and ∅ if j ≥ n − 1.

(7.4)

In this way we are reduced to estimating these differences 1-variable-at-a-time, which, in light of Lemma 6.1.1 suffices. In the proofs of the 1-dimensional estimates the only facts about the data we use are contained in the estimates in (5.21) and (5.32). This makes it possible to use these arguments to prove estimates in higher dimensions “one variable at a time.” The only other fact we use is that if f (x 1 , . . . , x n ) is an absolutely integrable function, such that for some j we know that, for any x j , x j , the 1-dimensional integral ∞

f (x j , z j +1 , x j )dz j = 0,

0

107

(7.5)

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then Fubini’s theorem implies that ∞

∞ ···

0

f (z)dz 1 · · · dz n = 0.

(7.6)

0

While we cannot simply quote the 1-dimensional estimates, using formulæ like that in (7.3), we can reduce the proof of an estimate in higher dimensions to the estimation of a product of 1-dimensional integrals. These integrals are in turn estimated in the lemmas stated here in the previous chapter. Using the “1-variable-at-a-time” approach we prove the higher dimensional estimates in several stages; we begin by considering the “pure corner” case where m = 0, and then turn to the Euclidean case, where n = 0. The Euclidean case is of course classical. In the next chapter we state the results we need for the case of general (0, m) and the estimates on the 1-dimensional solution Euclidean heat kernel needed to prove them. Finally, in Chapter 9 we do the general case, where n and m can assume arbitrary non-negative values. We first consider the homogeneous Cauchy problem L b,0 v(x, t) = 0 in ⺢n+ × (0, ∞) and v(x, 0) = f (x). Here b is a vector in solution is given by

⺢n+ .

(7.7)

If f is bounded and continuous, then the unique bounded ∞

v(x, t) =

··· 0

∞ ' n 0

b

kt j (x j , z j ) f (z)d z,

(7.8)

j =1

from which it is clear that |v(x, t)| ≤  f Ꮿ0 (⺢n+ ) .

(7.9)

For fixed x, v(x, t) extends analytically in t to define a function in S0 . We next turn to estimating the solution, u, of the inhomogeneous problem: ⎤ ⎡ N  ⎣ ∂t − [x j ∂x2j + b j ∂x j ]⎦ u = g, (7.10) j =1

vanishing at t = 0. Proposition 4.2.4 shows that the unique bounded solution is given by the integral:  t ∞ u(x, t) =

··· 0

0

∞ ' n 0

b

ks j (x j , z j )g(z, t − s)d zds.

(7.11)

j =1

It is quite easy to see that, for any k ∈ ⺞0 and 0 < γ < 1, the operator ∂t − L b,0 k,2+γ k,γ maps data with compact support in C W F (⺢n+ × [0, T ]) to C W F (⺢n+ × [0, T ]). Our k,γ aim, once again, is to prove that, for data with compact support in C W F (⺢n+ × [0, T ]), k,2+γ the solution belongs to C W F (⺢n+ ×[0, T ]). As in the 1-dimensional case, when k = 0 we do not need to assume that the data has compact support.

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7.1 THE CAUCHY PROBLEM We begin with the somewhat simpler homogeneous Cauchy problem. k,γ

P ROPOSITION 7.1.1. Fix k ∈ ⺞0 , 0 < R, and b ∈ ⺢n+ . Let f ∈ ᏯW F (⺢n+ ), and let v be the unique solution, given in (7.8), to (∂t − L b,0 )v = 0 v(x, 0) = f (x).

(7.12)

If k > 0, then assume that f is supported in N and x ≤ R}. B R+ (0) = {x : x ∈ ⺢+

For 0 < γ < 1 there a constant Ck,γ ,b,R so that vW F,k,γ ≤ Ck,γ ,b,R  f W F,k,γ ,

(7.13)

k,2+γ

and, if f ∈ ᏯW F (⺢n+ ), then vW F,k,2+γ ≤ Ck,γ ,b,R  f W F,k,2+γ .

(7.14)

For fixed γ , the constants Ck,γ ,b,R are uniformly bounded for 0 ≤ b ≤ B. If k = 0, then the constants are independent of R. P ROOF. Suppose that we have proved the estimates above with constants C which, for any B, are uniformly bounded for 0 < b j < B. As shown in the proof of Proposition 6.0.10, the case where b j = 0, for one or more values j, can be treated by choosing a sequence < bn > so that bn, j > 0 for all n and lim bn, j = b j . n→∞

(7.15)

We let v bn (x, t) denote the solutions with the given initial data f. Given that the estimates in the lemma have been proved for each bn , Proposition 5.2.8 shows that the sequence < v bn > contains subsequences convergent with respect to the topology on 0,γ 

ᏯW F , for any 0 < γ  < γ . If t >  > 0, then these solutions also converge uniformly in Ꮿm , for any m > 0. Hence the limit satisfies the limiting diffusion equation with the

given initial data; the uniqueness of such solutions shows that any convergent subse0,γ  quence has the same limit. Thus < v bn > itself converges in ᏯW F to v b , the solution in the limiting case. This implies that v b also satisfies the estimates in the proposition. This reasoning applies equally well to all the function spaces under consideration. Thus it suffices to consider the case where b j > 0 for j = 1, . . . , n, which we henceforth assume. In the sequel we use C to denote positive constants that may depend on γ and b, which are uniformly bounded so long as 0 < γ < 1 is fixed and, for j = 1, . . . , n, 0 < b j ≤ B, for any fixed B. The solution is given by formula (7.8). We observe that v(x 1 , . . . , x n , t) − v(y1 , . . . , yn , t) = v(x 1 , . . . , x n , t) − v(x 1 , . . . , x n−1 , yn , t)+ v(x 1 , . . . , x n−1 , yn , t) − v(x 1 , . . . , x n−2 , yn−1 , yn , t) + · · · + v(x 1 , y2 , . . . , yn , t) − v(y1 , . . . , yn , t).

(7.16)

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Hence it is enough to show that for each 1 < k ≤ n we have: |v(x 1 , . . . , x k−1 , x k , yk+1 , . . . , yn , t) − v(x 1 , . . . , x k−1 , yk , yk+1 , . . . , yn , t)| ≤ C f W F,0,γ ρs (x, y)γ .

(7.17)

It suffices to assume that x and y differ in exactly one coordinate, which we can choose to be n. For an n-vector x we let x  = (x 1 , . . . , x n−1 ).

(7.18)

The proof is simply a matter of recapitulating the steps in the 1-dimensional case, and showing how the n-dimensional case can be reduced to this case. We first do the k = 0 case, which does not require additional hypotheses on f, and then do the k > 0 case assuming that f has bounded support. The first step is to consider the special case v(x  , x n , t) − v(x  , 0, t). Using the fact that ∞ ktb (x, y)d y = 1, (7.19) 0

we see that 



∞

v(x , x n , t) − v(x , 0, t) = 

··· 0

(ktbn (x n , z n )

∞ n−1 ' 0

b

kt j (x j , z j )×

j =1

− ktbn (0, z n ))( f (z  , z n ) −

 f (z  , 0)) dz n d z  .

(7.20)

This follows because, for any z  we have: ∞ (ktbn (x n , z n ) − ktbn (0, z n )) f (z  , 0)dz n = 0.

(7.21)

0

Using the triangle inequality, the positivity of the kernels, the obvious estimate | f (x) − f ( y)| ≤ 2 f W F,0,γ ρs (x, y)γ ,

(7.22)

and (7.19), we obtain the estimate 



∞

|v(x , x n , t) − v(x , 0, t)| ≤ 2 f W F,0,γ

γ

|ktbn (x n , z n ) − ktbn (0, z n )|z n2 dz n . (7.23)

0 γ

Lemma 6.1.4 shows that integral is bounded by C x n2 , showing as before that for any c < 1, there is a C so that if yn < cx n , then √ √ (7.24) |v(x  , x n , t) − v(x  , yn , t)| ≤ C f W F,0,γ | x n − yn |γ .

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For the second step we show that

γ

|∂xn v(x  , x n , t)| ≤ C x n2

−1

γ

 f W F,0,γ

λ1− 2 1+λ

γ

1 2

= Ct 2 −1

 f W F,0,γ 1

1 + λ2

,

(7.25)

where λ = x n /t. Taking advantage of the fact that, for all x n ≥ 0 and t > 0, we have

∞

∂xn ktbn (x n , z n )dz n = 0,

(7.26)

0

it follows that

∞



∂xn v(x , x n , t) =

··· 0

∞ n−1 ' 0

b

kt j (x j , z j )×

j =1

∞



 ∂xn ktbn (x n , z n )( f (z  , z n ) − f (z  , x n )) dz n d z  .

(7.27)

0

As before it follows easily that



∞

|∂xn v(x , x n , t)| ≤ C f W F,0,γ

√ √ |∂xn ktbn (x n , z n )|| z n − x n |γ dz n .

(7.28)

0

An application of Lemma 6.1.10 suffices to complete the proof of (7.25). Integrating (7.25), we can now verify that (7.24) holds so long as λ is bounded. For the last step we fix 0 < c < 1, and consider cx n < yn < x n , and λ → ∞. For this case we need to find an analogue of the rather complicated formula in (6.77),

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which is again straightforward: 

∞



v(x , x n , t) − v(x , yn , t) = 

··· 0

∞ n−1 ' 0

b

kt j (x j , z j )×

j =1

ktbn (x n , z n )( f (z  , z n ) − f (z  , x n ))dz n +

J



ktbn (yn , z n )( f (z  , yn ) − f (z  , z n ))dz n +

J



ktbn (x n , z n )( f (z  , yn ) − f (z  , x n ))dz n +

Jc ∞



ktbn (x n , z n )( f (z  , x n ) − f (z  , yn ))dz n +

0

[ktbn (x n , z n ) − ktbn (yn , z n )]( f (z  , z n ) − f (z  , yn ))dz n d z  .

(7.29)

Jc

Recall that J = [α, β], where √ √ √ 3 yn − x n α= 2



β=

√ √ 3 x n − yn . 2

(7.30)

Using the triangle inequality repeatedly, and (7.19), we see that each term reduces to 1 bj one appearing in the 1-d argument multiplied by n−1 j =1 k t (x j , z j ), from the fact that √ √ | f (z  , x) − f (z  , y)| ≤ 2 f W F,0,γ | x − y|γ .

(7.31)

It follows immediately that the first four z n -integrals contribute terms bounded by a √ √ constant times  f W F,0,γ | x n − yn |γ . This leaves just the last integral over J c . This term is estimated by  √ √ 2 f W F,0,γ |ktbn (x n , z n ) − ktbn (yn , z n )|| z n − yn |γ dz n . (7.32) Jc

For c > 1/9 we may apply Lemma 6.1.7 to this term, and the estimate in (7.24) follows once again. This completes the spatial part of the k = 0 case. We now turn to the estimate of v(x, t)−v(x, s); we begin with the case ct < s < t, for a 0 < c < 1. By definition we have: ∞ v(x, t) − v(x, s) = 0

⎤ ⎡ ∞ ' n n ' bj bj ··· ⎣ kt (x j , z j ) − ks (x j , z j )⎦ f (z)d z. 0

j =1

j =1

(7.33)

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The difference of products can be represented as a telescoping sum: n '

b

kt j (x j , z j ) −

j =1

n '

b

ks j (x j , z j ) =

j =1

n  n−l  '

b

kt j (x j , z j ) ×

j =1

l=1

n '

b

ks j (x j , z j )×

j =n−l+2

  b b kt n−l+1 (x n−l+1 , z n−l+1 ) − ks n−l+1 (x n−l+1 , z n−l+1 ) ,

1 with the convention that, if q < p, then qp = 1. Recall that & zl = (z 1 , . . . , z n−l ) for 0 ≤ l ≤ n − 1 zl = (z n−l+2 , . . . , z n ) for 2 ≤ l ≤ n + 1,

(7.34)

(7.35)

with zl and zl equal to the empty set outside the stated ranges. For 1 ≤ l ≤ n, we have: ∞ 

 b b kt n−l+1 (x n−l+1 , z n−l+1 ) − ks n−l+1 (x n−l+1 , z n−l+1 ) ×

0

f (zl , x n−l+1 , zl )dz n−l+1 = 0. (7.36) Using these observations we can re-express v(x, t) − v(x, s) as ∞

∞ ···

v(x, t) − v(x, s) = 0

0

n  n−l  '



l=1

j =1

b

kt j (x j , z j ) ×

n '

b

ks j (x j , z j )×

j =n−l+2



b b kt n−l+1 (x n−l+1 , z n−l+1 ) − ks n−l+1 (x n−l+1 , z n−l+1 )

×

 [ f (z) − f (z l , x n−l+1 , z l )] d z.

(7.37)

Inserting absolute values, and using (7.19) repeatedly, we see that |v(x, t) − v(x, s)| ≤ ∞  n + + + bn−l+1 b + 2 f W F,0,γ (x n−l+1 , z n−l+1 ) − ks n−l+1 (x n−l+1 , z n−l+1 )+ +kt 0

l=1

√ √ × | z n−l+1 − x n−l+1 |γ dz n−l+1 .

(7.38)

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Applying Lemma 6.1.8 it follows that for any 0 < c < 1, there is a C, so that if ct < s < t, then γ |v(x, t) − v(x, s)| ≤ C f W F,0 γ |t − s| 2 . (7.39) To complete the proof of the proposition we need to consider only v(x, t) − v(x, 0); using (7.19) this can be expressed as ∞ v(x, t) − v(x, 0) =

··· 0

∞ ' n

f (z) − f (x) =

(7.40)

j =1

0

We rewrite

b

kt j (x j , z j )[ f (z) − f (x)]d z.

n−1     [ f (z l , x l+1 ) − f (z l+1 , x l+2 )].

(7.41)

l=0

Putting this expression into the integral above and repeatedly using (7.19), we obtain the estimate: n ∞  √ √ b |v(x, t) − v(x, 0)| ≤ 2 f W F,0,γ kt j (x j , z j )| z j − x j |γ d x j . (7.42) j =1 0

Applying Lemma 6.1.6 shows that there is a constant C so that γ

|v(x, t) − v(x, 0)| ≤ C f W F,0,γ t 2 .

(7.43)

As in the 1-dimensional case, this completes the proof that (7.39) holds for all x, s, t, and thereby the proof of (7.13) in the k = 0 case. If we assume that f is supported in B R+ (0), then the estimates in (7.13) for k > 0 follow from the k = 0 case by repeatedly applying Propositions 4.2.2 and 5.2.9. Many of the estimates needed to prove (7.14) with k = 0 follow from (7.13), Lemma 4.2.3 and applications of Propositions 4.2.2 and 5.2.9. We can use these results to show that n  ∇ x vW F,0,γ + ∂t vW F,0,γ + x i ∂x2i vW F,0,γ ≤ C f W F,0,2+γ . (7.44) i=1

To complete the proof in this case we need to similarly estimate the derivatives √ x i x j ∂xi ∂x j v with i = j (7.45) 0,γ

in ᏯW F (⺢n+ ×[0, ∞)). We can relabel so that i = 1 and j = 2. Using Proposition 4.2.2 we can express these derivatives as √

 x 1 x 2 ∂x1 ∂x2 v(x, t) = ⺢n−2 +

∞ ∞  0

0

x1 z1

1 2

n '

b

kt j (x j , z j )

j =3

ktb1 +1 (x 1 , z 1 )



x2 z2

1 2

√ ktb2 +1 (x 2 , z 2 ) z 1 z 2 ∂z1 ∂z2 f (z)d z.

(7.46)

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0,2+γ

Since f ∈ ᏯW F it is not immediately obvious that this is true, but can be obtained by a simple limiting argument. We let f  = f (x 1 + , x 2 + , x 3 , . . . , x n ),

(7.47)

and v  be the solution of the Cauchy problem with this initial data. For  > 0 it follows easily from Proposition 4.2.2 that √

 x 1 x 2 ∂x1 ∂x2 v  (x, t) = ⺢n−2 +

∞ ∞  0

0

x1 z1

1 2

n '

b

kt j (x j , z j )

j =3

ktb1 +1 (x 1 , z 1 )



x2 z2

1 2

√ ktb2 +1 (x 2 , z 2 ) z 1 z 2 ∂z1 ∂z2 f (z)d z.

(7.48)

√ For t > 0,√the left-hand side converges uniformly to x 1 x 2 ∂x1 ∂x2 v. Since the scaled derivative (z 1 + )(z 2 + )∂z1 ∂z2 f  (z) is uniformly bounded and converges to the √ limit z 1 z 2 ∂z1 ∂z2 f (z), and the kernel  1  1 x 2 2 b2 +1 x 1 2 b1 +1 kt (x 1 , z 1 ) kt (x 2 , z 2 ) (7.49) z1 z2 is absolutely integrable, we see that the limit can be taken inside the integral to give (7.46). The following lemma is used to bound these integrals. L EMMA 7.1.2. If 0 ≤ γ ≤ 1, and b > ν − γ2 > 0, then there is a constant Cb,φ , bounded for b ≤ B, and B −1 < b + γ2 − ν, so that, for t ∈ Sφ , where 0 < φ < π2 , we have the estimate ∞  ν γ γ x |ktb (x, y)|y 2 d y ≤ Cb,φ x 2 . (7.50) y 0

The lemma is proved in Appendix A. 0,2+γ Since f ∈ ᏯW F (⺢n+ ) it follows that γ γ √ | x 1 x 2 ∂x1 ∂x2 f (x)| ≤  f W F,0,2+γ min{x 12 , x 22 , 1};

applying this lemma shows that √ | x i x j ∂xi ∂x j v(x, t)| ≤ C f W F,0,2+γ .

(7.51)

(7.52)

We need to now establish the H¨older continuity of these derivatives. The argument used above for v applies directly to show the H¨older continuity in the (x 3 , . . . , x n ) variables, leaving only x 1 and x 2 . It clearly suffices to do the x 1 -case. We have the estimate: √ | x 1 x 2 ∂x1 ∂x2 v(x, t)| ≤  1 ∞ ∞   12 γ x1 x 2 2 b2 +1 b1 +1  f W F,0,2+γ kt (x 1 , z 1 ) kt (x 2 , z 2 )z 12 dz 1 dz 2 . (7.53) z1 z2 0

0

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Lemma 7.1.2 bounds both the z 1 - and z 2 -integrals and therefore γ γ √ | x 1 x 2 ∂x1 ∂x2 v(x, t)| ≤ C min{x 12 , x 22 } f W F,0,2+γ .

(7.54)

Note that this implies that lim

x i ∨x j →0+

√ | x i x j ∂xi ∂x j v(x, t)| = 0.

(7.55)

If c < 1, then this implies the estimate   √ √ | x 1 x 2 ∂x1 ∂x2 v(x, t) − x 1 x 2 ∂x1 ∂x2 v(x  , t)| ≤ C| x 1 − x 1 |γ  f W F,0,2+γ (7.56) for x 1 < cx 1 . Thus we are left to consider cx 1 < x 1 < x 1 . To simplify the notation in the following calculations, we let z˜ = (z 3 , . . . , z n ), b˜ = (b3 , . . . , bn ), f12 (z) = ∂z1 ∂z2 f (z), and ˜

x, z˜ ) = ktb (

n '

(7.57)

b

kt j (x j , z j ).

(7.58)

j =3

We have the formula √

x 1 x 2 ∂x1 ∂x2 v(x, t) −



x 1 x 2 ∂x1 ∂x2 v(x  , t) =

 ∞ ∞

 √

0 ⺢n−2 +

x 1 ktb1 +1 (x 1 , z 1 ) −

˜

ktb ( x, z˜ )ktb2 +1 (x 2 , z 2 )

0





x 1 ktb1 +1 (x 1 , z 1 )

We assume that

and let √ α = max

√



x2 z2

1 2

×

z 2 f 12 (z 1 , z 2 , z˜ )dz 1 dz 2 d z˜ .

x 1 ≤ 1 ≤ 1, 4 x1 ⎧  √ ⎨ 3 x 1 − x 1 ⎩

2

,0

⎫ ⎬ ⎭

and

(7.59)

(7.60)



β=

 √ 3 x 1 − x 1 2

.

(7.61)

Note that (7.60) implies that α > x 1 /4. We let J = [α, β], and observe that   √ √ | x 1 z 2 f 12 (x 1 , z 2 , z˜ ) − x 1 z 2 f 12 (x 1 , z 2 , z˜ )| ≤ 2 f W F,0,2+γ | x 1 − x 1 |γ . (7.62)

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This estimate implies that γ −1 √ | z 2 f 12 (x 1 , z 2 , z˜ )| ≤ 2C f W F,0,2+γ |x 1 2 .

(7.63)

To estimate the difference in(7.62) we dissect the z 1 -integral in a manner similar to that used in (6.77): √

x 1 x 2 ∂x1 ∂x2 v(x, t) −



x 1 x 2 ∂x1 ∂x2 v(x  , t) =

 ∞

 J



˜ b +1 ktb ( x, z˜ )kt 2 (x 2 , z 2 )

0 ⺢n−2 +



x2 z2

1 2

×

√ ktb1 +1 (x 1 , z 1 ) x 1 z 2 [ f 12 (z 1 , z 2 , z˜ ) − f 12 (x 1 , z 2 , z˜ )]dz 1 −  ktb1 +1 (x 1 , z 1 ) x 1 z 2 [ f 12 (z 1 , z 2 , z˜ ) − f12 (x 1 , z 2 , z˜ )]dz 1 +

J

 

√ ktb1 +1 (x 1 , z 1 ) x 1 z 2 [ f 12 (x 1 , z 2 , z˜ ) − f 12 (x 1 , z 2 , z˜ )]dz 1 +

Jc

 √ [ktb1 +1 (x 1 , z 1 ) x 1 z 2 − ktb1 +1 (x 1 , z 1 ) x 1 z 2 ][ f 12 (z 1 , z 2 , z˜ ) − f12 (x 1 , z 2 , z˜ )]dz 1 +

Jc

√

[ x 1 z 2 f 12 (x 1 , z 2 , z˜ ) −



x 1 z 2 f 12 (x 1 , z 2 , z˜ )]

dz 2 d z˜ .

(7.64)

We denote the terms on the right-hand side by I, I I, I I I, I V, and V . The terms I, I I, I I I, and V can be estimated fairly easily. For V we apply the estimate in (7.62) and Lemma 7.1.2 to conclude that  √ |V | ≤ 2 f W F 0,2+γ | x 1 − x 1 |γ . (7.65) To handle I we observe that √ | x 1 z 2 [ f 12 (z 1 , z 2 , z˜ ) − f 12 (x 1 , z 2 , z˜ )]| √ √ √ √ ≤ | z 1 z 2 f12 (z 1 , z 2 , z˜ ) − x 1 z 2 f 12 (x 1 , z 2 , z˜ )| + | z 1 z 2 − x 1 z 2 || f 12 (z 1 , z 2 , z˜ )|  γ −1 √ √ √ γ √ 2 ≤ 2 f W F 0,2+γ | x 1 − z 1 | + z 1 | z 1 − x 1 | . (7.66) We note that γ −1 2

z1

√

| z1 −

√

+ 2 +1−γ √ √ γ ++ x 1 ++ x 1 | = | z 1 − x 1 | +1 − . z + 1

(7.67)

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For z 1 ∈ J the ratio

x1 ≤ 16, (7.68) z1  √ √ √ and the differences | z 1 − x 1 | and | z 1 − x 1 | are bounded above by a multiple of  √ | x 1 − x 1 |. Once again we can use Lemma 7.1.2 to see that  √ |I | ≤ 2C f W F 0,2+γ | x 1 − x 1 |γ . The same argument applies with minor modifications to show that  √ |I I | ≤ 2C f W F 0,2+γ | x 1 − x 1 |γ .

(7.69)

(7.70)

This argument also shows that  √ √ | x 1 z 2 [ f 12 (x 1 , z 2 , z˜ ) − f12 (x 1 , z 2 , z˜ )]| ≤ 2 f W F,0,2+γ | x 1 − x 1 |γ ,

(7.71)

which implies that  √ |I I I | ≤ 2C f W F 0,2+γ | x 1 − x 1 |γ .

(7.72)

This leaves only the term of type I V. We rewrite this term as + 3 ++ 2  + + b1 +1 x + x1 (x 1 , z 1 ) − ktb1 +1 (x 1 , z 1 ) 1 ++ × I V = ++kt z1 z1 + + Jc √ z 1 z 2 | f 12 (z 1 , z 2 , z˜ ) − f 12 (x 1 , z 2 , z˜ )|dz 1 .

(7.73)

Arguing as in (7.67) we see that √ | z 1 z 2 [ f 12 (z 1 , z 2 , z˜ ) − f 12 (x 1 , z 2 , z˜ )]| ≤    √ √ γ  γ −1  2 f W F,0,2+γ | z 1 − x 1 | + (x 1 ) 2 | z 1 − x 1 | .

(7.74)

We complete the estimate of I V with the following lemma: L EMMA 7.1.3. If J = [α, β], with α, β are given by (7.61), assuming that x 1 , x 1 satisfy (7.60), and b > 0, 0 < γ ≤ 1, and 0 < φ < π2 , there is a Cb,φ so that if t ∈ Sφ , then, + 3 ++ 2  +   + + b+1 +k (x 1 , z 1 ) x 1 − k b+1 (x  , z 1 ) x 1 + |√z 1 − x  |γ dz 1 ≤ Cb,φ |√x 1 − x  |γ . t 1 1 1 + t + z1 z1 + + Jc

(7.75)

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Applying the lemma to the expression in (7.73) completes the proof that  √ | x i x j ∂xi ∂x j v(x, t) − x i x j ∂xi ∂x j v(x  , t)| ≤ C f W F,0,2+γ ρs (x, x  )γ .

119

(7.76)

We now consider the H¨older continuity in time for these derivatives. We re-write this difference as √ x 1 x 2 ∂x1 ∂x2 [v(x, t) − v(x, 0)] =  ∞ ∞ √ ˜ x1 x2 ktb ( x, z˜ )ktb1 +1 (x 1 , z 1 )ktb2 +1 (x 2 , z 2 )[ f 12 (z) − f 12 (x)]dz 1 dz 2 d z˜ , 0 ⺢n−2 +

0

(7.77) where f 12 = ∂z1 ∂z2 f. We re-write the difference f 12 (z) − f 12 (x) as 2 n z1 z2  [ f 12 (zl , z l , x l ) − f 12 (z l , xl , x l )]+ f 12 (z) − f 12 (x) = z1 z2 j =3

[ f 12 (z 1 , z 2 , x n )

− f 12 (z 1 , x 2 , x n )] + [ f 12 (z 1 , x 2 , x n ) − f 12 (x 1 , x 2 , x n )].

(7.78)

Substituting from the sum in (7.78) into (7.77), we see that the terms for l = 3, . . . , n are each bounded by: ∞ ∞ ∞  f W F,0,2+γ

b +1

kt 1 0

0

b +1

(x 1 , z 1 )kt 2

0

2

(x 2 , z 2 )×

√ √ x 1 x 2 bl kt (xl , z l )| xl − z l |γ dz l dz 1 dz 2 . z1 z2

(7.79)

From Lemma 7.1.2 and Lemma 6.1.6 we see that these terms are bounded by a constant γ times  f W F,0,2+γ t 2 . We re-write [ f 12 (z 1 , z 2 , x n ) − f 12 (z 1 , x 2 , x n )] as 1 √ √ [( z 1 x 2 − z 1 z 2 ) f 12 (z 1 , z 2 , x n )+ [ f 12 (z 1 , z 2 , x n ) − f 12 (z 1 , x 2 , x n )] = √ z1 x2 √ √ ( z 1 z 2 f12 (z 1 , z 2 , x n ) − z 1 x 2 f 12 (z 1 , x 2 , x n )]. (7.80) The right-hand side is estimated by γ −1  f W F,0,2+γ √ √ √ √ [| x 2 − z 2 |z 2 2 + | x 2 − z 2 |γ ]. √ z1 x2

(7.81)

The contribution of this term is therefore bounded by ∞ ∞ 2  f W F,0,2+γ 0

0

x 1 b1 +1 b +1 k (x 1 , z 1 )kt 2 (x 2 , z 2 )× z1 t γ −1 √ √ √ √ [| x 2 − z 2 |γ + | x 2 − z 2 |z 2 2 ]dz 1 dz 2 .

(7.82)

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√ √ Lemmas 7.1.2 and 6.1.6 show that the | x 2 − z 2 |γ -term is bounded by γ

C f W F,0,2+γ t 2 . To bound the contribution of the other term we apply L EMMA 7.1.4. For 0 ≤ b, 0 ≤ γ < 1, there is a constant Cb,φ , bounded for b ≤ B, so that for t ∈ Sφ , ∞

γ −1 γ √ √ |ktb+1 (x, y)|| y − x|y 2 d y ≤ Cb,φ |t| 2 .

(7.83)

0

The last term is re-written as √

x 1 x 2 [ f 12 (z 1 , x 2 , x n ) − f12 (x 1 , x 2 , x n )] = √ √ √ [ z 1 x 2 f 12 (z 1 , x 2 , x n ) − x 1 x 2 f 12 (x 1 , x 2 , x n )] + ( x 1 x 2 − z 1 x 2 ) f 12 (z 1 , x 2 , x n ), (7.84) √

which is estimated by γ −1 √ √ √ √  f W F 0,2+γ [| z 1 − x 1 |z 1 2 + | z 1 − x 1 |γ ].

(7.85)

These terms are estimated as in the previous case, showing that altogether there is a C so that γ √ √ | x i x j ∂xi ∂x j v(x, t) − x i x j ∂xi ∂x j v(x, 0)| ≤ C f W F,0,2+γ t 2 ,

(7.86)

which implies that for a c < 1, there is a C so that, if s < ct, then γ √ √ | x i x j ∂xi ∂x j v(x, t) − x i x j ∂xi ∂x j v(x, s)| ≤ C f W F,0,2+γ |t − s| 2 .

(7.87)

This leaves only the case ct < s < t, for a c < 1. We begin with the analogue of (7.37), √

∞ x 1 x 2 ∂x1 ∂x2 [v(x, t) − v(x, s)] =

··· 0

n '

∞

√

x1 x2

0

n−l n '  l=1

 b

kt j (x j , z j )×

j =1

     b b b ks j (x j , z j ) kt n−l+1 (x n−l+1 , z n−l+1 ) − ks n−l+1 (x n−l+1 , z n−l+1 ) ×

j =n−l+2

[ f 12 (z) − where

 b1 = b1 + 1,

f12 (zl , x n−l+1 , z l )]

 b2 = b2 + 1, and  b j = b j for j > 2.

 d z, (7.88)

(7.89)

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Each term in this sum with 1 ≤ l ≤ n − 2 is estimated by ∞ ∞ ∞   f W F,0,2+γ 0

0

x1 z1

1 2

b +1 kt 1 (x 1 , z 1 )



x2 z2

1 2

b +1

kt 2

(x 2 , z 2 )

0   bn−l+1 b × |kt (x n−l+1 , z n−l+1 ) − ks n−l+1 (x n−l+1 , z n−l+1 )| √ √ × | z n−l+1 − x n−l+1 |γ dz n−l+1 dz 2 dz 1 .

(7.90)

Lemmas 7.1.2 and 6.1.8 show that these terms are bounded by γ

C f W F,0,2+γ |t − s| 2 .

(7.91)

We now turn to l = n − 1, and n. These cases are essentially identical; we give the details for l = n − 1. The contribution of this term is bounded by   f W F,0,2+γ ⺢n−2 +

n ' j =3

b ks j (x j , z j )

∞ ∞ 0

√

b +1

x 1 x 2 kt 1

(x 1 , z 1 )×

0

+ + + + b2 +1 (x 2 , z 2 ) − ksb2 +1 (x 2 , z 2 )+ | f 12 (z) − f 12 (z 1 , x 2 , z n−1 )|dz 2 dz 1 d z n−1 . +kt

(7.92)

Proceeding as in (7.84) and (7.85), we see that √

x 1 x 2 | f 12 (z) − f 12 (z 1 , x 2 , z n−1 )| ≤ 2 γ −1 √ √ √ x1 √  f W F,0,2+γ [| x 2 − z 2 |z 2 2 + | x 2 − z 2 |γ ]. z1

(7.93)

Applying Lemma 7.1.2 shows that we are left to estimate  f W F,0,2+γ

∞ + + + b2 +1 + (x 2 , z 2 ) − ksb2 +1 (x 2 , z 2 )+ +kt 0 γ −1 √ √ √ √ × [| x 2 − z 2 |z 2 2 + | x 2 − z 2 |γ ]dz 2 .

(7.94)

Lemma 6.1.8 shows that ∞ + + √ γ √ + b2 +1 + (x 2 , z 2 ) − ksb2 +1 (x 2 , z 2 )+ | x 2 − z 2 |γ dz 2 ≤ C|t − s| 2 , +kt

(7.95)

0

leaving only ∞ + + √ γ −1 √ + b2 +1 + (x 2 , z 2 ) − ksb2 +1 (x 2 , z 2 )+ | x 2 − z 2 |z 2 2 dz 2 . +kt 0

This term is bounded in the following lemma:

(7.96)

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L EMMA 7.1.5. For 0 ≤ γ < 1, 1 ≤ b, and 0 < c < 1, there is a constant Cb , so that if ct < s < t, then ∞ + + √ γ √ γ −1 + b + +kt (x, z) − ksb (x, z)+ | x − z|z 2 dz ≤ Cb |t − s| 2 .

(7.97)

0

The proof of the lemma is in Appendix A. Applying this result completes the proof that γ √ | x 1 x 2 ∂x1 ∂x2 [v(x, t) − v(x, s)]| ≤ C f W F,0,2+γ |t − s| 2 . (7.98) 0,γ

The fact that ∂t v = L b,0 v allows us to deduce that ∂t v ∈ ᏯW F (⺢n+ × [0, ∞)) and satisfies the same estimates as the spatial derivative. This finishes the proof of (7.14) in the k = 0 case. We can now proceed as we did in the proof of (7.13) for k > 0, applying Proposition 4.2.2 to commute derivatives past the kernel functions. We now assume that f has support in B R+ (0), which allows the use of Proposition 5.2.9 to estimate the resultant data. This reduces the proof of (7.14) for k > 0, to the k = 0 case, which thereby completes the proof of the proposition.  7.2 THE INHOMOGENEOUS CASE We now turn to estimating the solution of the inhomogeneous problem in a n-dimensional corner. Let g ∈ ᏯW F,0,γ (⺢n+ × [0, T ]), and let u denote the solution to ∂t u −

N  [x j ∂x2j + b j ∂x j ]u = g,

(7.99)

j =1

which vanishes at t = 0. According to Proposition 4.2.4, it is given by the integral:  t ∞ ···

u(x, t) = 0

0

∞ ' n 0

b

ks j (x j , z j )g(z, t − s)d zds.

(7.100)

j =1

P ROPOSITION 7.2.1. Fix 0 < γ < 1, 0 < R, k ∈ ⺞0 , and (b1 , . . . , bn ) ∈ ⺢n+ . Let k,γ g ∈ ᏯW F (⺢n+ × [0, T ]), and let u be the unique solution, given in (7.11) to (7.99), with u(x, 0) = 0. If k > 0, then assume that g is supported in B R+ (0) × [0, T ]. The solution k,2+γ

u ∈ ᏯW F (⺢n+ × [0, T ]);

(7.101)

there is a constant Ck,γ ,b,R so that uW F,k,2+γ ,T ≤ Ck,γ ,b,R (1 + T )gW F,k,γ ,T .

(7.102)

For fixed γ , the constants Ck,γ ,b,R are uniformly bounded for 0 ≤ b < B. If k = 0, then the constants are independent of R.

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P ROOF. As before it suffices to assume that b j > 0 for j = 1, . . . , n. The estimates we prove below have constants C, which, for any B, are uniformly bounded if 0 < b j < B. The case where b j = 0, for one or more values j, is again treated by choosing a sequence < bn > so that bn, j > 0 for all n and lim bn, j = b j . n→∞

(7.103)

We let < u bn > denote the solutions with the given data g. Given that the estimates in the lemma have been proved for each bn , we see that Proposition 5.2.8 and uniqueness 0,2+γ  imply that for 0 < γ  < γ , the sequence < u bn > converges, in ᏯW F , to u b , the solution in the limiting case. This implies that u b also satisfies the estimates in the proposition. This reasoning applies equally well to all the function spaces under consideration. It therefore suffices to consider the case where b j > 0 for j = 1, . . . , n, which we henceforth assume. As in the proof of Proposition 6.0.11 we note that with t −∞ u  (x, t) =

··· 0

∞ ' n

0

b

ks j (x j , z j )g(z, t − s)d zds,

(7.104)

j =1

0

the solution u is the uniform limit of u  . The functions u  are smooth where t > 0, and we can show as before that, for 0 <  < t, we have ∂xk u  (x, t) = t −∞ √

0

∞ ' n b · · · ∂xk ks j (x j , z j )[g(z, t − s) − g(zk , x k , z k )]d zds

0

j =1

0

x k xl ∂xl ∂xk u  (x, t) = t −∞ ··· 0

0

∞ √

x k x l ∂ xl ∂ x k

n '

b

ks j (x j , z j )[g(z, t − s) − g(zk , x k , z k )]d zds.

j =1

0

(7.105) 0,γ

Assume that g ∈ ᏯW F (⺢n+ ×[0, T ]), for a 0 < γ < 1. Using Lemma 6.1.10 for the first derivatives, the mixed derivatives where k = l, and Lemma 6.1.15, when k = l, we can again show that these derivatives converge, as  → 0+ , uniformly on ⺢n+ × [0, T ]. This shows that u has continuous first partial x-derivatives on ⺢n+ × [0, T ], and continuous second x-derivatives on (0, ∞)n × [0, T ], with √ lim x k xl ∂xl ∂xk u(x, t) = 0. (7.106) xl ∨x k →0+

This also shows that we can allow  → 0+ , in the expressions for these derivatives in (7.105), to obtain absolutely convergent expressions for the corresponding derivatives of u. Finally we argue as before to show that ∂t u = L b,0 u + g,

(7.107)

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and therefore the t-derivative of u is continuous and u satisfies the desired equation. Note that t u(x, t) = v s (x, t − s)ds, (7.108) 0

where v s (x, t) is the solution to [∂t − L b,0 ]v s = 0 with v s (x, 0) = g(x, s).

(7.109)

This relation allows us to use estimates on the solution to the Cauchy problem to derive bounds on u. From the positivity of the heat kernel and (7.19) it is immediate that |u(x, t)| ≤ gW F,0,γ t.

(7.110)

To establish the Lipschitz continuity of u we integrate (7.25) to conclude that, for each 1 ≤ j ≤ n, we have the estimate: γ

|∂x j u(x, t)| ≤ CgW F,0,γ t 2

(7.111)

and therefore γ

|u(x j , x j , x j , t) − u(x j , y j , x j , t)| ≤ CgW F,0,γ t 2 |x j − y j |.

(7.112)

Thus we can also integrate the estimate in (7.24) with respect to t to see that √ √ |u(x j , x j , x j , t) − u(x j , y j , x j , t)| ≤ CgW F,0,γ t| x j − y j |γ .

(7.113)

The estimates, proved below on the first and second derivatives, show that γ

|L b,0 u(x, t)| ≤ CgW F,0,γ t 2 ,

(7.114)

thus the equation [∂t − L b,0 ]u(x, t) = g(x, t) implies that, for t1 < t2 , we have γ

|u(x, t2 ) − u(x, t1 )| ≤ C(1 + t22 )gW F,0,γ |t2 − t1 |.

(7.115)

Our next task is to establish the H¨older estimate for the first spatial-derivatives. There is a small twist in the higher dimensional case: we use one argument to estimate |∂x j u(x j , x j , x j , t) − ∂x j u(x j , y j , x j , t)|, and a rather different argument to estimate |∂xl u(x j , x j , x j , t) − ∂xl u(x j , y j , x j , t)|, for l = j. The former follows exactly as in the 1-dimensional case; we show that there is a constant so that for 1 ≤ l ≤ n, γ

γ

|xl ∂x2l u(x, t)| ≤ CgW F,0,γ min{t 2 , xl2 }. This estimate implies that

lim xl ∂x2l u(x, t) = 0.

xl →0+

(7.116) (7.117)

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The proof of (7.116) follows simply from  t ∞ xl ∂x2l u(x, t)

=

··· 0

0

∞ ' n 0

b

ks j (x j , z j )×

j =l

xl ∂x2l ksbl (xl , z l )[g(z, t − s) − g(zl , xl , z l , t − s)]d zds. Putting absolute values inside the integral, using the estimates  |g(x, t) − g( y, s)| ≤ 2gW F,0,γ [ρs (x, y) + |t − s|]γ ,

(7.118)

(7.119)

and (7.19), we see that (7.116) follows from Lemma 6.1.14. Integrating ∂x2j u and applying (7.116), we see that √ √ |∂x j u(x j , x j , x j , t) − ∂x j u(x j , y j , x j , t)| ≤ CgW F,0,γ | x j − y j |γ .

(7.120)

To do the “off-diagonal” case we use Lemma 6.1.5. To estimate |∂xl u(x m , x m , x m , t) − ∂xl u(x m , ym , x m , t)|, for l = m, we observe that ∂xl u(x m , x m , x m , t)

− ∂xl u(x m , ym , x m )

 t ∞ =

··· 0

0

∞ ' n 0

b

ks j (x j , z j )×

j =l,m

[ksbm (x m , z m ) − ksbm (ym , z m )]∂xl ksbl (xl , z l )[g(z, t − s) − g(zl , xl , z l , t − s)]d zds. (7.121) Putting absolute values into the integral and using (7.119), and (7.19), gives: |∂xl u(x m , x m , x m , t) − ∂xl u(x m , ym , x m )| ≤ gW F,0,γ ×  t ∞ ∞ 0

0

√ √ |ksbm (x m , z m ) − ksbm (ym , z m )||∂xl ksbl (xl , z l )|| z l − xl |γ dz m dz l ds.

0

(7.122) If ym = 0, then applying (6.32) we see that this is estimated by t CgW F,0,γ 0

γ

s 2 −1

x m /s ds, 1 + x m /s γ

(7.123)

which is easily seen to be bounded by CgW F,0,γ x m2 . Applying (6.72) we see that, if 0 < c < 1, then there is a constant C so that for ym < cx m , we have √ √ |∂xl u(x m , x m , x m , t) − ∂xl u(x m , ym , x m )| ≤ CgW F,0,γ | x m − ym |γ . (7.124)

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We are therefore reduced to considering cx m < ym < x m , for a c < 1. If we use (6.33) it follows that |∂xl u(x m , x m , x m , t) − ∂xl u(x m , ym , x m )| ≤ CgW F,0,γ × ⎛ √x −√ y ⎞ m m t √ γ s √ √ ⎠ ds. s 2 −1 ⎝ x − y 1 + m√s m

(7.125)

0

√

We split this into an integral from 0 to ( x m −

√

ym )2 and the rest, to obtain:

|∂xl u(x m , x m , x m , t) − ∂xl u(x m , ym , x m , t)| ≤ CgW F,0,γ × ⎤ ⎡ √ √ 2 ( x m− ym ) √ t √  γ γ x m − ym ⎥ ⎢ ds ⎦ . s 2 −1 + s 2 −1 √ ⎣ s √ √

(7.126)

( x m − ym )2

0

Performing these integrals shows that (7.124) holds in this case as well. To complete the analysis of ∂x j u(x, t) we need to show that there is a constant so that γ |∂x j u(x, t2 ) − ∂x j u(x, t1 )| ≤ CgW F,0,γ |t2 − t1 | 2 . (7.127) This follows immediately from the 1-dimensional argument. Using (7.111), we see that for any c < 1, there is a C so that this estimate holds for t1 < ct2 . As in the 1-dimensional case, we now assume that t1 < t2 < 2t1 . Without loss of generality we can take j = n; use (6.111) to re-express ∂xn u(x, t2 ) − ∂xn u(x, t1 ) as ∂xn u(x, t2 ) − ∂xn u(x, t1 ) = t 2 −t1∞

···

∞ n−1 '

b

ks j (x j , z j )∂xn ksbn (x n , z n )×

0 0 0 j =1 [g(zn , z n , t2 − s) − g(z n , z n , t1 2t1 −t2∞

∞ ···

0

0

⎡

∂xn ⎣

n ' j =1

0

b

kt2j−s (x j , z j ) −

[g(zn , z n , s) − t1 ∞ ∞ n−1 '

··· 2t1 −t2 0

0

j =1

− s)]dz n d z n ds+ n ' j =1

⎤

b

kt1j−s (x j , z j )⎦ ×

g(z n , x n , s)]dz n d z n ds+ b

kt2j−s (x j , z j )∂xn ktb2n−s (x n , z n )× [g(zn , z n , s) − g(z n , x n , s)]dz n d z n ds.

(7.128)

In the first integral we replace g(z n , z n , t j −s) with g(z n , z n , t j −s)− g(z n , x n , t j −s), for j = 1, 2, and then apply Lemma 6.1.10, as in the 1-dimensional case, to show that

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this term is bounded by the right-hand side of (7.127). A similar argument is applied to estimate the third integral. To handle the second term we use formula (7.34) to conclude that / n 0 n ' b ' bl l ∂xn kt2 (xl , z l ) − kt1 (xl , z l ) = l=1

l=1 n 



∂xn

m=1

 n−m '

ktb2l (xl , z l ) ×

l=1

n '

ktb1l (xl , z l )×

l=n−m+2



b b kt2n−m+1 (x n−m+1 , z n−m+1 ) − kt1n−m+1 (x n−m+1 , z n−m+1 )

. (7.129)

To estimate the contribution to the second integral coming from the term in (7.129) with m = 1, we observe that √ √ |g(zn , z n , s) − g(z n , x n , s)| ≤ 2gW F,0,γ | x n − z n |γ , (7.130) and apply Lemma 6.1.12. The contributions of the other terms are bounded by 2t1 −t2∞ ∞

b

b

|kt2j−s (x j , z j ) − kt1j−s (x j , z j )|×

CgW F,0,γ 0

0

0

√ √ |∂xn ktb1n−s (x n , z n )|| x n − z n |γ dz j dz n ds.

(7.131)

We use Lemma 6.1.10 and Lemma 6.1.9 to see that, upon setting σ = t1 − s, this integral is bounded by t1 CgW F,0,γ t2 −t1

γ −1

γ σ 2 (t2 − t1 ) ≤ CgW F,0,γ |t2 − t1 | 2 . √ √ ( σ + x n )(t2 − t1 + σ )

(7.132)

This completes the proof that there is a constant C so that γ

|∂x j u(x, t1 ) − ∂x j u(x, t2 )| ≤ CgW F,0,γ |t2 − t1 | 2 .

(7.133)

The fact that there is a constant C so that 1

|∇ x u(x 1 , t1 ) − ∇ x u(x 2 , t2 )| ≤ CgW F,0,γ [ρs (x 1 , x 2 ) + |t2 − t1 | 2 ]γ

(7.134)

now follows from the foregoing estimates and Lemma 6.1.1. An estimate showing the boundedness of |x j ∂x2j u(x, t)| is given in (7.116). We can use Lemma 6.1.10 to prove an analogous estimate for the mixed partial derivatives. Arguing as above, we easily establish that, for j = k, x j and x k both positive, we have  t ∞ ∞ |∂x j ∂xk u(x, t)| ≤ gW F,0,γ

b

|∂xk ks j (x k , z k )|×

0 0 0 bj  |∂x j ks (x j , z j )||g(z j , z j , z j , t − s)

− g(z j , x j , z j , t − s)|dz j dz k ds.

(7.135)

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Lemma 6.1.10 applies to show that this quantity is bounded by t CgW F,0,γ 0

γ

s 2 −1 ds , √ x j xk

(7.136)

which implies that γ √ | x j x k ∂x j ∂xk u(x, t)| ≤ CgW F,0,γ t 2 .

(7.137)

All that remains to complete the estimate of spatial derivatives is the proof of the H¨older continuity of the second derivatives of u. We begin by proving the H¨older continuity of x j ∂x2j u. The estimates in (7.116) and (6.72) show that for any c < 1, there is a C so that if y j < cx j , then √ √ |x j ∂x2j u(x j , x j , x j , t)− y j ∂x2j u(x j , y j , x j , t)| ≤ CgW F,0,γ | x j − y j |γ . (7.138) Thus in the “diagonal” case we only need to consider cx j < y j < x j . The proof in this case follows exactly as in the 1-dimensional case; we establish the H¨older continuity of L b j ,x j u, which is sufficient, as we have already done so for the first derivatives. To do this we express the difference: L b j ,x j u(x j , x j , x j , t) − L b j ,x j u(x j , y j , x j , t),

(7.139)

using (6.129) in the j -variable, much like the formula in (7.29). The estimate for each term in (6.129) carries over to the present situation to immediately establish that (7.138) holds for a suitable C, for all pairs (x j , y j ). To finish the spatial estimate in this case we need to consider the “non-diagonal” situation. With j = m, we express this difference as x j ∂x2j u(x m , x m , x m , t) − x j ∂x2j u(x m , ym , x m , t) =  t ∞ ··· 0

0

∞ ' 0 l = j,m

b

ksbl (xl , z l )x j ∂x2j ks j (x j , z j )[ksbm (x m , z m ) − ksbm (ym , z m )]× [g(z j , z j , z j , t − s) − g(zj , x j , z j , t − s)]d zds.

(7.140)

From this formula it follows that |x j ∂x2j u(x m , x m , x m , t) − x j ∂x2j u(x m , ym , x m , t)| ≤  t ∞ ∞ 2gW F,0,γ 0

0

√ √ b |x j ∂x2j ks j (x j , z j )|| x j − z j |γ ×

0

|ksbm (x m , z m ) − ksbm (ym , z m )]|dz j dz m ds. This case is completed by employing Lemmas 6.1.5 and 6.1.15.

(7.141)

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We begin with the case that ym = 0; Lemmas 6.1.5 and 6.1.15 in (7.141) show that |x j ∂x2j u(x m , x m , x m , t) − x j ∂x2j u(x m , 0, x m , t)| ≤ t

γ

s 2 −1

2gW F,0,γ 0

x m /s ds. 1 + x m /s

(7.142)

γ

This is easily seen to be bounded by CgW F,0,γ x m2 . We are therefore left to consider the case cx m < ym < x m , for any c < 1. We now use the second estimate in Lemma 6.1.5 to see that |x j ∂x2j u(x m , x m , x m , t) − x j ∂x2j u(x m , ym , x m , t)| ≤ t gW F,0,γ

s

√

γ 2 −1

0

√ x m − ym √ s √ √ x m − ym √ 1+ s

ds.

(7.143)

An elementary argument shows that the right-hand side is bounded by √ √ CgW F,0,γ | x m − ym |γ .

(7.144)

This completes the proof of the spatial part of the H¨older estimates for x j ∂x2j u(x, t). We next turn to the time estimate. From (7.116) it follows that γ

|x j ∂x2j u(x, t)| ≤ CgW F,0,γ t 2 .

(7.145) γ

This shows that x j ∂x2j u(x, t) tends uniformly to zero like t 2 . Applying (6.72) we see that for c < 1, there is a C so that, if s < ct, then γ

|x j ∂x2j u(x, t) − x j ∂x2j u(x, s)| ≤ CgW F,0,γ |t − s| 2 .

(7.146)

We are therefore left to consider the case t1 < t2 < 2t1 . To handle this case we begin

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with the formula from (6.143): L b j u(x, t2 ) − L b j u(x, t1 ) = t 2 −t1∞

··· 0

∞ '

b

ksbl (xl , z l )L b j ks j (x j , z j )[g(z, t2 − s) − g(z, t1 − s)]d zds+

0 l = j

0

2t1 −t2∞

∞ ···

0

0

L b j ,x j

/ n ' l=1

0

× [g(z, s) − t1

∞

∞ ···

2t1 −t2 0

'

0 l = j

ktb2l−s (xl , z l ) −

n '

0 ktb1l−s (xl , z l )

l=1

g(zj , x j , z j , s)]d zds+ b

kt2l−s (xl , z l )L b j kt2j−s (x j , z j )[g(z, s) − g(z j , x j , z j , s)]d zds. b

(7.147) In the first integral, as in the 1-dimensional cases, we use the estimates √ √ |g(z, tq − s) − g(z j , x j , z j , tq − s)| ≤ 2gW F,0,γ | x j − z j |γ ;

(7.148)

here q = 1, 2. In the last integral in (7.147) we use the estimate √ √ |g(z, s) − g(zj , x j , z j , s)| ≤ 2gW F,0,γ | x j − z j |γ .

(7.149)

This immediately reduces these cases to 1-dimensional cases, and these terms are thereγ fore bounded by CgW F,0,γ |t2 − t1 | 2 . To handle the second term we use formula (7.34) to conclude that / n 0 n ' b ' bl l L b j ,x j kt2 (xl , z l ) − kt1 (xl , z l ) = l=1

l=1 n 



L b j ,x j

 n−m '

m=1

b

kt2l (xl , z l ) ×

l=1

b kt2n−m+1 (x n−m+1 , z n−m+1 )

n '

b

kt1l (xl , z l )×

l=n−m+2

 .

b − kt1n−m+1 (x n−m+1 , z n−m+1 )

(7.150)

There are now two types of terms: those with n − m + 1 = j, and the term with n − m + 1 = j. In all cases we use the estimate in (7.149). With this understood, the term with n − m + 1 = j immediately reduces to the 1-dimensional case. Terms where n − m + 1 = j are bounded by 2t1 −t2∞ ∞

I = gW F,0,γ

b

b

|kt2l−s (xl , z l ) − kt1l−s (xl , z l )|× 0

0

0

√ √ b |L b j ktqj−s (x j , z j )|| x j − z j |γ dz l dz j ds;

(7.151)

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here q = 1 or 2. Using Lemmas 6.1.10, 6.1.9 and 6.1.15 we see that t1 I ≤ CgW F,0,γ

γ

(s + (q − 1)τ ) 2 −1

t2 −t1

τ ds , s+τ

(7.152)

with τ = t2 − t1 . The case q = 1 clearly produces a larger value. In this case we set w = s/τ, obtaining t1

I ≤ CgW F,0,γ τ

γ 2

τ

γ

w 2 −1

1

dw , 1+w

(7.153)

which completes the proof that, for j = 1, . . . , n, we have: γ

|L b j u(x, t2 ) − L b j u(x, t1 )| ≤ CgW F,0,γ |t2 − t1 | 2 .

(7.154)

To finish the proof of the proposition we need to show that the mixed derivatives x j xl ∂x j ∂xl u are H¨older continuous. There are two cases depending upon whether the variable that is allowed to vary is one of x j , xl or not. The latter case is immediate from lemmas we have already proved. Let m = j or l, then we easily see that

√

√ √ | x j xl ∂x j ∂xl u(x m , x m , x m , t) − x j xl ∂x j ∂xl u(x m , ym , x m , t)| ≤  t ∞ ∞ ∞ |ksbm (x m , z m ) − ksbm (ym , z m )|×

gW F,0,γ |

√

0 0 0 0 √ √ bj x j ∂x j ks (x j , z j )|| xl ∂xl ksbl (xl , z l )|| z l

−

√

yl |γ dz m dz l dz j ds.

(7.155)

We first let ym = 0 and use the first estimate in Lemma 6.1.5 to bound the z m -integral, and Lemma 6.1.10 to estimate the other two. This shows that this expression is bounded by  t  γ xm gW F,0,γ s 2 −1 ds. (7.156) s + xm 0

γ 2

This is bounded by CgW F,0,γ x m , which allows us to restrict to the case that, for a c < 1, we have the estimate cx m < ym < x m . Applying the other estimate in Lemma 6.1.5 we easily deduce that √ √ | x j xl ∂x j ∂xl u(x m , x m , x m , t) − x j xl ∂x j ∂xl u(x m , ym , x m , t)| ≤ √ √ CgW F,0,γ | x m − ym |γ . (7.157)

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Now suppose that m = l and yl = 0. In this case we see that √ | x j xl ∂x j ∂xl u(xl , xl , x l , t)| ≤  t ∞ ∞ gW F,0,γ 0

0

√ √ √ √ b | x j ∂x j ks j (x j , z j )|| xl ∂xl ksbl (xl , z l )|| z l − xl |γ dz l dz j ds.

0

(7.158) We apply Lemma 6.1.10 to see that this is bounded by  t ( √ γ −1 ) ( √ − 12 ) xjs xl s 2 ds. √ √ √ √ s + xl s + xj

(7.159)

0

An elementary argument shows that γ γ γ √ | x j xl ∂x j ∂xl u| ≤ CgW F,0,γ min{x j2 , xl2 , t 2 }.

(7.160)

This estimate implies that √

lim

x j ∨xl

→0+

x j xl ∂x j ∂xl u(x, t) = 0.

(7.161)

In light of (6.72) all that remains is to consider cxl < yl < xl , for a 0 < c < 1, for which we require an estimate of the quantity: ∞

√ √ √ √ | x 1 ∂x ktb (x 1 , y) − x 2 ∂x ktb (x 2 , y)|| x 1 − y|γ d y.

(7.162)

0

We now show how to use (6.40), and the estimate in Lemma 6.1.11, to prove the spatial √ H¨older estimate for x j xl ∂x j ∂xl u, with respect to x j and xl . √ √ | x j xl ∂x j ∂xl u(xl , xl , x l , t) − x j yl ∂x j ∂xl u(x l , yl , x l , t)| ≤  t ∞ ∞ 2gW F,0,γ √

0

0

√ b | x j ∂x j ks j (x j , z j )|×

0

√ √ yl ∂xl ksbl (yl , z l )|| z l − yl |γ dz l dz j ds.

(7.163)

We apply Lemmas 6.1.10 and 6.1.11 to see that this integral is bounded by √ √  | xl − yl | t γ s −1 √ √ ≤ C s2 | xl − yl | 1+ s 0 ⎡ √ √ 2 ⎤ | xl− yl | t γ γ −3 √ √ ⎢ ⎥ C⎣ s 2 −1 + s 2 | xl − yl |ds ⎦ .

(7.164)

|

xl ∂xl ksbl (xl , z l )

0

−

√

√ √ | xl − yl |2

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√ √ Here we implicitly assume that t > | xl − yl |2 ; if this is not the case then only the first integral on the right side of (7.164) is needed. In either case we easily see that the √ √ right-hand side is bounded by C| xl − yl |γ . To finish the proof of the proposition all that remains is to show that these derivatives are H¨older continuous with respect to time. The estimates in (7.137) and (6.72) show that if 0 < c < 1, then there is a C so that for 0 < t1 < ct2 , we have the estimate γ √ √ | x j xl ∂x j ∂xl u(x, t2 ) − x j xl ∂x j ∂xl u(x, t1 )| ≤ CgW F,0,γ |t2 − t1 | 2 . (7.165) Thus we are left to consider the case t1 < t2 < 2t1 . To that end we express √

x j xl ∂x j ∂xl [u(x, t2 ) − u(x, t1 )] = t 2 −t1∞

∞ '

··· 0

0

√ √ b ksbm (x m , z m ) x j ∂x j ks j (x j , z j ) xl ∂xl ksbl (xl , z l )×

0 m = j,l g(zj , x j , z j , t2

− s)) + (g(zj , x j , z j , t1 − s) − g(z, t1 − s))]d zds+ / n 0 2t1 −t2∞ ∞ n ' b ' √ bm m ··· x j x l ∂ x j ∂ xl kt2 −s (x m , z m ) − kt1 −s (x m , z m ) ×

[(g(z, t2 − s) −

0

0

m=1

0

m=1

g(z, s)d zds+ t1 ∞ ··· 2t1 −t2 0

∞ ' 0 m = j,l

√ √ b b ktb2m−s (x m , z m ) x j ∂x j kt2j−s (x j , z j ) xl ∂xl kt2l−s (xl , z l )× [g(z, s) − g(z j , x j , z j , s)]d zds.

(7.166)

In the first integral, as in the 1-dimensional case, we use the estimates √ √ |g(z, tq − s) − g(z j , x j , z j , tq − s)| ≤ 2gW F,0,γ | x j − z j |γ ;

(7.167)

here q = 1, 2. In the last integral in (7.166) we use the estimate √ √ |g(z, s) − g(z j , x j , z j , s)| ≤ 2gW F,0,γ | x j − z j |γ .

(7.168)

Applying Lemma 6.1.10 we see that these terms are bounded by 2(t2 −t1 )

C 0

γ √ x j xl s 2 −1 γ √ √ √ ≤ C|t2 − t1 | 2 . √ ( x j + s)( xl + s)

(7.169)

All that remains is to estimate the second integral in (7.166), where once again we employ formula (7.34). In each of the terms which arise, we can replace g(z, s) with either [g(z, s) − g(zj , x j , z j , s)], or [g(z, s) − g(zl , xl , z l , s)], without changing the values of these integrals. With this understood, there are only five essentially different cases to consider, depending upon which terms in the products on the right-hand side of (7.34) are differentiated. We let τ = t2 − t1 ; the cases requiring consideration are integrands with terms of the form:

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I. √ √ √ √ b | x j ∂x j ks j (x j , z j ) xl ∂xl [kτbl+s (xl , z l ) − ksbl (xl , z l )]|| x j − z j |γ , (7.170) II. √ √ √ √ bj | x j ∂x j kτ +s (x j , z j ) xl ∂xl [kτbl+s (xl , z l ) − ksbl (xl , z l )]|| x j − z j |γ , (7.171) III. With m = j or l √ √ b | x j ∂x j ks j (x j , z j ) xl ∂xl ksbl (xl , z l )×

√ √ [kτbm+s (x m , z m ) − ksbm (x m , z m )]|| x j − z j |γ ,

(7.172)

IV. With m = j or l √ √ bj | x j ∂x j kτ +s (x j , z j ) xl ∂xl ksbl (xl , z l )×

√ √ [kτbm+s (x m , z m ) − ksbm (x m , z m )]|| x j − z j |γ ,

(7.173)

V. With m = j or l √ √ bj b | x j ∂x j kτ +s (x j , z j ) xl ∂xl kτ l+s (xl , z l )×

√ √ [kτbm+s (x m , z m ) − ksbm (x m , z m )]|| x j − z j |γ .

(7.174)

Applying Lemmas 6.1.10 and 6.1.9 we see that the integrals of types III, IV and V are all bounded by ∞ γ −1 ∞ γ −1 γ γ s 2 τ ds σ 2 dσ = Cτ 2 ≤ Cτ 2 , C (7.175) τ +s 1+σ τ

1

leaving just the terms of types I and II. These are estimated using Lemma 6.1.10 and Lemma 6.1.13. Both of these terms are bounded by ∞ C τ

γ √ ∞ x j xl s 2 −1 τ ds γ γ √ √ √ ≤ C s 2 −2 τ ds ≤ Cτ 2 . √ ( x j + s)( xl + s)(τ + s)

(7.176)

τ

This completes the proof that γ √ √ | x j xl ∂x j ∂xl u(x, t2 ) − x j xl ∂x j ∂xl u(x, t1 )| ≤ CgW F,0,γ |t2 − t1 | 2 .

(7.177)

Using the estimates on the spatial derivatives, and the differential equation (7.99), we 0,γ easily establish that ∂t u ∈ ᏯW F (⺢n+ ×⺢+ ) satisfies the desired estimates. The estimates in (7.117) and (7.161) show that the appropriate scaled second derivatives tend to zero

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135

along portions of b ⺢n+ . The argument applied in the 1-dimensional case to show that (see equations (6.118) to (6.126)) lim [|u(x, t)| + |∂x u(x, t)| + |x∂x2 u(x, t)|] = 0,

x→∞

(7.178)

applies mutatis mutandis to show that  √ | x k xl ∂xk ∂xl u(x, t)|] = 0.

lim [|u(x, t)| + |∇ x u(x, t)| +

x→∞

(7.179)

k,l

One merely needs to observe that, if  R (x) =

n '

ϕ R,R (x j ),

(7.180)

j =1

then: 0,γ

1. [1 −  R (x)]g tends to zero in ᏯW F (⺢n+ × [0, T ]), as R → ∞. 2. For any fixed R the solution  t ∞ u 0R (x, t)

=

... 0

0

∞ ' n 0

b

ks j (x j , y j ) R ( y)g( y, t − s)d yds

(7.181)

j =1

along with all derivatives, tends rapidly to zero as x → ∞. These observations and the various H¨older estimates established above imply that we can apply Lemma 5.2.7 to conclude that 0,2+γ

u ∈ ᏯW F (⺢n+ × [0, T ]).

(7.182)

This completes the proof of the proposition in the k = 0 case. As in the 1j dimensional case, we can use Proposition 4.2.4 to commute derivatives ∂t ∂ xα past the kernel function in the integral representation. Assuming that g has support in B R+ (0) × [0, T ] allows us to apply Proposition 5.2.9 to bound the resultant data in terms of gW F,k,γ . Hence we can apply the estimates in the k = 0 case to establish the estimates in (7.102) for all k ∈ ⺞.  7.3 THE RESOLVENT OPERATOR As in the 1-dimensional case we can define the resolvent operator as the Laplace transform of the heat kernel. For μ ∈ S0 , we have the formula 1

  ∞ R(μ) f (x) = lim

→0+

··· 

0

∞ ' n 0

j =1

b

kt j (x j , z j ) f (z)d ze−t μ dt.

(7.183)

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Using the asymptotic expansion for the 1-dimensional factors it follows easily that for each fixed x ∈ ⺢n+ , R(μ) f (x) is an analytic function of μ. Applying Cauchy’s theorem we can easily show that, so long as Re[μeiθ > 0], we can rewrite this as: 1

  ∞ ···

R(μ) f (x) = lim

→0+



0

∞ ' n 0

j =1

b

ksejiθ (x j , z j ) f (z)d ze−e

iθ μs

eiθ ds.

(7.184)

This shows that R(μ) f (x) extends analytically to ⺓ \(−∞, 0]. We close this section by stating a proposition summarizing the properties of R(μ) as an operator on the H¨older k,γ spaces ᏯW F (⺢n+ ). The proof is deferred to the end of Chapter 9 where the analogous result covering all model operators is proved. P ROPOSITION 7.3.1. The resolvent operator R(μ) is analytic in the complement of (−∞, 0], and is given by the integral in (7.184) provided that Re(μeiθ ) > 0. For α ∈ (0, π], there are constants Cb,α so that if α − π ≤ arg μ ≤ π − α,

(7.185)

then for f ∈ Ꮿ0b (⺢n+ ) we have R(μ) f  L ∞ ≤

Cb,α  f L ∞ , |μ|

(7.186)

with Cb,π = 1. Moreover, for 0 < γ < 1, there is a constant Cb,α,γ so that if f ∈ 0,γ ᏯW F (⺢n+ ), then Cb,α,γ  f W F,0,γ . (7.187) R(μ) f W F,0,γ ≤ |μ| k,γ

k,2+γ

If for a k ∈ ⺞0 , and 0 < γ < 1, f ∈ ᏯW F (⺢n+ ), then R(μ) f ∈ ᏯW F (⺢n+ ), and we have (μ − L b )R(μ) f = f. (7.188) 0,2+γ

If f ∈ ᏯW F (⺢n+ ), then

R(μ)(μ − L b ) f = f.

There are constants Cb,k,α so that, for μ satisfying (7.185), we have  1  f W F,k,γ . R(μ) f W F,k,2+γ ≤ Cb,k,α 1 + |μ| For any B > 0, these constants are uniformly bounded for 0 ≤ b < B.

(7.189)

(7.190)

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Chapter Eight ¨ Holder Estimates for Euclidean Models The Euclidean model problems are given by ∂t u( y, t) −

m  j =1

∂ y2j u( y, t) = g( y, t) with u( y, 0) = f ( y).

(8.1)

The 1-dimensional solution kernel is |x−y|2

kte (x, y)

e 4t = √ , 4πt

(8.2)

and the solution to the equation in (8.1), vanishing at t = 0, is given by  t ∞ u( y, t) =

··· 0 −∞

∞ ' m

−∞ j =1

kte−s (y j , z j )g(z, s)d zds.

(8.3)

The solution to the homogeneous initial value problem with v( y, 0) = f ( y) is given by ∞ ∞ ' m ··· kte (y j , z j ) f (z)d z. (8.4) v( y, t) = −∞

−∞ j =1

For fixed y, v( y, t) extends analytically in t to define a function in S0 . The H¨older estimates for the solutions of this problem are, of course, classical. In this chapter we state the estimates and the 1-dimensional kernel estimates needed to prove them. ¨ 8.1 HOLDER ESTIMATES FOR SOLUTIONS IN THE EUCLIDEAN CASE The solutions of the problems ∂t v( y, t) −

m  j =1

∂ y2j v( y, t) = 0 with v( y, t) = f ( y) ∈ Ꮿk,γ (⺢m )

(8.5)

and ∂t u( y, t) −

m  j =1

∂ y2j u( y, t) = g( y, t) ∈ Ꮿk,γ (⺢m × ⺢+ ) with u( y, t) = 0, 137

(8.6)

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are well known to satisfy H¨older estimates. These can easily be derived from the 1dimensional kernel estimates, which are stated in the following subsection, much as in the degenerate case, though with considerably less effort. For the homogeneous Cauchy problem we have: P ROPOSITION 8.1.1. Let k ∈ ⺞0 and 0 < γ < 1. The solution v to (8.5) with initial data f ∈ Ꮿk,γ (⺢m ), given in (8.4), belongs to Ꮿk,γ (⺢m × ⺢+ ). There are constants C so that vk,γ ≤ C f k,γ . (8.7) For the inhomogeneous problem, with zero initial data, we have: P ROPOSITION 8.1.2. Let k ∈ ⺞0 and 0 < γ < 1. The solution, u, to (8.6) with g ∈ Ꮿk,γ (⺢m × ⺢+ ), is given in (8.3); it belongs to Ꮿk+2,γ (⺢m × ⺢+ ). There are constants C so that uk+2,γ ,T ≤ C(1 + T )gk,γ ,T . (8.8) The proofs of these propositions are in all essential ways identical to the proofs of Propositions 7.1.1 and 7.2.1, respectively, where the 1-dimensional kernel estimates from Chapter 6.1 are replaced by those given below in Chapter 8.2. The Euclidean arguments are a bit simpler, as there is no spatial boundary, and hence the special arguments needed, in the degenerate case, as x j → 0 are not necessary. The k > 0 estimates follow easily from the k = 0 estimates using Proposition 4.2.2, in the n = 0 case. The details of these arguments are left to the interested reader. As noted above, these results are classical, and complete proofs can be found in [29]. We can also define the resolvent operator R(μ) f, as the Laplace transform of the heat kernel. For μ ∈ S0 , we have the formula 1

  ∞ R(μ) f ( y) = lim

→0+

··· 

0

∞ ' n 0

kte (y j , w j ) f (w)dwe−t μ dt.

(8.9)

j =1

Using the asymptotic expansion for the 1-dimensional factors it follows easily that for each fixed y ∈ ⺢m , R(μ) f ( y) is an analytic function of μ. Applying Cauchy’s theorem we can easily show that, so long as Re[μeiθ > 0], we can rewrite this as: 1

  ∞ R(μ) f ( y) = lim

→0+

··· 

0

∞ ' n 0

j =1

e −e kse iθ (y j , w j ) f (w)dwe

iθ μs

eiθ ds.

(8.10)

This shows that R(μ) f ( y) extends analytically to ⺓ \(−∞, 0]. We close this section by stating a proposition summarizing the properties of R(μ) as an operator on the H¨older spaces Ꮿk,γ (⺢m ). The proof is deferred to the end of Chapter 9 where the analogous result covering all model operators is proved.

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P ROPOSITION 8.1.3. The resolvent operator R(μ) is analytic in the complement of (−∞, 0], and is given by the integral in (8.10) provided that Re(μeiθ ) > 0. For α ∈ (0, π], there are constants Cα so that if α − π ≤ arg μ ≤ π − α, then for f ∈

Ꮿ0b (⺢m )

(8.11)

we have R(μ) f  L ∞ ≤

Cα  f L ∞ , |μ|

(8.12)

with Cπ = 1. Moreover, for 0 < γ < 1, there is a constant Cα,γ so that if f ∈

Ꮿ0,γ (⺢m ), then

Cα,γ  f 0,γ . (8.13) |μ| For k ∈ ⺞0 , and 0 < γ < 1, if f ∈ Ꮿk,γ (⺢m ), then R(μ) f ∈ Ꮿ2+k,γ (⺢m ), and, we have (μ − L 0,m )R(μ) f = f. (8.14) R(μ) f 0,γ ≤

If f ∈ Ꮿ2,γ (⺢m ), then

R(μ)(μ − L 0,m ) f = f.

There are constants Ck,α so that, for μ satisfying (8.11), we have  1  f k,γ . R(μ) f k+2,γ ≤ Ck,α 1 + |μ|

(8.15)

(8.16)

Remark. As before the solution v( y, t) to the Cauchy problem can be expressed as a contour integral:  1 v( y, t) = [R(μ) f ]( y)eμt dμ. (8.17) 2πi α,R

From this representation it follows that v extends analytically in t to S0 . Moreover, for t ∈ S0 we see that v(·, t) belongs to Ꮿ2,γ (⺢m ). Hence by the semi-group property v(·, t) ∈ Ꮿ∞ (⺢m ). 8.2 1-DIMENSIONAL KERNEL ESTIMATES The 1-dimensional kernel estimates can easily be used to prove the H¨older estimates stated in the previous subsection. They form essential components of the proofs of the H¨older estimates for the general model problems, considered in the next chapter. The proofs of these estimates are elementary, largely following from the facts that the kernel, kte (x, y), is a function of (x − y)2/t, which extends analytically to ⺢2 × S0 . As in the degenerate case, we have ∞ kte (x, y)d y = 1 for x ∈ ⺢, t ∈ S0 . −∞

The proofs of the following classical results are left to the reader.

(8.18)

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8.2.1 Basic Kernel Estimates L EMMA 8.2.1. For 0 ≤ γ < 1, 0 < φ ≤ ∞

π 2,

there is a Cφ so that, for t ∈ Sφ , γ

|kte (x, y)||x − y|γ d y ≤ Cφ |t| 2 .

(8.19)

−∞

L EMMA 8.2.2. For 0 < φ ≤

π 2,

there is a constant Cφ so that for t ∈ Sφ ⎛

∞ |kte (x 2 , z) − kte (x 1 , z)|dz ≤ −∞

We set αe =

|x√ 2 −x 1 | |t | Cφ ⎝ 2 −x 1 | 1 + |x√ |t |

⎞ ⎠.

(8.20)

3x 1 − x 2 3x 2 − x 1 and βe = . 2 2

(8.21)

L EMMA 8.2.3. Let J = [αe , βe ], as defined in (8.21). For 0 < γ < 1, 0 < φ ≤ there is a Cφ so that for t = |t|eiθ ∈ Sφ 

|kte (x 2 , y) − kte (x 1 , y)||y − x 1 |γ d y ≤ Cφ |x 2 − x 1 |γ e− cos θ

(x 2 −x 1 )2 2|t|

.

π 2,

(8.22)

Jc

L EMMA 8.2.4. For 0 < γ < 1 and c < 1 there is a C such that if c < s/t < 1, then ∞ + e + γ +k (x, y) − k e (x, y)+ |x − y|γ d y ≤ C|t − s| 2 . (8.23) t s −∞

Without an upper bound on 0 < t/s, we have the estimate ∞

+ e + +k (x, y) − k e (x, y)+ d y ≤ C t s

−∞



 t/s − 1 . 1 + [t/s − 1]

(8.24)

8.2.2 First Derivative Estimates L EMMA 8.2.5. For 0 ≤ γ < 1, and 0 < φ ≤ π2 , there is a Cφ so that for t ∈ Sφ we have ∞ √ γ −1 √ |∂x kte (x, y)|| y − x|γ d y ≤ Cφ |t| 2 . (8.25) 0

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L EMMA 8.2.6. For 0 < γ < 1, 0 < φ ≤ t ∈ Sφ , ∞

141 π 2,

there is a constant Cφ so that for

√ √ |∂x kte (x 1 , y) − ∂x kte (x 2 , y)|| x 1 − y|γ d y ≤

0



Cφ |t|

γ −1 2

√ √  | x2 − x1 | √ |t |  √ √ . | x2 − x1 | √ 1+ |t |

(8.26)

L EMMA 8.2.7. For 0 ≤ γ < 1, and 0 < τ, we have for s ∈ [τ, ∞) that there is a constant C so that γ −1 ∞ τs 2 . (8.27) |∂x kτe +s (x, y) − ∂x kse (x, y)||x − y|γ d y < C (τ + s) −∞

8.2.3 Second Derivative Estimates L EMMA 8.2.8. For 0 ≤ γ < 1, 0 < φ ≤ π2 , there is a Cφ so that for t ∈ Sφ we have the estimate ∞ γ |∂x2 kte (x, y)||x − y|γ d y ≤ Cφ |t| 2 −1 . (8.28) −∞

This implies that, if γ > 0, then |t | ∞ e |∂x2 kse iθ (x, y)||x 0 −∞

L EMMA 8.2.9. For 0 < φ ≤ |t |

π 2,

γ

− y| d yds ≤



2Cφ γ



γ

|t| 2 .

(8.29)

there is a Cφ so that for t ∈ Sφ

(x −x )2 + e + +∂ y k iθ (x 2 , αe ) − ∂ y k e iθ (x 2 , βe )+ ds ≤ Cφ e− cos θ 2 |t| 1 , se se

(8.30)

0

where αe and βe are defined in (8.21). L EMMA 8.2.10. For 0 < γ < 1, 0 < φ ≤ π2 , there is a Cφ so that for |θ | ≤ and J = [αe , βe ], with the endpoints given by (8.21), we have |t | βe

0 αe

−φ

e γ γ |∂x2 kse iθ (x 2 , y)||y − x 2 | d yds ≤ C φ |x 2 − x 1 |

0 αe

|t | βe

π 2

(8.31) e γ γ |∂x2 kse iθ (x 1 , y)||y − x 1 | d yds ≤ C φ |x 2 − x 1 | .

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These estimates follow from the more basic L EMMA 8.2.11. For 0 < γ < 1, 0 < φ ≤ π2 , there is a Cφ so that for |θ | < π2 − φ and J = [αe , βe ], with the endpoints given by (8.21), we have ⎧ γ βe ⎨s 2 −1 if s < (x 2 − x 1 )2 2 e γ γ +1 |∂x kseiθ (x 2 , y)||y − x 2 | d y ≤ Cφ |x2 −x1 | (8.32) if s ≥ (x 2 − x 1 )2 . ⎩ 3 αe

s2

L EMMA 8.2.12. For 0 < γ < 1, 0 < φ ≤ π2 , there is a Cφ so that for |θ | < and J = [αe , βe ], with the endpoints given by (8.21), we have |t |  0

e 2 e γ γ |∂x2 kse iθ (x 2 , y) − ∂ x k seiθ (x 1 , y)||y − x 1 | d yds ≤ C φ |x 2 − x 1 | .

π 2

−φ

(8.33)

Jc

This follows from the more basic L EMMA 8.2.13. For 0 < γ < 1, 0 < φ ≤ π2 , there is a Cφ so that for |θ | < π2 − φ and J = [αe , βe ], with the endpoints given by (8.21), we have + +  + x 2 − x 1 + − cos θ (x2 −x1 )2 γ e 2 e γ +e 2 −1 + 8s |∂x2 kse (x , y) − ∂ k (x , y)||y − x | d y ≤ C s √ . 2 1 1 φ iθ x seiθ + s + Jc

(8.34) Finally we have L EMMA 8.2.14. For 0 ≤ γ < 1, 0 < τ and τ < s, there is a constant C so that ∞

γ

|∂x2 kτe +s (x, y) − ∂x2 kse (x, y)||x − y|γ d y ≤ C|τ |s 2 −2 .

(8.35)

−∞

8.2.4 Large t Behavior To prove estimates on the resolvent, and to study the off-diagonal behavior of the heat kernel in many variables, it is useful to have estimates on the derivatives of kte (x, y) valid for t bounded away from zero. L EMMA 8.2.15. For j ∈ ⺞ and 0 < φ ≤ π2 there is a constant C j,φ so that if t ∈ Sφ , then ∞ C j,φ j |∂x kte (x, y)|d y ≤ . (8.36) j 2 |t| −∞ The proof of this lemma is in the Appendix.

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Chapter Nine ¨ Holder Estimates for General Models We now turn to the task of estimating solutions to heat equations defined by the operators of the form: n m   L b,m = [x j ∂x2j + b j ∂x j ] + ∂ y2k . (9.1) j =1

k=1

The general model operator on ⺢n+ × ⺢m , denoted L b,m , is labeled by a non-negative n-vector b = (b1 , . . . , bn ), and m, the dimension of the corner. We use x-variables to denote points in ⺢n+ and y-variables to denote points in ⺢m . If we have a function of these variables f (x, y) then, as before, we estimate differences f (x 2 , y2 ) − f (x 1 , y1 ) 1-variable-at-a-time. We first observe that f (x 2 , y2 ) − f (x 1 , y1 ) = [ f (x 2 , y2 ) − f (x 1 , y2 )] + [ f (x 1 , y2 ) − f (x 1 , y1 )]; (9.2) each term in brackets can then be written as a telescoping sum: f (x 2 , y2 ) − f (x 1 , y1 ) = ⎫ ⎧ n−1 ⎬ ⎨ [ f (x 2j , x 2j +1 , x 1 , y2 ) − f (x 2j , x 1j +1 , x 1 , y2 )] + j j ⎭ ⎩ j =0 &m−1 6  1 2 2 1 1 2 2 1 [ f (x , yl , yl+1 , yl ) − f (x , yl , yl+1 , yl )] , (9.3) l=0

where x j and x j are defined in (7.4). We say that terms in the first sum have a “variation in an x-variable,” and terms in the second have a “variation in a y-variable.” We only need to deal with terms that have a variation in one or the other type of variable, and this simplifies the proofs for the general case considerably. In this chapter we prove H¨older estimates for the solutions on ⺢n+ × ⺢m × ⺢+ to the homogeneous Cauchy problem [∂t − L b,m ]v(x, y, t) = 0 with v(x, y, 0) = f (x, y),

(9.4)

and the inhomogeneous problem [∂t − L b,m ]u(x, y, t) = g(x, y, t) with u(x, y, 0) = 0. 143

(9.5)

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The solution to the homogeneous initial value problem with v(x, y, 0) = f (x, y) is given by ∞ v(x, y, t) =

···

∞ ' n

m '

ktbl (xl , wl )

(9.6)

j =1

−∞ l=1

−∞ = κtb,m

kte (y j , z j ) f (w, z)d zdw

f.

For fixed (x, y), v(x, y, t) extends analytically in t to define a function in S0 . The solution to the inhomogeneous problem is given by the operator K tb,m defined by  t ∞ u(x, y, t) = =

··· 0 −∞ K tb,m g.

∞ ' n

l ktb−s (xl , wl )

m '

−∞ l=1

j =1

kte−s (y j , z j )g(w, z, s)d zdwds

(9.7)

As before, for t > 0, this expression should be understood as u(x, y, t) = lim u  (x, y, t),

(9.8)

→0+

where t − ∞ u  (x, y, t) =

··· 0 −∞

∞ ' n

l ktb−s (xl , wl )

−∞ l=1

m ' j =1

kte−s (y j , z j )g(w, z, s)d zdwds.

We also note that if we let v s (x, y, t) denote the solution to the Cauchy problem: [∂t − L b,m ]v s = 0 with v s (x, y, 0) = g(x, y, s), then

(9.9)

(9.10)

t v s (x, y, t − s)ds.

u(x, y, t) =

(9.11)

0

˙ 0 (⺢n+ × ⺢m ), and μ ∈ S0 , by The resolvent operator R(μ) f is defined, for f ∈ Ꮿ 1

 R(μ) f (x, y) = lim

→0+

e−μt v(x, y, t)dt.

(9.12)



As in the earlier cases, this is an analytic function of μ ∈ S0 . By deforming the contour we can replace this representation with 1

 R(μ) f (x, y) = lim

→0+



e−μse v(x, y, seiθ )eiθ ds, iθ

(9.13)

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which converges if Re[μeiθ ] > 0. This analytically extends R(μ) f (x, y) to complex numbers μ ∈ ⺓ \ (−∞, 0]. As in the previous two chapters, the estimates herein are all proved by reduction to 1-variable kernel estimates. 9.1 THE CAUCHY PROBLEM We begin with estimates for the homogeneous Cauchy problem. P ROPOSITION 9.1.1. Let k ∈ ⺞0 , 0 < R, (b1 , . . . , bn ) ∈ ⺢n+ , and 0 < γ < 1. k,γ Suppose that the initial data f lies in ᏯW F (⺢n+ × ⺢m ); if k > 0, assume that f is supported in B R+ (0) × ⺢m . Then the solution v to (9.4), with initial data f, given k,γ in (9.6), belongs to ᏯW F (⺢n+ × ⺢m × ⺢+ ). There are constants C b,γ ,R so that vW F,k,γ ≤ C b,γ ,R  f W F,k,γ . k,2+γ

(9.14)

k,2+γ

If f ∈ ᏯW F (⺢n+ × ⺢m ), then v belongs to ᏯW F (⺢n+ × ⺢m × ⺢+ ). There are constants C b,γ ,R so that vW F,k,2+γ ≤ C b,γ ,R  f W F,k,2+γ .

(9.15)

For B > 0, these constants are uniformly bounded for b ≤ B1, and if k = 0, then the constants are independent of R. P ROOF. For the k = 0, b > 0, the estimates in (9.14) follow as in the proof of Proposition 7.1.1, via the 1-variable-at-a-time method. The cases where two “x” (or parabolic) variables differ follow, essentially verbatim, as in the proof of Proposition 7.1.1, from the lemmas in Chapter 6.1. The new cases in the proof of this proposition are those involving the “y”- (or Euclidean) variables. As noted after the statement of Proposition 8.1.1, these cases follow, mutatis mutandis, via the arguments used in the proof of Proposition 7.1.1. The estimates for the kernel ktb must be replaced with estimates for the 1-dimensional, Euclidean heat kernel. These are stated in Section 8.2. As these cases also arise in the proof of Proposition 9.2.1, to avoid excessive repetition, we forego giving the details now, and leave them for the proof of the next proposition. As before all the constants in these estimates are uniformly bounded for bounded 0 < b ≤ B. Applying the compactness result in Proposition 5.2.8, we can allow entries of b to tend to zero, obtaining the unique limiting solution with all the desired estimates for these cases as well. If we assume that f is supported in B R+ (0) × ⺢m , then the estimates in (9.14) for the H¨older spaces with k > 0 follow from the k = 0 results, and Propositions 4.2.2 and 5.2.9. As in the proof of Proposition 7.1.1, some additional estimates are needed to establish (9.15). We begin with the k = 0 case. Applying Proposition 4.2.2 to commute 0,γ derivatives through the integral kernel, we see that estimates for ᏯW F -norm of ∇ x v, ∇ y v, x j ∂x2j v, j = 1, . . . , n and ∂ yk ∂ yl v, 1 ≤ k, l ≤ m follow from (9.14). To estab0,γ

lish (9.15), we need only estimate the ᏯW F -norm of √ √ x i ∂xi ∂ yk v and x i x j ∂xi ∂x j v.

(9.16)

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To estimate given by √

√

x i ∂xi ∂ yk v, we can relabel so that i = n, and k = m; this derivative is 



x n ∂xn ∂ ym v =



κtb ,m−1 (x n−1 , w n−1 ; ym−1 , z m−1 )×

⺢m−1 ⺢n−1 +

∞ ∞ 2 0 −∞

x n bn +1 √ kt (x n , wn )kte (ym , z m ) wn ∂wn ∂zm f (w, z)dwn dz m dwn−1 d z m−1 , wn (9.17)

where 

κtb ,m−1 (x n−1 , wn−1 ; ym−1 , z m−1 ) =

n−1 '

b

kt j (x j , w j )

m−1 '

j =1

kte (yk , z k ).

(9.18)

k=1

Applying Lemma 7.1.2 we see that √ | x n ∂xn ∂ ym v| ≤ C f W F,0,2+γ .

(9.19)

0,2+γ

Since f ∈ ᏯW F , we know that γ √ | x n ∂xn ∂ ym f (x, y)| ≤  f W F,0,2+γ x n2 .

(9.20)

Using this estimate in (9.17) and applying Lemma 7.1.2 shows that γ √ | x n ∂xn ∂ ym v| ≤ C f W F,0,2+γ x n2 ,

establishing that lim

xn

√

→0+

x n ∂xn ∂ ym v = 0.

(9.21) (9.22)

The H¨older continuity of this derivative in the y-variables follows by re-expressing the difference, √ √ x n ∂xn ∂ ym v(x, y, t) − x n ∂xn ∂ ym v(x, y, t), (9.23) as a sum of terms like those appearing in (7.29). We then apply estimates from Lemmas 8.2.2 and 8.2.3 to the terms in this sum, along with Lemma 7.1.2, to conclude that √ √ | x n ∂xn ∂ ym v(x, y, t) − x n ∂xn ∂ ym v(x, y, t)| ≤ C f W F,0,2+γ ρe ( y, y)γ . (9.24) The argument to establish the estimates √ √ | x n ∂xn ∂ ym v(x j , x 1j +1 , x j , y, t) − x n ∂xn ∂ ym v(x j , x 2j +1 , x j , y, t)| ≤ + +γ  + + C f W F,0,2+γ + x 1j +1 − x 2j +1 + , with j ≤ n − 2, (9.25)

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is essentially identical, with Lemmas 8.2.2 and 8.2.3 replaced by Lemmas 6.1.5 and 6.1.7. The only remaining spatial estimate is (9.25) with j = n −1. The estimate in (9.21) implies that for any c < 1, there is a C so that, if x n1 < cx n2, then √ √ | x n ∂xn ∂ ym v(x n−1 , x n1 , y, t) − x n ∂xn ∂ ym v(x n−1 , x n2 , y, t)| ≤ +  ++γ + 1 + C f W F,0,2+γ + x n − x n2 ++ ,

(9.26)

leaving only the case cx n2 < x n1 < x n2 . Using an obvious modification of (7.64), and essentially the same argument as appears after (7.64), we can prove this estimate as well. To complete the estimates of this derivative we need to show that it is H¨older continuous in the time variable. The proof of this estimate is a small modification of that used to prove (7.98). We begin with    √ x n ∂xn ∂ ym [v(x, y, t)−v(x, y, 0)] = κtb ,m−1 (x n−1 , wn−1 ; ym−1 , z m−1 )× ⺢m−1 ⺢n−1 +

∞ ∞ 0

√

x n ktbn +1 (x n , wn )kte (ym , z m )[ f nm (w, z) − f nm (x, y)]dwn dz m dwn−1 d z m−1 ,

0

(9.27) where fnm = ∂wn ∂zm f. We rewrite the difference, [ f nm (w, z) − f nm (x, y)], as a telescoping sum like that in (9.3): ⎧ ⎫ n−1 ⎨ ⎬ f nm (w, z)− f nm (x, y) = [ f nm (wj , w j +1 , x j , z) − f nm (w j , x j +1 , x j , z)] + ⎩ ⎭ j =0 &m−1 6      [ f nm (x, z l , z l+1 , yl ) − fnm (x, zl , yl+1 , yl )] . (9.28) l=0

Each term in the second sum is estimated by √ x n | f nm (x, z l , z l+1 , yl ) − f nm (x, z l , yl+1 , yl )| ≤  f W F,0,2+γ |z l+1 − yl+1 |γ . (9.29) Each term in the first sum, except for j = n − 1, is estimated by √ √ √ x n | f nm (wj , w j +1 , x j , z)− f nm (wj , x j +1 , x j , z)| ≤  f W F,0,2+γ | x j +1 − w j +1 |γ . (9.30) The remaining case is estimated by √

√ √ x n | f nm (wn−1 , wn , z) − f nm (wn−1 , x n , z)| ≤ | x n − wn || f nm (wn−1 , wn , z)|+ √ √ | wn f nm (wn−1 , wn , z) − x n fnm (wn−1 , x n , z)| γ −1 √ √ √ √ ≤  f W F,0,2+γ [| x n − wn |wn 2 + | x n − wn |γ ]. (9.31)

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Using these estimates in (9.27) we apply Lemmas 6.1.6, 7.1.4 and 8.2.1 to deduce that γ √ | x n ∂xn ∂ ym [v(x, y, t) − v(x, y, 0)]| ≤ C f W F,0,2+γ t 2 . (9.32) As usual, this shows that for 0 < c < 1, there is a C so that if s < ct, then γ √ | x n ∂xn ∂ ym [v(x, y, t) − v(x, y, s)]| ≤ C f W F,0,2+γ |t − s| 2 .

(9.33)

We are left with the case ct < s < t, which again closely follows the pattern of the proof of (7.98). As before we use an analogue of (7.37): √ x n ∂xn ∂ ym [v(x, y, t) − v(x, y, s)]    n  n−l n  '  '  √ b b e,m κt ( y, z) kt j (x j , w j ) × ks j (x j , w j )× = xn l=1

⺢n+ ⺢m



j =1

j =n−l+2

   b b kt n−l+1 (x n−l+1 , wn−l+1 ) − ks n−l+1 (x n−l+1 , wn−l+1 ) ×  [ f nm (w, z) − f nm (wl , x n−l+1 , wl , z)] + ˜

κsb,0 (x, w) *

m  m−q  '

kte (y j , z j ) ×

m '

kse (y j , z j )×

q=1 j =1 j =m−q+2 , e e kt (ym−q+1 , z m−q+1 ) − ks (ym−q+1 , z m−q+1 ) ×

[ f nm (w, z) − where

f nm (w, z q , ym−q+1 , z q )]

 dwd z, (9.34)

 bn = bn + 1. b j = b j for j = 1, . . . , n and 

(9.35)

Each term in the second sum is estimated by an integral of the form ∞ ∞ 2  f W F,0,2+γ 0

0

x n bn +1 k (x n , wn )× wn s |kte (y, z) − kse (y, z)||y − z|γ dzdwn .

(9.36)

Lemmas 7.1.2 and 8.2.4 show that these terms are bounded by γ

C f W F,0,2+γ |t − s| 2 .

(9.37)

Every term in the first sum, with l = 1, is bounded by an integral of the form ∞ ∞ 2  f W F,0,2+γ 0

0

x n bn +1 k (x n , wn )× wn s √ √ b b |kt j (x j , w j ) − ks j (x j , w j )|| x j − w j |γ dw j dwn .

(9.38)

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Lemmas 7.1.2 and 6.1.8 show that these terms are bounded by γ

C f W F,0,2+γ |t − s| 2 .

(9.39)

This leaves just the l = 1 case, which is bounded by ∞  f W F,0,2+γ

|ktbn +1 (x n , wn ) − ksbn +1 (x n , wn )|×

0 1−γ √ √ √ √ [| x n − wn |γ + | x n − wn |wn 2 ]dwn .

(9.40) γ

Lemmas 6.1.8 and 7.1.5 show that this is also bounded by C f W F,0,2+γ |t − s| 2 , thereby completing the proof that γ √ | x n ∂xn ∂ ym (v(x, y, t) − v(x, y, s)]| ≤ C f W F,0,2+γ |t − s| 2 . (9.41) √ This brings us to the H¨older estimates for x i x j ∂xi ∂x j v. The proofs here are quite similar to the analogous result in Proposition 7.1.1. The proof that  √ | x i x j ∂xi ∂x j v(x, y, t) − x˜i x˜ j ∂xi ∂x j v( x, y, t)| ≤ Cρs (x, x)γ , (9.42) follows exactly as before. Using Lemma 7.1.2, working 1-variable-at-a-time, we also easily establish √ √ | x i x j ∂xi ∂x j v(x, y, t) − x i x j ∂xi ∂x j v(x, y, t)| ≤ Cρe ( y, y)γ . (9.43) The H¨older continuity in time follows as in Proposition 7.1.1, while incorporating the √ Euclidean variables as in the previous case, i.e., x j ∂x j ∂ yl v. Finally we observe that, 0,γ

as ∂t v = L b,m v, and we have established that L b,m v ∈ ᏯW F (⺢n+ × ⺢m × [0, ∞)), the same is true of ∂t v. This completes the proof of (9.15) in the k = 0 case. Assuming that f is supported in B R+ (0) × ⺢m , applying Propositions 4.2.2 and 5.2.9, we can easily deduce (9.15) when k > 0 from the k = 0 case.  9.2 THE INHOMOGENEOUS PROBLEM We now turn to the inhomogeneous problem. P ROPOSITION 9.2.1. For any k ∈ ⺞0 , (b1 , . . . , bn ) ∈ ⺢n+ , 0 < R, and 0 < γ < 1, we let g ∈ Ꮿk,γ (⺢n+ × ⺢m × ⺢+ ). If 0 < k, then assume that g is supported in B R+ (0) × ⺢m × [0, T ]. The solution u to (9.5), with right-hand side g, given in (9.7), belongs to Ꮿk,2+γ (⺢n+ × ⺢m × ⺢+ ). There are constants Ck,b,γ ,R so that uW F,k+2,γ ,T ≤ Ck,b,γ ,R (1 + T )gW F,k,γ ,T .

(9.44)

The tangential first derivatives satisfy a stronger estimate; there is a constant C so that, if T ≤ 1, then γ ∇ y uW F,0,γ ,T ≤ C T 2 gW F,0,γ ,T . (9.45) The constants are uniformly bounded for b ≤ B1, and independent of R if k = 0.

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P ROOF. As before we begin by assuming that 0 < b, and k = 0. Using the 1variable-at-a-time method, any estimate of the variation in an x-variable of a derivative in the x-variables alone, or the variation in a y-variable of a derivative in the y-variables alone, follows easily from the lemmas in Sections 6.1 and 8.2. The maximum principle and (9.11) show that |u(x, y, t)| ≤ tg L ∞ (⺢n+ ×⺢m ×[0,t ]) .

(9.46)

We use the representation in (9.11) and Proposition 9.1.1 to deduce that, for t ≤ T, |u(x 1 , y1 , t) − u(x 2 , y2 , t)| ≤ Ctρ((x 1 , y1 ), (x 2 , y2 ))γ gW F,0,γ ,T .

(9.47)

As before, estimates of the second derivatives (see equations (6.105) and (6.110) and Lemma 8.2.8) show that γ

|L b,m u(x, y, t)| ≤ CgW F,0,γ t 2 .

(9.48)

Integrating the equation, ∂t u = L b,m u + g, in t we can therefore show that there is a constant C so that, if t1 < t2 ≤ T, then  γ γ 2 +1 2 +1 |u(x, y, t2 ) − u(x, y, t1 )| ≤ CgW F,0,γ ,T |t2 − t1 | + |t2 − t1 | . (9.49) These results show that there is a constant C so that γ

γ

uW F,0,γ ,T ≤ C T 1− 2 (1 + T 2 )gW F,0,γ ,T .

(9.50)

9.2.1 First Derivative Estimates Using the estimates proved above, we can easily show that γ

|∂x j u(x, y, t)| ≤ CgW F,0,γ t 2 and |∂ yl u(x, y, t)| ≤ CgW F,0,γ t

γ +1 2

.

(9.51)

The first estimate follows by the argument used to prove (7.111). We indicate how the second estimate is proved. The standard limiting argument shows that  t ∞ ∂ ym u(x, y, t) =

∞ ∞ ···

···

∞ ' n

b

ks j (x j , w j )

'

kse (yl , z l )×

l =m −∞ j =1 0 0 0 −∞ e   ∂ ym ks (ym , z m )[g(w, z, t − s) − g(w, z m−1 , ym , z m−1 , t

− s)]dwd zds.

(9.52)

Putting in absolute values we see that  t ∞ |∂ ym u(x, y, t)| ≤ 2gW F,0,γ 0

0

|∂ ym kse (ym , z m )||ym − z m |γ dz m ds.

(9.53)

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Lemma 8.2.5 shows that t |∂ ym u(x, y, t)| ≤ CgW F,0,γ

s

γ −1 2

γ +1

ds = CgW F,0,γ

0

2t 2 . γ +1

(9.54)

We note that by integrating these estimates for ∇ x, y u we obtain a Lipschitz estimate for u itself: √ γ |u(x 2 , y2 , t) − u(x 1 , y1 , t)| ≤ CgW F,0,γ t 2 (1 + t)(x 2 , y2 ) − (x 1 , y1 ), (9.55) though these estimates are not directly relevant to estimating uW F,0,2+γ ,T . The arguments used to prove (7.120) and (7.124) apply, essentially verbatim, to show that |∂x j u(x 2 , y, t) − ∂x j u(x 1 , y, t)| ≤ CgW F,0,γ ρs (x 1 , x 2 )γ

(9.56)

provided ρs (x 1 , x 2 ) is bounded by 1. For ρe ( y1 , y2 ) < 1 we have γ

|∂ yl u(x, y2 , t) − ∂ yl u(x, y1 , t)| ≤ Ct 2 gW F,0,γ ρe ( y1 , y2 )γ . To prove this we can assume that differ in the lth entry, then

y2

−

y1

has exactly one non-zero entry. If

(9.57) y1

and y2

|∂ yl u(x, y2 , t) − ∂ yl u(x, y1 , t)| ≤  t ∞ gW F,0,γ

|∂ yl [kse (yl2 , z l ) − kse (yl1 , z l )]||yl1 − z l |γ dz l ds.

(9.58)

0 −∞

Applying Lemma 8.2.6 we see that this integral is bounded by   |yl2 −yl1 | t √  s γ −1  ds.  C s 2 |yl2 −yl1 | √ 1+ 0 s

(9.59)

An elementary calculation shows that if ρe ( y1 , y2 ) < 1, then this integral is bounded γ by a constant times t 2 ρe ( y1 , y2 )γ . If y1 and y2 differ in a coordinate other than the lth, then Lemmas 8.2.2 and 8.2.5 show that the estimate for |∂ yl u(x, y2 , t) − ∂ yl u(x, y1 , t)| reduces again to the integral in (9.59). We are therefore left to consider the off-diagonal cases: |∂x j u(x, y2 , t) − ∂x j u(x, y1 , t)| and |∂ yl u(x 2 , y, t) − ∂ yl u(x 1 , y, t)|. We can again assume that x 2 − x 1 and y2 − y1 each have exactly one non-zero entry, which we can assume is the first. We first consider  t ∞ ∞ 2

1

|∂x j u(x, y , t) − ∂x j u(x, y , t)| ≤ 2gW F,0,γ √

b

|∂x j ks j (x j , w j )|× 0

0 −∞

√ | x j − w j |γ |kse (y12 , z 1 ) − kse (y11 , z 1 )|dw j dz 1 ds.

(9.60)

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Applying Lemmas 8.2.2 and 6.1.10 shows that this is bounded by ⎛ ⎞ |y12 −y11 | t √ γ s ⎜ ⎟ |∂x j u(x, y2 , t) − ∂x j u(x, y1 , t)| ≤ CgW F,0,γ s 2 −1 ⎝ ⎠ ds. (9.61) |y12 −y11 | √ 1 + 0 s An elementary estimate and Lemma 6.1.1 shows that therefore |∂x j u(x, y2 , t) − ∂x j u(x, y1 , t)| ≤ CgW F,0,γ ρe ( y1 , y2 )γ .

(9.62)

To estimate |∂ ym u(x 2 , y, t) − ∂ ym u(x 1 , y, t)|, first bound |∂xq ∂ yl u(x, y, t)|. By relabeling, it suffices to consider q = 1, for which we use the expression  t ∞ ∂x1 ∂ ym u(x, y, t) =

∞ ∞ ···

···

∞ '

−∞ j =1 0 0 0 −∞ b1 e ∂x1 ks (x 1 , w1 )∂ ym ks (ym , z m )[g(w, z, t − s) −

b

ks j (x j , w j )

'

kse (yl , z l )×

l =m

g(x 1 , w0 , z, t

− s)]dwd zds.

(9.63)

Putting in absolute values and using the standard estimate for the difference g(w, z, t − s) − g(x 1 , w0 , z, t − s), gives the bound: |∂x1 ∂ ym u(x, y, t)| ≤  t ∞ ∞ 2gW F,0,γ

√ √ |∂x1 ksb1 (x 1 , w1 )∂ ym kse (ym , z m )|| x 1 − w1 |γ dw1 dz m ds.

0 −∞ 0

(9.64) Applying Lemmas 6.1.10 and 8.2.5 we see that t

γ

s 2 −1 ds √ √ x1 + s

|∂x1 ∂ ym u(x, y, t)| ≤ CgW F,0,γ 0

t ≤ CgW F,0,γ √

γ 2

x1

(9.65)

.

By integrating the last expression we see that +  ++ γ + |∂ ym u(x 12 , x 0 , y, t) − ∂ ym u(x 11 , x 0 , y, t)| ≤ CgW F,0,γ t 2 ++ x 12 − x x1 ++ .

(9.66)

This estimate implies that, for x 1 , x 2 with ρs (x 1 , x 2 ) ≤ 1, we have γ

|∂ ym u(x 12 , x 0 , y, t) − ∂ ym u(x 11 , x 0 , y, t)| ≤ CgW F,0,γ t 2 ρs (x 1 , x 2 )γ ,

(9.67)

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completing the proof of the spatial part of (9.45). To complete the estimates of the first derivatives we need to bound the difference |∇u(x, y, t2 ) − ∇u(x, y, t1 )|. From the estimates in (9.51), we see that for t1 , t2 ≤ T, and any 0 < c < 1, there is a C T so that if t1 < ct2 , then γ

|∇x u(x, y, t2 ) − ∇x u(x, y, t1 )| ≤ C T gW F,0,γ |t2 − t1 | 2 |∇ y u(x, y, t2 ) − ∇ y u(x, y, t1 )| ≤ C T gW F,0,γ |t2 − t1 |

γ +1 2

(9.68)

.

As usual, this reduces us to consideration of the case that ct2 < t1 < t2 . For this argument we fix a 12 < c < 1, and use a slightly different argument depending upon whether we are estimating an x-derivative or a y-derivative. The x-derivatives are done very much like the estimates in Chapter 7 beginning with (7.128). For example, to estimate the x n -derivative we use the representation ∂xn u(x, y, t2 ) − ∂xn u(x, y, t1 ) = t 2 −t1 ∞

∞ ∞ ···

0

···

−∞ 0 0 j =1  [g(wn , wn , z, t2 − s) −

∞ ∞ ···

−∞

kse (y j , z j )

···

−∞ 0 m ' j =1

∞  ' m 0

j =1

kte1 −s (y j , z j )

n−1 '

g(wn , wn , z, t1

kte2 −s (y j , z j )

n−1 '

b

ks j (x j , w j )∂xn ksbn (x n , wn )×

j =1

−∞

2t1 −t2 ∞

0

∞ ' m

n−1 ' j =1

− s)]dwn dwn d zds+ b

kt2j−s (x j , w j )∂xn ktb2n−s (x n , wn )− ×

b kt1j−s (x j , w j )∂xn ktb1n−s (x n , wn )

j =1  [g(wn , wn , z, s) − g(wn , x n , z, s)]dwn dwn d zds+ t1 ∞ ∞ ∞ ∞ ' m n−1 ' bj ··· ··· kte2 −s (y j , z j ) kt2 −s (x j , w j )∂xn ktb2n−s (x n , wn )× j =1 −∞ 0 2t1 −t2 −∞ 0 j =1

[g(wn , wn , z, s) − g(wn , x n , z, s)]dwn dwn d zds.

(9.69)

The first and last terms are estimated exactly as before. To estimate the second integral we use the analogue of the expression in (7.129), first observing that ⎫ ⎧ m n m n ⎬ ⎨' ' ' ' b b ∂xn kte2 −s (y j , z j ) kt2l−s (xl , wl ) − kte1 −s (y j , z j ) kt1l−s (xl , wl ) = ⎭ ⎩ j =1 l=1 j =1 l=1 / n 0 ' m n ' b ' bl e l kt2 −s (y j , z j ) kt2 −s (xl , wl ) − kt1 −s (xl , wl ) + ∂xn j =1

⎡ ⎣

l=1 m ' j =1

kte2 −s (y j , z j )

−

l=1 m ' j =1

⎤

kte1 −s (y j , z j )⎦

n ' l=1

 .

b kt1l−s (xl , wl )

(9.70)

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We use the expansion in (7.34) to replace the differences of products on the right-hand side of (9.70) with terms containing a single term of the form b

b

[kt2l−s (xl , wl ) − kt1l−s (xl , wl )] or

(9.71)

[kte2 −s (y j , z j ) − kte1 −s (y j , z j )]. If we always use the estimate √ √ |g(wn , wn , z, s) − g(wn , x n , z, s)| ≤ 2gW F,0,γ | wn − x n |γ ,

(9.72)

then we see that there are three types of terms that must be bounded: I.

2t1 −t2∞ ∞

0

0

|ktb2l−s (xl , wl )−ktb1l−s (xl , wl )||∂xn ktb2n−s (x n , wn )|

0

√

| xn −

(9.73)

√ γ wn | dwl dwn ds,

II. 2t1 −t2∞

0

III.

√ √ |∂xn [ktb2n−s (x n , wn ) − ktb1n−s (x n , wn )]|| x n − wn |γ dwn ds,

(9.74)

0

2t1 −t2 ∞ ∞

0

|kte2 −s (y j , z j )−kte1 −s (y j , z j )||∂xn ktb1n−s (x n , wn )|

−∞ 0

(9.75)

√ √ | x n − wn |γ dwl dz j ds.

Terms of types I and II were shown, in the proof of (7.133), to be bounded by a constant γ times |t2 −t1 | 2 , leaving just the term of type III. Using Lemma 6.1.10 and Lemma 8.2.4 we see that these terms are bounded by t1

γ

γ

(t2 − t1 )σ 2 −2 dσ ≤ C|t2 − t1 | 2 .

(9.76)

t2 −t1

The argument for estimating the differences |∂ y j u(x, y, t2 ) − ∂ y j u(x, y, t1 )|

(9.77)

is essentially identical, though the results are a bit different. We are free to assume that j = m and use the analogue of (9.69) with ∂xn replaced with ∂ ym and the difference g(wn , wn , z, s) − g(wn , x n , z, s) replaced by g(w, z m , z m , s) − g(w, z m , ym , s).

(9.78)

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The contributions of the first and third integrals are then bounded by 2(t2 −t1 ) ∞

2gW F,0,γ

|∂ ym kse (ym , z m )||ym − z m |γ dz m ds.

(9.79)

−∞

0

Applying Lemma 8.2.5 we see that this integral is bounded by 2(t2 −t1 )

s

C

γ −1 2

ds =

0

γ +1 2 [2|t2 − t1 |] 2 , 1+γ

(9.80)

which suffices to prove the desired estimate. This leaves the analogue of the second integral in (9.69), which we expand using the analogue of (9.70) and (7.34), replacing ∂xn with ∂ ym . We need to estimate three types of terms: I. 2t1 −t2∞ ∞

0

|ktb2l−s (xl , wl )−ktb1l−s (xl , wl )||∂ ym kte2 −s (ym , z m )||ym −z m |γ dz m dwl ds,

0 −∞

(9.81)

II. 2t1 −t2 ∞

0

|∂ ym [kte2−s (ym , z m ) − kte1 −s (ym , z m )]||ym − z m |γ dz m ds,

(9.82)

−∞

III. 2t1 −t2∞ ∞

0

0

|kte2 −s (y j , z j )−kte1 −s (y j , z j )||∂ ym kte1 −s (ym , z m )||ym −z m |γ dz m dz j ds.

0

(9.83)

Lemma 6.1.9 and Lemma 8.2.5 show that terms of type I are bounded by 2t1 −t2

0

(t2 − t1 )(t1 − s) (t2 − s)

γ −1 2

ds

2t1 −t2

≤

(t2 − t1 )(t1 − s)

γ −3 2

ds ≤ C|t2 − t1 |

γ +1 2

. (9.84)

0

Using Lemma 8.2.7 we see that the terms of type II are bounded by C|t2 − t1 |

γ +1 2

.

(9.85)

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The second estimate in Lemma 8.2.4 and Lemma 8.2.5 show that terms of type III are also bounded by t1 γ −1 γ +1 s 2 (t2 − t1 )ds C ≤ C|t2 − t1 | 2 . (9.86) t2 − t1 + s t2 −t1

Thus we see that there is a constant C so that we have: |∇ y u(x, y, t2 ) − ∇ y u(x, y, t1 )| ≤ CgW F,0,γ |t2 − t1 |

γ +1 2

.

(9.87)

|∇ y u(x, y, t2 ) − ∇ y u(x, y, t1 )| ≤ Ct22 gW F,0,γ |t2 − t1 | 2 ,

(9.88)

If t1 < t2 , then (9.86) also gives the estimate 1

γ

completing the proof of (9.45) as well as the proof that, if t1 , t2 < T, then there is a constant C so that the first derivatives satisfy |∇ x, y u(x 2 , y2 , t2 ) − ∇ x, y u(x 1 , y1 , t1 )| ≤ γ

C(1 + T 2 )gW F,0,γ [ρs (x 1 , x 2 ) + ρe ( y1 , y2 ) +



|t2 − t1 |]γ .

(9.89)

9.2.2 Second Derivative Estimates This brings us to the second derivatives. As it is essentially the same as the 1-dimensional case, Lemma 6.1.14 suffices to prove the bounds γ

γ

|xl ∂x2l u(x, y, t)| ≤ CgW F,0,γ min{xl2 , t 2 },

(9.90)

for l = 1, . . . , n. The calculations between (7.158) and (7.160) suffice to prove that for 1 ≤ l, k ≤ n, we have the estimates γ γ γ √ | xl x k ∂xl ∂xk u(x, y, t)| ≤ CgW F,0,γ min{xl2 , x k2 , t 2 }.

(9.91)

Using Lemmas 8.2.5 and 8.2.8, we easily derive the estimates γ

|∂ y j ∂ yk u(x, y, t)| ≤ CgW F,0,γ t 2 ,

(9.92)

where 1 ≤ j, k ≤ m. Using Lemmas 6.1.10 and 8.2.5 we can also show that γ γ √ | xl ∂xl ∂ y j u(x, y, t)| ≤ CgW F,0,γ min{xl2 , t 2 },

(9.93)

for 1 ≤ l ≤ n and 1 ≤ j ≤ m. To complete the spatial part of the estimate, we need to show that the second derivatives are H¨older continuous. As before, the earlier arguments suffice to show that + +  + 2 2 + + x x ∂x ∂x u(x 2 , y, t) − x 1 x 1 ∂x ∂x u(x 1 , y, t)+ ≤ l k l k + l k l k + CgW F,0,γ ρs (x 1 , x 2 )γ and |∂ y j ∂ yk u(x, y2 , t) − ∂ y j ∂ yk u(x, y1 , t)| ≤ CgW F,0,γ ρe ( y1 , y2 )γ .

(9.94)

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Thus we are left to estimate √ √ | xl x k ∂xl ∂xk u(x, y2 , t) − xl x k ∂xl ∂xk u(x, y1 , t)| and |∂ y j ∂ yk u(x 2 , y, t) − ∂ y j ∂ yk u(x 1 , y, t)|, √ and the mixed derivatives xl ∂xl ∂ y j u. We begin with the quantities in (9.95), by considering √ √ | xl x k ∂xl ∂xk u(x, y2 , t) − xl x k ∂xl ∂xk u(x, y1 , t)|,

(9.95)

with k = l. Without loss of generality we can assume k = 1, l = 2 and therefore y2 − y1 = (y12 − y11 , 0, . . . , 0). With these assumptions, using the observation that ∞

∂x1 ksb1 (x 1 , w1 )g(x 1, w 0 , z, t − s)dw1 = 0,

(9.96)

0

for all values of (w0 , z, t − s), we get the estimate √ √ | x 1 x 2 ∂x1 ∂x2 u(x, y2 , t) − xl x k ∂xl ∂xk u(x, y1 , t)| ≤  t ∞ ∞ ∞ 2gW F,0,γ 0

0

√ √ √ √ | x 1 ∂x1 ksb1 (x 1 , w1 ) x 2 ∂x2 ksb2 (x 2 , w2 )|| x 1 − w1 |γ ×

0 −∞

|kse (y12 , z 1 ) − kse (y11 , z 1 )|dw1 dw2 dz 1 ds.

(9.97)

Applying Lemmas 6.1.10 and 8.2.2, we see that the integral is estimated by ⎞ ⎛ ⎞ ⎛ |y12 −y11 | |y12 −y11 | γ √ −1 t √ t √ √ −1 γ x1s 2 x2 s s s ⎟ ⎜ ⎟ ⎜ ds ≤ C s 2 −1 ⎝ ds, C √ √ ⎝ 2 −y 1 | ⎠ 2 −y 1 | ⎠ |y |y 1 + x 1 /s 1 + x 2 /s 1 + 1√ 1 1√ 1 1 + 0 0 s s (9.98) and that ⎛ ⎞ |y12 −y11 |2 |y12 −y11 | t  t √ γ γ γ −3 s ⎜ ⎟ −1 −1 s2 ⎝ s 2 ds + |y12 − y11 |s 2 ds ⎠ ds ≤ 2 1 |y −y | 1 + 1√s 1 (9.99) 0 0 |y 2 −y 1 |2 1

1

2 |y 2 − y11 |γ , ≤ γ (1 − γ ) 1 which completes this case. We next consider this situation with k = l; we can take k = l = 1, with y2 − y1 as before. Using the fact that ∞ 0

∂x21 ksb1 (x 1 , w1 )g(x 1, w 0 , z, t − s)dw1 = 0,

(9.100)

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for all values of (w0 , z, t − s), we get the estimate |x 1 ∂x21 u(x, y2 , t) − x 1 ∂x21 u(x, y1 , t)| ≤  t ∞ ∞ 2gW F,0,γ 0

√ √ |x 1 ∂x21 ksb1 (x 1 , w1 )|| x 1 − w1 |γ ×

0 −∞

|kse (y12 , z 1 ) − kse (y11 , z 1 )|dw1 dz 1 ds.

(9.101)

We now apply Lemma 6.1.15 and Lemma 8.2.2 to see that this integral is bounded by ⎞ ⎛ |y12 −y11 | γ t √ √ −1 x1s 2 s ⎟ ⎜ ds C √ ⎝ √ 2 −y 1 | ⎠ |y x 1 + s 1 + 1√ 1 0 s ⎡ ⎤ 2 |y1 −y11 |2  t ⎢ ⎥ γ γ −3 ≤C⎢ s 2 −1 ds + |y12 − y11 |s 2 ds ⎥ ⎣ ⎦ 0

|y12 −y11 |2

≤

2C |y 2 − y11 |γ . γ (1 − γ ) 1

(9.102)

We have implicitly assumed that t > |y12 − y11 |2 ; if this is not the case, then one gets a single term in (9.99) and (9.102). Otherwise the argument is identical. This completes the spatial part of the H¨older estimate for the second x-derivatives. We now turn to |∂ y j ∂ yk u(x 2 , y, t) − ∂ y j ∂ yk u(x 1 , y, t)| by considering j = k. We can assume that j = 1, k = 2, and x 2 − x 1 = (x 12 − x 11 , 0, . . . , 0). We first need to consider the case where x 11 = 0. We use the fact that ∞

∂ y1 kse (y1 , z 1 )[g(w, z 1 , z 0 , t − s) − g(w, y1 , z 0 , t − s)]dz 1 = 0

(9.103)

−∞

to see that |∂ y j ∂ yk u(x 2 , y, t) − ∂ y j ∂ yk u(x 1 , y, t)| ≤  t ∞ ∞ ∞ 2gW F,0,γ

|∂ y1 kse (y1 , z 1 )∂ y2 kse (y2 , z 2 )||y1 − z 1 |γ ×

0 −∞ −∞ 0

|ksb1 (x 12 , w1 ) − ksb1 (0, w1 )|dw1 dz 1 dz 2 ds.

(9.104)

Applying Lemma 8.2.5 and the first estimate in Lemma 6.1.5 we see that the integral is estimated by  t γ2 −1 2 γ s x 1 ds 4 (x 12 ) 2 . ≤ (9.105) 2 γ (2 − γ ) x1 + s 0

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Applying (6.72), this estimate implies that for any 0 < c < 1, there is a constant C so that if x 11 < cx 12 , then 2

1

|∂ y j ∂ yk u(x , y, t) − ∂ y j ∂ yk u(x , y, t)| ≤ CgW F,0,γ

+ + + 2  1 +γ + x − x + . 1+ + 1

(9.106)

We are therefore reduced to the case cx 12 < x 11 < x 12 . In this case we have |∂ y j ∂ yk u(x 2 , y, t) − ∂ y j ∂ yk u(x 1 , y, t)| ≤  t ∞ ∞ ∞ 2gW F,0,γ

|∂ y1 kse (y1 , z 1 )∂ y2 kse (y2 , z 2 )||y1 − z 1 |γ ×

0 −∞ −∞ 0

|ksb1 (x 12 , w1 ) − ksb1 (x 11 , w1 )|dw1 dz 1 dz 2 ds, (9.107) which can be estimated using Lemma 8.2.5 and the second estimate in Lemma 6.1.5. These lemmas show that the integral is bounded by

C

⎛

 x 12 − x 11 √ ⎜ γ s   s 2 −1 ⎜ ⎝ x 12 − x 11 √ 0 1+ s ⎡   2 ( x 1 − x 11 )2

t

⎢ ⎢ C⎢ ⎣





⎞ ⎟ ⎟ ds ≤ ⎠

γ

s 2 −1 ds +

t s

  ( x 12 − x 11 )2

0

γ −3 2

⎤ + + + 2  1+ ⎥ + x − x + ds ⎥ ⎥ 1+ + 1 ⎦

+ + + 2  1 +γ 2 + ≤ x − x 1 ++ . (9.108) γ (1 − γ ) + 1

The case j = k = 1 follows exactly the same pattern. We use the fact that ∞

∂ y21 kse (y1 , z 1 )g(w, y1 , z 0 , t − s)dz 1 = 0

(9.109)

−∞

for any values of (w, z 0 , t − s), and the estimate (8.28) to see that ∞

γ

|∂ y21 kse (y1 , z 1 )||y1 − z 1 |γ ≤ Cs 2 −1 .

(9.110)

−∞

From this point the argument used for the case j = k can be followed verbatim. We   2 2 have again implicitly assumed that t > x 1 − x 11 . If this is not the case, then

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we get only the first term in the second line of (9.108); otherwise the argument is unchanged. To complete the spatial estimates we need only show that the mixed partial deriva√ tives xl ∂xl ∂ y j u(x, y, t) are H¨older continuous. Without loss of generality we can take j = l = 1. As usual we can assume that the points of evaluation differ in a single coordinate. We start by considering variations in the x-variables. There are two cases to consider: (1) The x-variable differs in the first slot, (2) The x-variable differs in another slot. For case 1, we first need to take x 11 = 0. In this case we see that the second estimate in (9.93) and (6.72) imply that, if 0 < c < 1, then there is a C so that, for x 11 < cx 12, we have the estimate + +  + 2 + + x ∂x ∂ y u(x 2 , y, t) − x 1 ∂x ∂ y u(x 1 , y, t)+ ≤ CgW F,0,γ 1 1 1 + 1 1 1 +

+ + + 2  1 +γ + x − x + . 1+ + 1 (9.111) We are therefore left to consider cx 12 < x 11 < x 12 . For this case we see that + +  + 2 + + x ∂x ∂ y u(x 2 , y, t) − x 1 ∂x ∂ y u(x 1 , y, t)+ ≤ 1 1 1 + 1 1 1 + CgW F,0,γ

+  t ∞ ∞ +  + 2 + + x ∂x k b1 (x 2 , w1 ) − x 1 ∂x k b1 (x 1 , w1 )+ × 1 1 1 s + 1 1 s 1 + 0

0 −∞

+ + + 2 √ +γ + x − w1 + |∂ y k e (y1 , z 1 )|dw1 dz 1 ds. 1 s + + 1

(9.112)

Applying Lemmas 8.2.5 and 6.1.11 we see that this integral is bounded by ⎛ ++ t C 0

γ

s 2 −1

 ++ ⎞ + x 2− x 1+ + 1 1+ ⎝ ⎠ √ s

⎛ ++

 ++ ⎞ ds. + x 2 − x 1+ + 1 1+ ⎠ √ 1+⎝ s

(9.113)

+ + + 2  1+ + As before we easily establish that, when + x 1 − x 1 ++ < 1, this is bounded by a + + + 2  1 +γ + constant times + x 1 − x 1 ++ . We now turn to the case that x 2 − x 1 is non-zero in the j th entry where j > 1. As in the previous case, we need to first consider x 1j = 0. In this case the difference is

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estimated by + +  + 1 + + x ∂x ∂ y u(x 2 , y, t) − x 1 ∂x ∂ y u(x 1 , y, t)+ ≤ 1 1 1 + 1 1 1 + 2gW F,0,γ

+γ + +  t ∞ ∞ ∞ + + 1 + ++ + x ∂x k b1 (x 1 , w1 )+ + x 1 − √w1 + × + 1 1 s 1 + ++ 1

0 0 0 −∞ bj 2 b |ks (x j , w j ) − ks j (0, w j )||∂ y1 kse (y1 , z 1 )|dz 1 dw j dw1 dt.

(9.114)

Applying Lemmas 6.1.10, 8.2.5 and the first estimate in Lemma 6.1.5 we see that this integral is estimated by t x 2j ds γ γ s 2 −1 ≤ C(x 2j ) 2 . (9.115) 2 s + xj 0

Applying (6.72) we are reduced to consideration of the case cx 2j < x 1j < x 2j , for a 0 < c < 1. In this case we use the second estimate in Lemma 6.1.5 to see that the replacement for (9.115) is (  ) t C

| x 2j − x 1j | √ s

γ

s 2 −1

0

( 1+

  ) ds | x 2j − x 1j | √ s

 +γ + + + ≤ C + x 2j − x 1j + .

(9.116)

This completes the proof that + +  + 2 + + x ∂x ∂ y u(x 2 , y, t) − x 1 ∂x ∂ y u(x 1 , y, t)+ ≤ CgW F,0,γ ρs (x 1 , x 2 )γ . (9.117) l l j + l l j + +√ + √ We are left to consider + x 1 ∂x1 ∂ y1 u(x, y2 , t) − x 1 ∂x1 ∂ y1 u(x, y2 , t)+ , where, as before, we need to distinguish between the case that the y-variables differ in the first coordinate and in other coordinates. If y2 − y1 = (y12 − y11, 0, . . . , 0), then this difference is estimated by  t ∞ ∞ 2gW F,0,γ 0

√ | x 1 ksb1 (x 1 , w1 )|×

0 −∞

|∂ y1 kse (y12 , z 1 ) − ∂ y1 kse (y11 , z 1 )||y11 − z 1 |γ dz 1 dw1 ds. Lemmas 6.1.10 and 8.2.6 show that this integral is estimated by   |y12 −y11 | t √  +γ + s γ + +  ds ≤ C +y12 − y11 + .  s 2 −1 |y12 −y11 | √ 1+ 0 s

(9.118)

(9.119)

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If the y-variables differ in another coordinate, then the difference of second derivatives is estimated by  t ∞ ∞ ∞ 2gW F,0,γ

√ | x 1 ksb1 (x 1 , w1 )|×

0 −∞ −∞ e 1 |∂ y1 ks (y1 , z 1 )||y11 0

− z 1 |γ |kse (y 2j , z j ) − kse (y 1j , z j )|dz j dz 1 dw1 ds.

(9.120)

We now apply Lemmas 6.1.10, 8.2.5, and 8.2.2 to see that the integral is bounded by  2 1 |y j −y j | √ t +γ + s γ + +  2 1  ds ≤ C +y 2j − y 1j + . s 2 −1 (9.121) |y j −y j | √ 1+ 0 s This completes the proof that + +  + 1 + + x ∂x ∂ y u(x, y2 , t) − x 1 ∂x ∂ y u(x, y1 , t)+ ≤ CgW F,0,γ ρe ( y1 , y2 )γ . (9.122) l l j + l l j + To finish the proof of the proposition we need to establish the H¨older continuity, in the time-variable, of the second spatial-derivatives of u. Using (6.72) along with the estimates in (9.91), (9.92), and (9.93), we see that for any 0 < c < 1, there is a constant C so that, if t1 < ct2 , then γ √ √ | xl x k ∂xl ∂xk u(x, y, t2 ) − xl x k ∂xl ∂xk u(x, y, t1 )| ≤ CgW F,0,γ |t2 − t1 | 2 γ √ √ (9.123) | xl ∂xl ∂ y j u(x, y, t2 ) − xl ∂xl ∂ y j u(x, y, t1 )| ≤ CgW F,0,γ |t2 − t1 | 2 γ

|∂ yl ∂ y j u(x, y, t2 ) − ∂ yl ∂ y j u(x, y, t1 )| ≤ CgW F,0,γ |t2 − t1 | 2 . We are left to consider these differences for ct2 < t1 < t2 , where we assume that 1 2 < c < 1. For all these cases we use an expansion like that in (9.69), with the operator ∂xn replaced by the appropriate second order operator. √ We first treat the pure x-derivatives, xl x k ∂xl ∂xk u, where l = k. Without loss of generality we can assume that k = n. By replacing the difference g(wn , wn , z, t2 − s) − g(wn , wn , z, t1 − s) with g(wn , wn , z, t2 − s) − g(wn , x n , z, t2 − s) + g(wn , x n , z, t1 − s) − g(wn , wn , z, t1 − s) (9.124) in the first integral in the analogue of (7.128), we see that it is estimated by t 2 −t1∞ ∞

CgW F,0,γ 0

0

0

√

x n xl |∂xn ksbn (x n , wn )∂xl ksbl (xl , wl )|× √ √ | wn − x n |γ dwl dwn ds.

(9.125)

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Using Lemma 6.1.10 the integral in this term is estimated by t 2 −t1

C 0

√

γ

xl x n s 2 −1 ds √ √ √ ≤ C|t2 − t1 |γ . √ ( xl + s)( x n + s)

(9.126)

The last integral in the analogue of (7.128) is easily seen to be bounded by 2(t2 −t1 )

C t2 −t1

√

γ

xl x n s 2 −1 ds √ √ √ ≤ C|t2 − t1 |γ . √ ( xl + s)( x n + s)

(9.127)

This leaves only the second integral in the analogue of (7.128), which we replace by a sum of terms using the analogue of (9.70) and (7.34). All the possible terms that arise from the analogue of the first term on the right-hand side of (9.70) are enumerated γ in (7.170)–(7.174), and shown to be bounded by CgW F,0,γ |t2 − t1 | 2 . The second term on the right-hand side of (9.70) produces an additional type of term: 2t1 −t2 ∞ ∞ ∞

|kte2 −s (y j , z j ) − kte1 −s (y j , z j )|× 0

√

−∞ 0

0

√ √ x n xl |∂xn ktb1n−s (x n , wn )∂xl ktb1l−s (xl , wl )|| wn − x n |γ dwl dwn dz j ds.

(9.128)

Lemma 6.1.10 and the second estimate in Lemma 8.2.4 show that this integral is bounded by t1 t2 −t1

γ √ t1 γ γ xl x n (t2 − t1 )s 2 −1 ds √ √ √ ≤ |t2 − t1 |s 2 −2 ds ≤ C|t2 − t1 | 2 . √ ( xl + s)( x s + s)(t2 − t1 + s)

t2 −t1

(9.129) Now we need to consider the case k = l = n. For these cases we are free to replace x n ∂x2n with L bn ,xn . Most of the terms that arise in this case have been treated in the proof of Proposition 7.2.1. The only new type of term arises from expanding the second term on right-hand side of the analogue of (9.70) in the second integral. These are of the form 2t1 −t2 ∞ ∞

|kte2 −s (y j , z j ) − kte1 −s (y j , z j )|× 0

−∞ 0

√ √ |x n ∂x2n ktb1n−s (x n , wn )|| wn − x n |γ dwn dz j ds.

(9.130)

Lemma 8.2.4 and Lemma 6.1.15 show that this term is bounded by t1 C t2 −t1

γ

γ x n (t2 − t1 )s 2 −1 ds ≤ C|t2 − t1 | 2 . (s + x n )(t2 − t1 + s)

(9.131)

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This completes the proof that γ √ | xl x k ∂xl ∂xk [u(x, y, t2 ) − u(x, y, t1 )]| ≤ CgW F,0,γ |t2 − t1 | 2 .

(9.132)

The verification that ∂ y j ∂ yk u(x, y, t) satisfies the same estimate is essentially identical, simply interchanging estimates for ksb with estimates for kse and vice versa. We leave the details to the interested reader. To conclude the proof of Proposition 9.2.1 in the k = 0 case, we verify that γ √ | xl ∂xl ∂ y j [u(x, y, t2 ) − u(x, y, t1 )]| ≤ CgW F,0,γ |t2 − t1 | 2 .

To prove this estimate we use the expression in (7.128) with ∂xn replaced by The first and third integrals are estimated by 2(t2 −t1 ) ∞ ∞

x n ∂xn ∂ ym .

√ |∂ ym kse (ym , z m ) x n ∂xn ksbn (x n , wn )|×

−∞ 0

0

(9.133) √

√ √ | x n − wn |γ dz m dwn ds.

(9.134)

We use Lemma 6.1.10 and 8.2.5 to see that this integral is bounded by 2(t2 −t1 ) √

C 0

γ

γ x n s 2 −1 ds √ ≤ C|t2 − t1 | 2 . √ xn + s

(9.135)

Two cases arise in the estimation of the contribution of first term on the right-hand side of (9.70). In the first case we get terms of the form: 2t1 −t2 ∞ ∞

|∂ ym kte2 −s (ym , z m )|× 0

−∞ 0

√ √ √ | x n ∂xn [ktb2n−s (x n , wn ) − ktb1n−s (x n , wn )]|| x n − wn |γ dz m dwn ds.

(9.136)

Applying Lemmas 8.2.5 and 6.1.13 we see that this term is bounded by t1 t2 −t1

γ √ t1 γ γ x n (t2 − t1 )s 2 −1 ds √ ≤ (t2 − t1 )s 2 −2 ds ≤ C|t2 − t1 | 2 . √ ( x n + s)(t2 − t1 + s)

(9.137)

t2 −t1

For the second case we have terms of the form 2t1 −t2 ∞ ∞ ∞

0

−∞ 0

√ |∂ ym kte2 −s (ym , z m )|| x n ∂xn ktb1n−s (x n , wn )|×

0

√ √ |ktb2l−s (xl , wl ) − ktb1l−s (xl , wl )|| x n − wn |γ dz m dwl dwn ds

(9.138)

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165

and 2t1 −t2 ∞ ∞ ∞

0

−∞ 0

√ |∂ ym kte2 −s (ym , z m )|| x n ∂xn ktb2n−s (x n , wn )|×

0

√ √ |ktb2l−s (xl , wl ) − ktb1l−s (xl , wl )|| x n − wn |γ dz m dwl dwn ds.

(9.139)

Applying Lemmas 6.1.10, 8.2.5 and 6.1.9 shows that these terms are estimated by t1 √ C t2 −t1

γ −1

1

x n s 2 (t2 − t1 + s)− 2 (t2 − t1 )ds ≤ √ √ ( x n + s)(t2 − t1 + s) t1 C

γ

γ

(t2 − t1 )s 2 −2 ds ≤ C|t2 − t1 | 2 . (9.140)

t2 −t1

Two cases also arise in the estimation of the contribution of second term on the right-hand side of (9.70). For the first case we get 2t1 −t2 ∞ ∞

|∂ ym kte2 −s (ym , z m ) − ∂ ym kte1 −s (ym , z m )|× 0

−∞ 0

√ √ √ | x n ∂xn ktb1n−s (x n , wn )|| x n − wn |γ dz m dwn ds.

(9.141)

Lemma 6.1.10 and Lemma 8.2.7 show that this integral is estimated by the expression in (9.137). For the second case we get 2t1 −t2 ∞ ∞ ∞

0

−∞ 0

0

√ |∂ ym kte2 −s (ym , z m )|| x n ∂xn ktb1n−s (x n , wn )|×

√ √ |kte2−s (yk , z k ) − kte1 −s (yk , z k )|| x n − wn |γ dz m dz k dwn ds

(9.142)

and 2t1 −t2 ∞ ∞ ∞

0

−∞ 0

√ |∂ ym kte1 −s (ym , z m )|| x n ∂xn ktb1n−s (x n , wn )|×

0

√ √ |kte2 −s (yk , z k ) − kte1 −s (yk , z k )|| x n − wn |γ dz m dz k dwn ds.

(9.143)

Using Lemmas 6.1.10, 8.2.5 and 8.2.4 we see that these terms are estimated by γ −1 1 t1 √ t1 γ γ x n s 2 s − 2 (t2 − t1 )ds ≤C C (t2 − t1 )s 2 −2 ds ≤ C|t2 − t1 | 2 . (9.144) √ √ ( x n + s)(t2 − t1 + s)

t2 −t1

t2 −t1

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This completes the proof that the second derivatives satisfy the appropriate H¨older estimates: there is a constant C uniformly bounded for 0 < b < B so that +  +  + 2 2 + + x x ∂x ∂x u(x 2 , y2 , t2 ) − x 1 x 1 ∂x ∂x u(x 1 , y1 , t1 )+ , max l k l k l k l k + + 1≤k,l≤n; 1≤i, j ≤m + +  + 2 + + x ∂x ∂ y u(x 2 , y2 , t2 ) − x 1 ∂x ∂ y u(x 1 , y1 , t1 )+ , l l j + l l j +  |∂ y j ∂ yi u(x 2 , y2 , t2 ) − ∂ y j ∂ yi u(x 1 , y1 , t1 )|  γ  ≤ CgW F,0,γ ρs (x 1 , x 2 ) + ρe ( y1 , y2 ) + |t2 − t1 | .

(9.145)

To prove the H¨older continuity of ∂t u(x, y, t) we simply use the equation ∂t u =

n  l=1

L b j ,x j u +

m  j =1

∂ y2j u + g,

(9.146)

and the H¨older continuity of the expression appearing on the right-hand side of this relation. Arguing as before we can use Proposition 5.2.8 to allow components of b to tend to zero, and thereby extend these estimates to the case that 0 ≤ b. Using the estimates for scaled second derivatives (9.90), (9.91), and (9.93), along with Proposition 4.2.5, 0,2+γ we argue as before to apply Lemma 5.2.7 and show that u ∈ ᏯW F (⺢n+ × ⺢m ×[0, T ]). This completes the proof of the proposition in the k = 0 case. For the k > 0 cases we assume that g is supported in B R+ (0) × ⺢m × [0, T ]. Using Propositions 4.2.4 and 5.2.9, as before, we easily obtain the desired conclusions for all k ∈ ⺞, and thereby complete the proof of the Proposition 9.2.1.  9.3 OFF-DIAGONAL AND LONG-TIME BEHAVIOR We next consider a general result describing the off-diagonal behavior of the solution kernel for (9.5). This result is important in the perturbation theory that follows in the next chapter. n m P ROPOSITION 9.3.1. Let ϕ, ψ ∈ Ꮿ∞ c (⺢+ × ⺢ ), and assume that

distW F (supp ϕ, supp ψ) = η > 0.

(9.147)

Let 0 < B and 0 ≤ b j ≤ B. For any k ∈ ⺞0 , the map b ᏷ϕ,ψ,t : g → ψ(w)K tb [ϕ(z)g(z, ·)]

defines a bounded operator b ˙ 0 (⺢n+ × ⺢m × [0, T ]) → Ꮿ ˙ k (⺢n+ × ⺢m × [0, T ]). ᏷ϕ,ψ,t :Ꮿ

(9.148)

There are positive constants cη , C, where C depends on k, η and B so that the operator cη

norm of this map is bounded, as T → 0+ , by Ce− T .

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This proposition is a consequence of estimates on the 1-dimensional kernels. For the degenerate models we have L EMMA 9.3.2. Let η > 0 and for x ∈ ⺢+ define the set √ √ Jx,η = {y ∈ ⺢+ : | x − y| ≥ η}.

(9.149)

For 0 < b ≤ B, 0 < φ ≤ π2 , and j ∈ ⺞0 there is a constant Cη, j,B,φ so that if t = |t|eiθ , where |θ | ≤ π2 − φ, then 

η2

j |∂x ktb (x, y)|d y

Jx,η

e− cos θ 2|t| ≤ Cη, j,B,φ . |t| j

(9.150)

For the Euclidean models we have L EMMA 9.3.3. Let η > 0 and for x ∈ ⺢ define the set Jx,η = {y ∈ ⺢ : |x − y| ≥ η}. For j ∈ ⺞0 , 0 < φ ≤ |θ | ≤ π2 − φ, then

π 2,

(9.151)

there is a constant Cη, j,φ so that if t = |t|eiθ , with the angle



η2

j |∂x kte (x, y)|d y

≤ Cη, j,φ

Jx,η

e− cos θ 8t j

|t| 2

.

(9.152)

The lemmas are proved in the Appendix. P ROOF OF P ROPOSITION 9.3.1. We need to consider integrals of the form  I ((x, y), t) = ⺢n+ ×⺢m

β

|∂ y ∂ xα

n '

b

kt i (x i , wi )

i=1

m '

kte (yl , z l )ϕ(w, z) f (w, z)|dwd z,

l=1

(9.153)

for (x, y) ∈ supp ψ, with |α| = q, |β| = p. For such (x, z) we let √ √ U(x, y), j = {(w, z) ∈ supp ϕ : | x j − w j | ≥ U(x, y), j = {(w, z) ∈ supp ϕ : |y j −n − z j −n | ≥

η } for 1 ≤ j ≤ n, n+m

η } for n + 1 ≤ j ≤ n + m. n+m (9.154)

Since distW F ((x, y), (w, z)) =

n m   √ √ | x j − wj| + |yl − z l |, j =1

l=1

(9.155)

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and distW F (supp ψ, supp ϕ) ≥ η, it follows that supp ϕ ⊂

m+n 7

U(x, y), j

(9.156)

j =1

and that these sets are measurable. Thus we have the estimate I ((x, y), t) ≤ ϕ f  L ∞



m+n  j =1 U

β

|∂ y ∂ xα

n '

m '

ktbi (x i , wi )

i=1

(x, y), j

kte (yl , z l )|dwd z. (9.157)

l=1

Let I j ((x, y), t) denote the integral in this sum over U(x, y), j . We observe that j −1

U(x, y), j ⊂ ⺢+

× Jx j ,

η n+m

n− j

× ⺢+

U(x, y), j ⊂ ⺢n+ × ⺢ j −n−1 × Jy j −n ,

× ⺢m for 1 ≤ j ≤ n,

η n+m

× ⺢m+n− j for n + 1 ≤ j ≤ n + m.

(9.158)

Applying the 1-dimensional estimates we see that, if 1 ≤ j ≤ n, then I j ((x, y), t) ≤ ∞ ∞  ϕ f  L ∞ · · · 0



0 ⺢m Jx

η j , n+m

|∂ xα

n '

β

b

kt i (x i , wi )∂ y

i=1

m '

8j d z, (9.159) kte (yl , z l )|dw j dw

l=1

8j is the volume form in ⺢n+ with dw j omitted. Lemmas 9.3.2, 9.3.3, where, as usual, dw 6.1.21, and 8.2.15 show that I j ((x, y), t) ≤ Cϕ f  L ∞

e

− cos θ

η2 2(n+m)2 t p

t q+ 2

.

(9.160)

A similar estimate applies for n + 1 ≤ j ≤ n + m, which, upon summing, shows that I ((x, y), t) ≤ Cϕ f  L ∞

e

− cos θ

η2 8(n+m)2 t p

t q+ 2

.

(9.161)

The estimate on the right-hand side is independent of (x, y) ∈ supp ψ, so we can integrate it to obtain T I ((x, y), t)dt ≤ Cϕ f  L ∞ e

− cos θ

η2 16(n+m)2 T

.

(9.162)

0

Coupling this with the Leibniz formula, the proposition follows easily from these estimates. 

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For each t > 0, we have defined the map f → K tb,m f, where K tb,m

 f (x, y) =

n '

ktbi (x i , wi )

⺢n+ ×⺢m i=1

m '

kte (yl , z l ) f (w, z)d zdw.

(9.163)

l=1

For any t > 0 and f ∈ L ∞ (⺢n+ × ⺢m ), it is clear that K tb,m f ∈ Ꮿ∞ (⺢n+ × ⺢m ). The 1-dimensional estimates (6.52) and (8.36) imply the following result: P ROPOSITION 9.3.4. For multi-indices α ∈ ⺞n0 and β ∈ ⺞m 0 , there are constants Cα,β so that  f L ∞ β . (9.164) |∂ xα ∂ y K tb,m f (x, y)| ≤ Cα,β |β| t |α|+ 2 √ √ √ If we let ( x) = ( x 1 , . . . , x n ), then we also have √  f L ∞ β |( x)α ∂ xα ∂ y K tb,m f (x, y)| ≤ Cα,β |α|+|β| . t 2

(9.165)

This proposition follows easily from (6.52), (6.53) and (8.36). We leave the details to the interested reader. 9.4 THE RESOLVENT OPERATOR We close this section by stating a proposition summarizing the properties of the resolk,γ vent operator, R(μ), as an operator on the H¨older spaces ᏯW F (⺢n+ ×⺢m ). As contrasted with the case of the heat equation, we do not need to assume that the data has compact support in the x-variables to prove estimates when k > 0. As before we use Propoβ sition 4.2.4 to commute differential operators of the form ∂ xα ∂ y past the heat kernel. j Since we are only proving spatial estimates we do not need to commute ∂t past the kernel, and hence do not encounter the need for weighted estimates on the data. P ROPOSITION 9.4.1. The resolvent operator R(μ) is analytic in the complement of (−∞, 0], and is given by the integral in (9.13) provided that Re(μeiθ ) > 0. For α ∈ (0, π], there are constants C b,α so that if α − π ≤ arg μ ≤ π − α,

(9.166)

˙ 0 (⺢n+ × ⺢m ) we have then for f ∈ Ꮿ R(μ) f  L ∞ ≤

C b,α  f L ∞ , |μ|

(9.167)

with C b,π = 1. Moreover, for 0 < γ < 1, and k ∈ ⺞0 , there are constants Ck,b,α,γ so k,γ that if f ∈ ᏯW F (⺢n+ × ⺢m ), then R(μ) f W F,k,γ ≤

Ck,b,α,γ  f W F,k,γ . |μ|

(9.168)

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We also have the estimates

/

∇ y R(μ) f W F,k,γ ≤ Ck,α

1 γ

|μ| 2 /

√  x · ∇ x R(μ) f W F,0,γ ≤ Ck,α ψ(x)x · ∇ x R(μ) f W F,k,γ ≤

√

+ 1 γ

 f W F,k,γ ,

γ +1 2

|μ|

|μ| 2 /

X Ck,α

0

1

+ 1 γ

|μ| 2

0

1 |μ|

(9.169)

γ +1 2

 f W F,0,γ ,

0 1  f W F,k,γ . + |μ|

(9.170)

Here ψ(x) is a smooth function with |∇ψ(x)| ≤ 1 and supp ψ ⊂ [0, X ]n . k,γ If for a k ∈ ⺞0 , and 0 < γ < 1, f ∈ ᏯW F (⺢n+ × ⺢m ), then k,2+γ

R(μ) f ∈ ᏯW F (⺢n+ × ⺢m ), and we have

(μ − L b,m )R(μ) f = f.

(9.171)

0,2+γ

If f ∈ ᏯW F (⺢n+ × ⺢m ), then R(μ)(μ − L b,m ) f = f.

(9.172)

There are constants C b,k,α so that, for μ satisfying (7.185), we have  1 R(μ) f W F,k,2+γ ≤ C b,k,α 1 +  f W F,k,γ . |μ|

(9.173)

For any B > 0, these constants are uniformly bounded for 0 ≤ b < B1. P ROOF. The proof of this proposition, with k = 0, is almost immediate from the proof of Proposition 9.2.1. If κtb,m (x, z; y, w) denotes the heat kernel for L b,m , then this proof estimated the integrals t   κsb,m (x, z; y, w)g(z; w, t − s)d zdwds. 0

⺢n+

(9.174)

⺢m

The only estimate on g that is used in these arguments is |g(x 1 , y1 , t) − g(x 2 , y2 , t)| ≤ gW F,0,γ ρ((x 1 , y1 ), (x 2 , y2 ))γ .

(9.175)

To prove the present theorem we consider integrals of the form ∞   0

⺢n+

⺢m

b,m −se κse iθ (x, z; y, w) f (z; w)e

iφ μ

d zdweiφ ds,

(9.176)

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171

where μ = |μ|e−iψ , with |ψ| ≤ π − α, for an 0 < α ≤ π. We can choose |φ| < π−α 2 so that α π α π (9.177) − + 0 by using the formulæ in Proposition 4.2.2 to commute β the derivatives ∂ xα ∂ y through the heat kernel and onto the data, f. As noted above, in this context there is no need for time derivatives, hence we do not need to assume that f has compact support in the x-variables. Next observe that we can use the single variable estimates in formulæ analogous to those in (7.105) to show that |∂x j R(μ) f (x, y)| ≤ C b,α

 f W F,0,γ |μ|

γ 2

 f W F,0,γ √ and | x j ∂x j R(μ) f (x, y)| ≤ C b,α , γ +1 |μ| 2 (9.181)

and |∂ yl R(μ) f (x, y)| ≤ C b,α

 f W F,0,γ |μ|

γ +1 2

.

(9.182)

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The simple 1-dimensional estimates also suffice to prove that |x j ∂x2j R(μ) f (x, y)| ≤ C b,α |x j ∂x2j

 f W F,0,γ γ

|μ| 2

(9.183)

γ 2

R(μ) f (x, y)| ≤ C b,α  f W F,0,γ x j ,

and |∂ y2l R(μ) f (x, y)| ≤ C b,α

 f W F,0,γ γ

|μ| 2

.

(9.184)

Using a formula like that in (7.158) we can show that  f W F,0,γ √ | x i x j ∂xi ∂x j R(μ) f (x, y)| ≤ C b,α γ |μ| 2  f W F,0,γ |∂ yl ∂ ym R(μ) f (x, y)| ≤ C b,α . γ |μ| 2

(9.185)

Finally, using an expression like that in (9.64), we can show that  f W F,0,γ √ . | x i ∂xi ∂ yl R(μ) f (x, y)| ≤ C b,α γ |μ| 2

(9.186)

This establishes that R(μ) f ∈ Ꮿ2W F (⺢n+ × ⺢m ). We can now use a standard integration by parts argument, see (4.52)–(4.53), to show that (μ − L b,m )R(μ) f = f. (9.187) 0,γ

As in the 1-d case, we demonstrate below that, if f ∈ ᏯW F (⺢n+ × ⺢m ), then 0,2+γ

R(μ) f ∈ ᏯW F (⺢n+ × ⺢m ), and therefore, by the open mapping theorem, to show that R(μ) is also a left inverse it suffices to show that the null-space (μ − L b,m ) is trivial for μ ∈ S0 . If there were a non-trivial eigenfunction f, for such a μ, then eμt f would solve the Cauchy problem, and grow exponentially with t. As this contradicts (9.14), it follows that this null-space 0,2+γ is empty. We can therefore conclude that if f ∈ ᏯW F (⺢n+ × ⺢m ), and μ ∈ S0 , then R(μ)(μ − L b,m ) f = f.

(9.188)

As the left-hand side is analytic in μ ∈ ⺓ \ (−∞, 0], it follows that this relation also holds in the complement of the negative real axis. It remains to establish the H¨older continuity of the first and second derivatives of R(μ) f. Equation (9.184) implies that if y and y differ only in the lth coordinate then |∂ yl R(μ) f (x, y) − ∂ yl R(μ) f (x, y )| ≤ C b,α

 f W F,0,γ γ

|μ| 2

|yl − yl |.

(9.189)

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As observed earlier, if y − y is supported in the mth place and m = l, then Lemmas 8.2.2 and 8.2.5 show that we have the bound: |∂ yl R(μ) f (x, y) − ∂ yl R(μ) f (x, y )| ∞ ≤ C b,α  f W F,0,γ

s 0 ∞

≤ C b,α  f W F,0,γ



γ −1 2

 | |ym −ym √ s

1+





 | |ym −ym √ s

 e− cos θ|μ|s ds (9.190)

γ

s 2 −1 |ym − ym |e− cos θ|μ|s ds,

0

from which it follows easily that, for ρe ( y, y ) < 1, we have: |∂ yl R(μ) f (x, y) − ∂ yl R(μ) f (x, y )| ≤ C b,α

 f W F,0,γ [ρe ( y, y )]γ γ

|μ| 2

.

(9.191)

To complete the estimate of the first y-derivatives, we need to bound the difference |∂ yl R(μ) f (x, y) − ∂ yl R(μ) f (x  , y)|. We can assume that x − x  is supported in the first slot. The derivation of (9.65) implies that ∞ |∂x1 ∂ yl R(μ) f (x, y)| ≤ C b,α  f W F,0,γ 0

γ

s 2 −1 e− cos θ|μ|s ds . √ √ x1 + s

(9.192)

Splitting the integral into the part from 0 to x 1 and the rest we see that  f W F,0,γ |∂x1 ∂ yl R(μ) f (x, y)| ≤ C b,α √ γ , x 1 |μ| 2

(9.193)

which upon integration implies that |∂ yl R(μ) f (x, y) − ∂ yl R(μ) f (x  , y)| ≤ C b,α

 f W F,0,γ ρs (x, x  ) γ

|μ| 2

.

(9.194)

As we only require an estimate when ρs (x, x  ) < 1, this shows that |∂ yl R(μ) f (x, y) − ∂ yl R(μ) f (x  , y)| ≤ C b,α

 f W F,0,γ ρs (x, x  )γ γ

|μ| 2

.

(9.195)

By commuting spatial derivatives past the kernel using Proposition 4.2.2, we obtain (9.169) for all k ∈ ⺞0 . √ To obtain an estimate for  x j ∂x j R(μ) f W F,0,γ , we integrate the estimate of √ | x j ∂x j v(·, t)| afforded by Lemma 6.1.10 to conclude that γ

√ | x j ∂x j R(μ) f (x, y)| ≤ Cα

x j2

γ

|μ| 2

.

(9.196)

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As usual this implies that for 0 < c < 1, there is a C so that if x j < cx j and x − x  is supported in the j th place, then +  ++γ +√ + + x − x j +  + j +√ + . (9.197) + x j ∂x j R(μ) f (x, y) − x j ∂x j R(μ) f (x  , y)+ ≤ C γ |μ| 2 To obtain a similar estimate when cx j < x j < x j , we use Lemma A.142. Integrating the estimates in (9.185) and (9.186) we easily complete the proof that, for ρ((x, y), (x  , y )) < 1, we have  |ρ((x, y), (x  , y ))|γ √ , (9.198) | x j ∂x j R(μ) f (x, y) − x j ∂x j R(μ) f (x  , y )| ≤ C γ |μ| 2 finishing the proof of the first estimate in (9.170). The second estimate is proved by using Proposition 4.2.2 to commute derivatives past the heat kernel, and applying the first estimate in (9.170) and the Leibniz formula, (5.62), to terms of the form: √ √ β ψ(x) x j x j ∂x j R(μ)∂ xα ∂ y f. √ This explains the appearance of the X . All other terms are of lower order and easily estimated. This completes the proof of the estimates in (9.170). We still need to establish the H¨older continuity of the unscaled first derivatives in the x-variables. By integrating the second estimate in (9.183) we can show that if x and x  differ only in the j th coordinate, then  √ |∂x j R(μ) f (x, y) − ∂x j R(μ) f (x  , y)| ≤ C b,α  f W F,0,γ | x j − x j |γ . (9.199) To do the off-diagonal cases, we assume that x − x  is supported in the mth slot, with j = m. If x m = 0, then by arguing as in (7.124) we see that |∂x j R(μ) f (x, y) − ∂x j R(μ) f (x  , y)|C b,α  f W F,0,γ ∞ γ x m /s e− cos θ|μ|s ds, × s 2 −1 1 + x m /s

(9.200)

0

γ

which is easily seen to be bounded by C b,α  f W F,0,γ x m2 . Applying (6.72) we see that, if 0 < c < 1, then there is a constant C b,α so that for cx m > x m , we have  √ |∂x j R(μ) f (x, y) − ∂x j R(μ) f (x  , y)| ≤ C b,α  f W F,0,γ | x m − x m |γ . (9.201) We are therefore reduced to considering cx m < x m < x m , for a c < 1. If we use (6.33) it follows that |∂x j R(μ) f (x, y) − ∂x j R(μ) f (x  , y)| ≤ C b,α  f W F,0,γ × ⎛ √ √  ⎞ xm − xm ∞ √ γ s ⎜ ⎟ −1 s2 ⎝ √  ⎠ e− cos θ|μ|s ds. √ xm − xm √ 1+ 0 s

(9.202)

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 √ We split this into an integral from 0 to ( x m − x m )2 and the rest, to obtain: |∂x j R(μ) f (x, y) − ∂x j R(μ) f (x  , y)| ≤ C b,α  f W F,0,γ × ⎤ ⎡ √ √  )2 ( xm − xm (√  )  ∞ ⎥ ⎢ γ γ x m − x m ⎢ 2 −1 + 2 −1 e− cos θ|μ|s ds ⎥ s s √ ⎦ . (9.203) ⎣ s √ √ 0  2 ( xm −

xm )

Performing these integrals shows that this term is also estimated by  √ |∂x j R(μ) f (x, y) − ∂x j R(μ) f (x  , y)| ≤ C b,α  f W F,0,γ | x m − x m |γ . y )|,

(9.204)

y

We now estimate |∂x j R(μ) f (x, y) − ∂x j R(μ) f (x, with y − supported at a single index, which we label 1. Arguing exactly as in the derivation of (9.61), we see that |∂x j R(μ) f (x, y) − ∂x j R(μ) f (x, y )| ∞ ≤ C b,α  f W F,0,γ

⎛ γ ⎜ s 2 −1 ⎝

0

|y12 −y11 | √ s

1+

|y12 −y11 | √ s

⎞ ⎟ − cos θ|μ|s ds. ⎠e

(9.205)

The same argument used to prove (9.204) can be employed to show that |∂x j R(μ) f (x, y) − ∂x j R(μ) f (x, y )| ≤ C b,α  f W F,0,γ ρe ( y, y )γ .

(9.206)

All that remains is to prove the H¨older continuity of the second derivatives. Using the second estimate in (9.183) and the argument used in the proof of (6.162) we can show that |x j ∂x2j R(μ) f (x j , x j , x j , y) − y j ∂x2j R(μ) f (x j , y j , x j , y)| ≤ √ √ Cb,α  f W F,0,γ | x j − y j |γ . (9.207) Arguing as in the derivation of (7.142), we see that |x j ∂x2j R(μ) f (x m , x m , x m , y) − x j ∂x2j R(μ)(x m , 0, x m , y)| ≤ ∞ Cb,α  f W F,0,γ

γ

s 2 −1 e− cos θ|μ|s

0

x m /s ds. 1 + x m /s

(9.208)

γ

As before the integral is bounded by a constant times x m2 , which suffices to prove the H¨older estimate for x m < cx m , for a fixed 0 < c < 1. If we fix such a c, then for cx m < x m < x m , the argument leading to (7.143) gives |x j ∂x2j R(μ) f (x m , x m , x m , y) − x j ∂x2j R(μ) f (x m , x m , x m , y)| ≤ √  √ xm − xm ∞ √ γ s Cb,α  f W F,0,γ s 2 −1 √  e− cos θ|μ|s ds. √ xm − xm √ 1+ 0 s

(9.209)

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The argument used to estimate the integral in (9.202) applies to show that |x j ∂x2j R(μ) f (x m , x m , x m , y) − x j ∂x2j R(μ) f (x m , x m , x m , y)| ≤  √ Cb,α  f W F,0,γ | x m − x m |γ .

(9.210)

To estimate |x j ∂x2j R(μ) f (x, y) − x j ∂x2j R(μ) f (x, y )|, we argue as in the derivation of (9.102) to see that, if y − y is supported in the first argument, then |x j ∂x2j R(μ) f (x, y) − x j ∂x2j R(μ) f (x, y )| ⎞ ⎛ |y12 −y11 | ∞ √ γ −1 √ x1s 2 s ⎟ − cos θ|μ|s ⎜ √ ⎝ ≤ Cb,α  f W F,0,γ ds. √ ⎠e x 1 + s 1 + |y12√−y11 | 0 s

(9.211)

As before, this integral is estimated by a constant times |y12 − y11 |γ , completing the proof that |x j ∂x2j R(μ) f (x, y) − x j ∂x2j R(μ) f (x  , y )| ≤ Cb,α  f W F,0,γ [ρ((x, y), (x  , y ))]γ .

(9.212)

The argument between (7.155) and (7.164) applies with small modifications (largely by replacing the upper limit in the s-integrations with infinity and the measure ds with e− cos θ|μ|s ds) to show that, with j = k, we have:  √ | x j x k ∂x j ∂xk R(μ) f (x, y) − x j x k ∂x j ∂xk R(μ) f (x  , y)| ≤ Cb,α  f W F,0,γ [ρs (x, x  )]γ .

(9.213)

Similarly, the derivation of the estimate in (9.97)–(9.98) shows that, if y − y is supported in the first slot, then √ √ | x j x k ∂x j ∂xk R(μ) f (x, y) − x j x k ∂x j ∂xk R(μ) f (x, y )| ≤ ⎛ ⎞ |y12 −y11 | ∞ √ γ s ⎜ ⎟ − cos θ|μ|s ds, Cb,α  f W F,0,γ s 2 −1 ⎝ ⎠e |y12 −y11 | √ 1 + 0 s

(9.214)

which completes the proof that  √ | x j x k ∂x j ∂xk R(μ) f (x, y) − x j x k ∂x j ∂xk R(μ) f (x  , y )| ≤ Cb,α  f W F,0,γ [ρ((x, y), (x  , y ))]γ .

(9.215)

As before we can use the analogous estimates for the Euclidean kernel to show that |∂ y j ∂ yk R(μ) f (x, y) − ∂ y j ∂ yk R(μ) f (x, y )| ≤ Cb,α  f W F,0,γ [ρe ( y, y )]γ .

(9.216)

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The argument between (9.104) and (9.110) applies with the usual small changes to show that |∂ y j ∂ yk R(μ) f (x, y) − ∂ y j ∂ yk R(μ) f (x  , y)| ≤ Cb,α  f W F,0,γ [ρs (x, x  )]γ . (9.217)

√ To estimate the mixed derivatives x j ∂x j ∂ yk R(μ) f (x, y), we slightly modify the argument between (9.111) and (9.122). In each case we are reduced to estimating an integral of one of the following two forms: ∞ 0

γ

s 2 −1

De− cos θ|μ|s ds or √ s+D

∞

γ

s 2 −1

0

De− cos θ|μ|s ds . s+D

(9.218)

γ

The first integral is estimated by Cγ D γ and the second by Cγ D 2 . Using these estimates we complete the proof that  √ | x j ∂x j ∂ yk R(μ) f (x, y) − x j ∂x j ∂ yk R(μ) f (x  , y )| ≤ Cb,α  f W F,0,γ [ρ((x, y), (x  , y ))]γ . (9.219) This completes the k = 0 case for b > 0. The constants are again uniformly bounded for 0 < b ≤ B1, and so we can apply Proposition 5.2.8 to extend these results to 0 ≤ b. Finally, to treat k > 0, we use Proposition 4.2.2 to commute the spatial derivatives past the heat kernel and follow the argument above to establish this theorem for arbitrary k ∈ ⺞. The only terms that require additional consideration are contributions to the left-hand side of (9.170) from terms of the form: β

∂ xα ∂ y x j ∂x j R(μ) f W F,0,γ , for f ∈ ᏯW F , k,γ

(9.220)

where α j = 0. In all other cases β

[∂ xα ∂ y , x j ∂x j ] = 0 and the estimate follows easily using Proposition 4.2.2. If α j = 0, then β

β

β

∂ xα ∂ y x j ∂x j = x j ∂x j ∂ xα ∂ y + α j ∂ xα ∂ y ,

(9.221)

which shows that β

β

∂ xα ∂ y x j ∂x j R(μ) f W F,0,γ ≤ x j ∂x j ∂ xα ∂ y R(μ) f W F,0,γ + R(μ) f W F,k,γ . (9.222) It now follows from Proposition 4.2.2 and the k = 0 case that this term is bounded by / 0 1 1  f W F,k,γ . (9.223) ≤ Cα γ + |μ| |μ| 2 This completes the proof of the proposition.



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Part III

Analysis of Generalized Kimura Diffusions

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Chapter Ten Existence of Solutions We now return to the principal goal of this monograph, the analysis of a generalized Kimura diffusion operator, L, defined on a manifold with corners, P. The estimates proved in the previous chapters for the solutions to model problems, along with the adapted local coordinates introduced in Chapter 2.2, allow the use of the Schauder method to prove existence of solutions to the inhomogeneous problem (∂t − L)w = g in P × (0, T ) with w(x, 0) = f.

(10.1)

Ultimately we will show that if k,2+γ

k,γ

f ∈ ᏯW F (P) and g ∈ ᏯW F (P × [0, T ]),

(10.2)

k,2+γ

then the unique solution w ∈ ᏯW F (P × [0, T ]). In this chapter we prove the basic existence result: T HEOREM 10.0.2. For k ∈ ⺞0 and 0 < γ < 1, if the data f, g satisfy (10.2), then k,2+γ equation (10.1) has a unique solution w ∈ ᏯW F (P × [0, T ]). There are constants Ck,γ , Mk,γ so that wW F,k,2+γ ,T ≤ Mk,γ exp(Ck,γ T )[gW F,k,γ ,T +  f W F,k,2+γ ].

(10.3)

k,2+γ

Remark. The hypothesis that f ∈ ᏯW F (P) is not what one should expect: the k,γ result suggested by the non-degenerate case would be that for f ∈ ᏯW F (P), there is k,2+γ k,γ a solution in ᏯW F (P × (0, ∞)) ∩ ᏯW F (P × [0, ∞)). This is true for the model problems. For basic applications to probability theory, Theorem 10.0.2 suffices. We resolve this issue, according to expectations, in Chapter 11. As we have done before, we write the solution w = v + u, where v solves the homogeneous Cauchy problem with v(x, 0) = f (x) and u solves the inhomogeneous problem with u(x, 0) = 0. Each part is estimated separately. In the early sections of this chapter we treat the k = 0 case, returning to the problem of higher regularity at the end. The issues with the support of the data that arose in the analysis of higher regularity for the model problems does not arise in the present context. This is because whenever a model solution operator appears as part of a parametrix it is always multiplied on the right by a smooth compactly supported function. Hence it can be regarded as acting on data with fixed compact support. 181

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With k = 0, we begin by proving the existence of u for t ∈ [0, T0 ], where T0 > 0 is independent of u. A similar argument establishes the existence of v. Using these arguments together, we obtain existence up to time T, and the estimate given in the theorem in the k = 0 case. Before delving into the details of the argument, we first give definitions for the WF-H¨older spaces on a general compact manifold with corners, and then a brief account of the steps involved in the existence proof. ¨ 10.1 WF-HOLDER SPACES ON A MANIFOLD WITH CORNERS k,γ

k,γ

We now give precise definitions for various function spaces, ᏯW F (P), ᏯW F (P×[0, T ]), etc. which we need to use. For the (0, γ )-case we could use an intrinsic definition, using the singular, incomplete metric, gW F , determined by the principal symbol of L, to define a distance function, d W F (x, y). We could then define the global (0, γ )-WF semi-norm by setting 1

[| f ]|W F,0,γ = sup

x = y

| f ( p1 ) − f ( p2 )| , d W F ( p 1 , p 2 )γ

(10.4)

0,γ

and a norm on ᏯW F (P) by letting  f W F,0,γ =  f  L ∞ ( P) + |[ f ]|W F,0,γ .

(10.5)

For computations it is easier to build the global norms out of locally defined norms. By Proposition 2.2.3, there are coordinate charts covering a neighborhood of b P in which the operator L assumes a simple normal form. At a point q ∈ b P of codimension n this coordinate system (x 1 , . . . , x n ; y1, . . . , ym ) is parametrized by a subset of the form l l C n,m (l) = [0, l 2 )n × (− , )m , (10.6) 2 2 where q ↔ (0n ; 0m ). Let U denote the open set centered at q covered by this coordinate patch and φ : C n,m (l) → U, the coordinate map. We call this a normal cubic coordinate or NCC patch centered at q. The parameter domain, C n,m (l), is called a “positive cube” of side length l in ⺢n+ × ⺢m . In these coordinates the operator L takes the form L=

n 

x i ∂x2i +

i=1





 ckl (x, y)∂ yk ∂ yl +

1≤k,l≤m

n 

bi (x, y)∂xi +

i=1

x i x j ai j (x, y)∂xi ∂x j +

1≤i = j ≤n

n  m 

x i bil (x, y)∂xi ∂ yl +

i=1 l=1

m 

dl (x, y)∂ yl .

(10.7)

l=1

The principal part of L at q is given by p

Lq =

n  i=1

x i ∂x2i +

 1≤k,l≤m

 ckl (0n , 0m )∂ yk ∂ yl +

n  i=1

bi (0n , 0m )∂xi .

(10.8)

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 (0 , 0 ) is positive definite and the coefficients {b (0 , 0 )} are nonThe matrix ckl n m i n m p negative. The estimates in the previous chapter show that L − L q is, in a precise sense, a residual term. If ψ ∈ Ꮿ∞ c (U ), and f is defined in U, then we can use the local definitions of the various WF norms to define the local WF norms:

ψ f U W F,k,γ = (ψ f ) ◦ φW F,k,γ

(10.9)

ψ f U W F,k,2+γ = (ψ f ) ◦ φW F,k,2+γ .

If g is defined in U ×[0, T ] then we similarly define the local (in space and time) norm: ψgU W F,k,γ ,T = (ψg)(φ, ·)W F,k,γ ,T

(10.10)

ψgU W F,k,2+γ ,T = (ψg)(φ, ·)W F,k,2+γ ,T .

D EFINITION 10.1.1. Let W = {(W j , φ j ) : j = 1, . . . , K } be a cover of b P by NCC charts, W0 ⊂⊂ int P, covering P \ ∪ Kj=1 W j , and let {ϕ j : j = 0, . . . , K } k,γ

be a partition of unity subordinate to this cover. A function f ∈ ᏯW F (P) provided k,γ k,γ (ϕ j f ) ◦ φ j ∈ ᏯW F (W j ), for each j. We define a global norm on ᏯW F (P) by setting  f W F,k,γ =

K 

W

(ϕ j f ) ◦ φ j W jF,k,γ .

(10.11)

j =0 k,2+γ

k,γ

There are analogous definitions for the spaces ᏯW F (P), ᏯW F (P × [0, T ]), and k,2+γ ᏯW F (P × [0, T ]). The corresponding norms are denoted by  f W F,k,2+γ , gW F,k,γ ,T , gW F,k,2+γ ,T . It is straightforward to show that different NCC covers define equivalent norms and therefore, in all cases, the topological vector spaces do not depend on the choice of NCC cover. Once we have fixed such a cover, then the definitions of the norm on 0,γ ᏯW F (P) in (10.5) and (10.11) are also equivalent. In fact, if U is an NCC patch of codimension n, with local coordinates (x 1 , . . . , x n ; y1 , . . . , ym ), then there is a constant C so that, for (x 1 , y1 ), (x 2 , y2 ) ∈ U, we have Cd W F ((x 1 , y1 ), (x 2 , y2 )) ≤ [ρs (x 1 , x 2 ) + ρe ( y1 , y2 )] ≤ C −1 dW F ((x 1 , y1 ), (x 2 , y2 )). (10.12) In the remainder of this chapter we fix the cover W. 10.1.1 Properties of WF-H¨older Spaces The details of the construction of the parametrix rely on some general results about the 0,γ 0,γ local function spaces ᏯW F (⺢n+ × ⺢m ), ᏯW F (⺢n+ × ⺢m × [0, T ]), for which it is useful

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to recall the local semi-norms [| f ]|W F,0,γ =

[|g|]W F,0,γ =

| f (x 1 , y1 ) − f (x 2 , y2 )| , 1 2 1 2 γ (x 1 , y 1 ) =(x 2 , y2 ) [ρs (x , x ) + ρe ( y , y )] 1

sup

|g(x 1 , y1 , t 1 ) − g(x 2 , y2 , t 2 )|  , 1 2 1 2 2 1 γ (x 1 , y1 ,t 1 ) =(x 2 , y2 ,t 2 ) [ρs (x , x ) + ρe ( y , y ) + |t − t |] 1

sup

(10.13)

(10.14)

and the Leibniz formula: 0,γ

0,γ

L EMMA 10.1.2. Suppose that f, g ∈ ᏯW F (⺢n+ × ⺢m ) or ᏯW F (⺢n+ × ⺢m × [0, T ]). The semi-norm of the product f g satisfies the estimate: [| f g|]W F,0,γ ≤  f  L ∞ [|g|]W F,0,γ + g L ∞ |[ f ]|W F,0,γ .

(10.15)

P ROOF. These estimates follow easily from the observation that, with w j = (x j , y j ), or w j = (x j , y j , t j ), j = 1, 2, we have | f (w1 )g(w1 ) − f (w2 )g(w2 )| ≤ [ρ(w1 , w 2 )]γ |[ f (w1 ) − f (w2 )]g(w1 )| + | f (w 2 )[g(w1 ) − g(w2 )]| , [ρ(w1 , w2 )]γ from which the assertions of the lemma are immediate.

(10.16) 

We also have a result about the behavior of WF norms under the scaling of cutoff functions. L EMMA 10.1.3. Suppose that f ∈ Ꮿ1c (⺢n+ × ⺢m ) has support in the positive cube [0, l 2 ]n × [−l, l]m . If  > 0 and we define  x y , (10.17) f  (x, y) = f , 2  then there is a constant Cl depending on the support of f so that  f  W F,0,γ ≤ Cl [ −γ + 1] f Ꮿ1 .

(10.18)

P ROOF. First observe that  f   L ∞ =  f  L ∞ ; so we only need to estimate [| f  ]|W F,0,γ . This estimate follows from the observation that  x y   x y  1 2 ρ = ρ((x 1 , y1 ), (x 2 , y2 )), , 1 , 2, 2 (10.19)  2   and therefore

+    + + x 2 y2 x 1 y1 + f − f , , + 2 2 | f  (x 2 , y2 ) − f  (x 1 , y1 )|   +   −γ      =  γ. y y [ρ((x 1 , y1 ), (x 2 , y2 ))]γ ρ x 21 , 1 , x 22 , 2

(10.20)

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Letting w j = x j / 2 , z j = y j /, for j = 1, 2, this becomes: | f (w2 , z 2 ) − f (w1 , z 1 )| | f  (x 2 , y2 ) − f  (x 1 , y1 )| =  −γ [ρ((x 1 , y1 ), (x 2 , y2 ))]γ [ρ((w2 , z 2 ), (w1 , z 1 ))]γ ∇ f  L ∞ [|w2 − w1 | + |z 2 − z 1 |] ≤  −γ [ρ((w2 , z 2 ), (w1 , z 1 ))]γ

(10.21)

where we used the mean value theorem on the right-hand side of (10.21). The second line in (10.21) is estimated by / 0 n  √ √ √ √  −γ ∇ f  L ∞ |z 2 − z 1 |1−γ + | w2l − w1l |1−γ | w2l + w1l | . (10.22) l=1

Taking the supremum of the quantity in the brackets in (10.22) for pairs (w1 , z 1 ), (w2 , z 2 ) lying in [0, 4l 2 ]n × [−2l, 2l]m , shows that there is a constant Cl , so that for such pairs: | f  (x 2 , y2 ) − f  (x 1 , y1 )| ≤ Cl  −γ ∇ f  L ∞ . (10.23) [ρ((x 1 , y1 ), (x 2 , y2 ))]γ This covers the case where both (x 1 , y1 ) and (x 2 , y2 ) lie in certain neighborhood of the supp f  . If neither point lies in supp f, then the numerator is zero. Hence the only case remaining is when (w1 , z 1 ) ∈ [0, l 2 ]n ×[−l, l]m , and (w2 , z 2 ) ∈ / [0, 4l 2 ]n ×[−2l, 2l]m . In this case the denominator in the first line of (10.21) is bounded below by l γ , and the numerator is bounded above by 2 f  L ∞ , which completes the proof of the lemma.  L EMMA 10.1.4. Suppose that f ∈ Ꮿ1 (⺢n+ × ⺢m ) and a ∈ Ꮿ1 (⺢n+ × ⺢m ) with support in a positive cube of side length l, and a(0, 0) = 0. There is a constant C, depending on l and the dimension, so that, if m = 0, then we have [|a f  ]|W F,0,γ ≤ C f Ꮿ1 aᏯ1  2−γ .

(10.24)

If m ≥ 1, then

(10.25) [|a f  ]|W F,0,γ ≤ C f Ꮿ1 aᏯ1  1−γ . √ √ √ If a is a Ꮿ1 -function of the variables ( x, y) = ( x 1 , . . . , x n , y1 , . . . , ym ), that is √ a(x, y) = A( x, y), where A ∈ Ꮿ1 (⺢n+ × ⺢m ),

(10.26)

then the estimate in (10.25) holds for a f  , with aᏯ1 replaced by AᏯ1 . P ROOF. We begin with the case m = 0. The triangle inequality shows that |a(x 1 ) f  (x 1 ) − a(x 2 ) f (x 2 )| ≤ [ρ(x 1 , x 2 )]γ |a(x 1 ) − a(x 2 )|| f  (x 1 )| |a(x 2 )|| f  (x 1 ) − f  (x 2 )| + . (10.27) [ρ(x 1 , x 2 )]γ [ρ(x 1 , x 2 )]γ

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We first assume that x 1 , x 2 ∈ supp f  . In this case the second term on the right-hand side of (10.27) is bounded by l 2 ∇a L ∞ [| f  ]|W F,0,γ ≤ Cl  2−γ aᏯ1  f Ꮿ1 ,

(10.28)

where we use Lemma 10.1.2 and (10.23) to bound |[ f ]|W F,0,γ , along with the fact that a(0) = 0. The first term is bounded by  f  L ∞ aᏯ1 |x 1 − x 2 | ≤ C 2−γ  f  L ∞ aᏯ1 . [ρ(x 1 , x 2 )]γ

(10.29)

This proves (10.24) when both x 1 , x 2 ∈ supp f  . Essentially the same argument applies if x 2 ∈ supp f , and x 1 / 2 ∈ [0, 4l 2]n \ supp f  , though only the second term on the right-hand side of (10.27) is non-zero. The final case we need to consider is x 2 ∈ supp f  , and x 1 / 2 ∈ / [0, 4l 2 ]n . For this case, the denominator in |a(x 2 )|| f  (x 1 ) − f  (x 2 )| [ρ(x 1 , x 2 )]γ

(10.30)

is bounded below by (l)−γ , and the numerator is bounded above by  2 aᏯ1  f  L ∞ ,

(10.31)

thus completing the proof of (10.24) in case m = 0. For the case m = 0, observe that |a(x1 , y1 ) f  (x 1 , y1 ) − a(x 2 , y2 ) f  (x 2 , y2 )| ≤ [ρ((x 1 , y1 ), (x 2 , y2 ))]γ |a(x 1 , y1 ) − a(x 2 , y2 )|| f  (x 1 , y1 )| + | f  (x 1 , y1 ) − f  (x 2 , y2 )||a(x2 , y2 )| . [ρ((x 1 , y1 ), (x 2 , y2 ))]γ (10.32) Note that supp f  ⊂ [0,  2l 2 ]n × [−l, l]m . If both points again belong to the supp f  , then the quantity on the right-hand side of (10.32) is bounded by (l)1−γ aᏯ1  f  L ∞ +  [| f  ]|W F,0,γ ≤ Cl  1−γ aᏯ1  f Ꮿ1 .

(10.33)

If now (x 2 , y2 ) ∈ supp f  , but (x 1 , y1 ) ∈ [0, 4 2l 2 ]n × [−2l, 2l]m \ supp f  , then only the second term on the right-hand side of (10.32) is non-zero; it is estimated by (l)aᏯ1 [| f ]|W F,0,γ ≤ Cl  1−γ aᏯ1  f Ꮿ1 .

(10.34)

/ [0, 4 2l 2 ]n × [−2l, 2l]m , then the Finally, if (x 2 , y2 ) ∈ supp f  , but (x 1 , y1 ) ∈ γ denominator is bounded below by (l) , and the numerator is bounded above by the product aᏯ1  f  L ∞ , which completes the proof in this case.

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√ The argument if a is a Ꮿ1 -function of ( x, y) is again quite similar. √ √ |A( x 1 , y1 ) f  (x 1 , y1 ) − A( x 2 , y2 ) f  (x 2 , y2 )| ≤ |[ρ((x 1 , y1 ), (x 2 , y2 ))]|γ √ √ √ |A( x 1 , y1 ) − A( x 2 , y2 )|| f  (x 1 , y1 )| + | A( x 2 , y2 )|| f  (x 1 , y1 ) − f (x 2 , y2 )| . |[ρ((x 1 , y1 ), (x 2 , y2 ))]|γ (10.35) We observe that

⎤ ⎡ n    |A( x 1 , y1 ) − A( x 2 , y2 )| ≤ ∇ A L ∞ ⎣ | x 1j − x 2j | + | y1 − y2 |⎦ 



j =1

(10.36)

= ∇ A L ∞ ρ((x 1 , y1 ), (x 2 , y2 )). If both points are in supp f  , then the right-hand side of (10.35) is bounded by . (10.37) ∇ A L ∞ [ρ((x 1 , y1 ), (x 2 , y2 ))]1−γ  f  L ∞ +  [| f  ]|W F,0,γ . Since both points are in supp f  , Lemma 10.1.2 shows that this is bounded by C 1−γ AᏯ1  f Ꮿ2 .

(10.38) 

The other cases follow similarly. 10.2 OVERVIEW OF THE PROOF

The domain P is assumed to be a manifold with corners of dimension N > 1. The boundary of P is a stratified space with bP =

M 7

j,

(10.39)

j =1

where  j is the (open) stratum of codimension j boundary points. From the definition of manifold with corners it follows that k =

M 7

j.

(10.40)

j =k

To prove the existence of a solution to the equation (10.1), we use an induction over M the maximal codimension of a stratum of b P. The argument begins by assuming that P is a manifold with boundary, i.e., M = 1. Using the estimates proved in the previous chapter we can easily show that there is a function ϕ ∈ Ꮿ∞ c (P), equal to 1 in a neighborhood of b P and an operator

t : Ꮿk,γ (P × [0, T ]) → Ꮿk,2+γ (P × [0, T ]), Q b WF WF

(10.41)

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so that where

t g = ϕg + (E 0,t + E 1,t )g, (∂t − L) Q b b b k,γ

k,γ

E b0,t , E b1,t : ᏯW F (P × [0, T ]) → ᏯW F (P × [0, T ])

(10.42) (10.43)

are bounded and E b1,t is a compact operator on this space, which tends to zero in norm as T tends to zero. If k = 0, then we can arrange for E b0,t to have norm as small as we please. Let U be a neighborhood of b P so that bU ∩ int P is a smooth hypersurface in P, and U ⊂⊂ ϕ −1 (1). The subset PU = P ∩ U c is a smooth compact manifold with boundary, and L  PU is a non-degenerate elliptic operator. We can double PU across U , which is a manifold without boundary. The operator L its boundary to obtain P U . The classical can be extended to a classically elliptic operator  L defined on all of P theory of non-degenerate parabolic equations on compact manifolds, without boundary, t [(1−ϕ)g] to the inhomogeneous applies to construct an exact solution operator u i = Q equation: U × [0, T ] L)u i = (1 − ϕ) g in P (∂t −  U . with u i ( p, 0) = 0, p ∈ P

(10.44)

k,γ  k,2+γ  This operator defines bounded maps from ᏯW F ( P U × [0, T ]) → Ꮿ W F ( P U × [0, T ]),  for any 0 < γ < 1 and k ∈ ⺞0 . Of course, in PU these spaces are equivalent to the U × [0, T ]) and Ꮿk+2,γ ( P U × [0, T ]), respectively. classical heat H¨older spaces Ꮿk,γ ( P To complete the construction when M = 1, choose ψ ∈ Ꮿ∞ c (PU ) so that ψ ≡ 1 on a neighborhood of the support of (1 − ϕ), and set

t g = Q

t g + Q

t g Q b i

(10.45)

where interior parametrix is defined to be t [(1 − ϕ)g].

ti = ψ Q Q

(10.46)

t [(1 − ϕ)g] are extended by zero to all of Here it is understood that (1 − ϕ)g and ψ Q  PU and P, respectively. Applying the operator gives

where

t g = g + (E 0,t + E 1,t )g + E ∞,t g, (∂t − L) Q b b i

(10.47)

ti [(1 − ϕ)g]. E i∞,t g = [ψ, L] Q

(10.48)

Since ψ ≡ 1 on a neighborhood of the support of (1 − ϕ) it follows from classical results that E i∞,t is a smoothing operator which tends to zero exponentially as T → 0+ . More generally, assume by induction that E i∞,t is a compact operator tending to zero, k,γ as T0 → 0, in the operator norm defined by ᏯW F (P × [0, T0 ]). If T0 is sufficiently small, then the operator E b0,t + E b1,t + E i∞,t = E t : ᏯW F (P × [0, T0 ]) −→ ᏯW F (P × [0, T0 ]) 0,γ

0,γ

(10.49)

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has norm strictly less than 1, and therefore (Id +E t ) is invertible. Thus the operator

t (Id +E t )−1 ᏽt = Q

(10.50)

is a right inverse to (∂t − L) up to time T0 and is a bounded map 0,γ

0,2+γ

ᏽt : ᏯW F (P × [0, T0 ]) → ᏯW F (P × [0, T0 ]).

(10.51)

At the end of this chapter we use a result from [14] to show that the Neumann series k,γ for (Id +E t )−1 converges in the operator norm topology of ᏯW F (P × [0, T0 ]), for any k ∈ ⺞. To handle the case of higher codimension boundaries we use the following induction hypotheses: [Inhomogeneous Case:] Let P be any manifold with corners such that the maximal codimension of b P is less than or equal to M, and let L be a generalized Kimura diffusion on P. We assume that the solution operator ᏽt of the initial value problem (∂t − L)u = g with u(x, 0) = 0

(10.52)

exists and has the following properties: 1. For k ∈ ⺞0 , 0 < T, and 0 < γ < 1, the maps k,γ

k,2+γ

(10.53)

k,γ

(10.54)

ᏽt : ᏯW F (P × [0, T ]) −→ ᏯW F (P × [0, T ])

are bounded. The maps k,γ

ᏽt : ᏯW F (P × [0, T ]) −→ ᏯW F (P × [0, T ])

tend to zero in norm as T → 0+ . 2. Let ψ1 , ψ2 ∈ Ꮿ∞ (P) be such that dist(supp ψ1 , supp ψ2 ) > 0. Then the operator k,γ

k,2+γ

ψ1 ᏽt ψ2 : ᏯW F (P × [0, T ]) −→ ᏯW F (P × [0, T ])

(10.55)

is compact, and its norm tends to zero as T → 0+ . We call this the small time localization property. [Homogeneous Case:] We also assume the existence of a solution operator ᏽt0 for the homogeneous Cauchy problem: (∂t − L)v = v with v(x, 0) = f,

(10.56)

with the following properties: 1. For k ∈ ⺞0 , 0 < T, and 0 < γ < 1, the maps k,2+γ

k,2+γ

ᏽt0 : ᏯW F (P) −→ ᏯW F (P × [0, T ])

are bounded.

(10.57)

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2. As t → 0+ , for f ∈ ᏯW F (P), and 0 <  γ < γ , we have that lim ᏽt0 f − f W F,k, γ = 0.

t →0+

(10.58)

3. If ψ1 , ψ2 ∈ Ꮿ∞ (P) have dist(supp ψ1 , supp ψ2 ) > 0, then the operator k,2+γ

k,2+γ

ψ1 ᏽt0 ψ2 : ᏯW F (P × [0, T ]) −→ ᏯW F (P × [0, T ])

(10.59)

is compact and tends to zero in norm as T → 0+ . To carry out the induction step we require the following basic geometric result, which is related to Lemma 2.1.7: T HEOREM 10.2.1. Let P be a compact manifold with corners with maximal codimension of b P equal to M ≥ 1, and L a generalized Kimura diffusion operator on P. Suppose that b P = 1 ∪ · · · ∪  M , (10.60) where each  j is the boundary component of b P of codimension j , and let U ⊂ P  so that the be a neighborhood of  M . There exists a compact manifold with corners P  is M − 1, with a generalized Kimura diffusion operator  maximal codimension of b P L  The subset PU = P ∩ U c is diffeomorphic to a subset of P  under a map defined on P.  which carries L U = L  P∩U c to  L ( P∩U c ) .   Remark. Informally we say that (PU , L U ) is embedded into ( P, L). An example of the doubling construction is shown in Figure 10.1.

Figure 10.1: The doubling construction used to remove the maximal codimension component of b P when P is an equilateral triangle. The proof of Theorem 10.2.1 is given later in this chapter. To carry out the induction    is a manifold step, we use Theorem 10.2.1 to embed (PU , L U ) into ( P, L), where P

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with corners, of codimension at most M − 1. The induction hypothesis shows that t for the equation (∂t −   In the there is an exact solution operator Q L) u = g on P. i sequel we refer to this as the interior term, which explain the i subscript. In the context of inductive arguments over the maximal codimension of b P, we use the adjective “interior” to refer to the things coming from parts of P disjoint from the maximal codimensional part of b P. t , We use the codimension M model operators to build a boundary parametrix, Q b t in a neighborhood of  M . Arguing much as in the codimension 1 case, we can glue Q i t to obtain an operator to Q b

t : Ꮿk,γ (P × [0, T ]) → Ꮿk,2+γ (P × [0, T ]), Q WF WF so that t = Id +E t , with E t : Ꮿ (P × [0, T ]) → Ꮿ (P × [0, T ]). (10.61) (∂t − L) Q WF WF k,γ

k,γ

As before, if k = 0, then we can arrange to have the norm of the error term E t bounded by any fixed δ < 1, as T → 0. Thus, for some T0 > 0, we obtain the exact solution t (Id +E t )−1 ; this operator defines a bounded operator for (P, L) by setting ᏽt = Q map 0,γ 0,2+γ ᏽt : ᏯW F (P × [0, T0 ]) → ᏯW F (P × [0, T0 ]). (10.62) In Section 10.4 we give the detailed construction of a boundary parametrix for the maximal codimension stratum of the boundary. Combining this with the estimates in Section 10.3 we verify the induction hypothesis in the base case that M = 1 and also the inductive step itself, which completes the proof for the k = 0 case. The estimates with k > 0 are left for the end of this chapter. 10.3 THE INDUCTION ARGUMENT To complete the proof of the theorem we need only verify the induction hypothesis. Assume that P is a manifold with corners so that the maximal codimension of b P is M + 1, and that L is a generalized Kimura diffusion operator on P. Using the estimates in the previous chapters we show in Section 10.4 that there is a function ϕ ∈ Ꮿ∞ c (P), that equals 1 on a small neighborhood of  M+1 , and vanishes outside a slightly larger

t , with the mapping properties in (10.41), so that, for neighborhood, and an operator Q b 0,γ g ∈ ᏯW F (P × [0, T ]), we have:

tb g = ϕg + (E 0,t + E 1,t )g. (∂t − L) Q b b k,γ

(10.63)

Here E b0,t and E b1,t are bounded maps of ᏯW F (P × [0, T ]), for any k ∈ ⺞0 and 0 < γ < 1. Below we show that for any δ > 0 we can construct E b0,t so that its norm, acting 0,γ on ᏯW F (P × [0, T ]), is less than δ and E b1,t is a compact map of this space to itself, which tends to zero in norm as T → 0. At the end of the chapter this is verified for k,γ ᏯW F (P × [0, T ]), with k ∈ ⺞.

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Let U be a neighborhood of  M+1 so that U ⊂⊂ int ϕ −1 (1); set PU = P ∩ U c and L U = L  PU .

(10.64)

 of maximal codimenWe now apply Theorem 10.2.1 to find a manifold with corners P, sion M, and a generalized Kimura diffusion operator  L so that (PU , L U ) is embedded   t to into ( P, L). The induction hypothesis implies that there is a solution operator Q  with the desired mapping properties with respect to the equation (∂t −  L) u = g on P  As before, we choose ψ ∈ Ꮿ∞ the WF-H¨older spaces on P. c (PU ) so that ψ ≡ 1 on a neighborhood of the support of (1 − ϕ) and define t [(1 − ϕ)g],

t g = ψ Q Q i

(10.65)

 and ψ Q t [(1 − ϕ)g] by where it is understood that we extend (1 − ϕ)g by zero to P, zero to P.

t = Q

t + Q

t , then If we let Q b i

t g = g + (E 0,t + E 1,t + E t )g, (∂t − L) Q i b b

(10.66)

t [(1 − ϕ)g]. The support of the kernel of E t is a where, as before, E it g = [ψ, L] Q i positive distance from the diagonal and therefore the induction hypothesis implies that this is again a compact operator, tending to zero, as T → 0+ , in the operator norms k,γ defined by ᏯW F (P × [0, T0 ]). If we choose T0 sufficiently small, then, with E t = k,γ (E b0,t + E b1,t + E it ), the operator Id +E t is invertible as a map from ᏯW F (P × [0, T0 ]) to itself. We set

t (Id +E t )−1 ᏽt = Q (10.67) to get a right inverse to (∂t − L) on the time interval [0, T0 ], which clearly has the correct mapping properties with respect to the WF-H¨older spaces on P. But for the construction of the boundary parametrix, which is done in Section 10.4, we can complete the proof of the induction step in this case by showing that ᏽt has the small time localization property. That is, if ϕ  , ψ  are smooth functions on P with dist(supp ϕ  , supp ψ  ) > 0,

(10.68)

ψ  ᏽt ϕ  : ᏯW F (P × [0, T ]) −→ ᏯW F (P × [0, T ])

(10.69)

then the operator k,γ

k,2+γ

is a compact operator that tends to zero in norm, as T → 0+ . k,γ The operator (Id +E t )−1 as a map from ᏯW F (P × [0, T ]) to itself is defined as a convergent Neumann series (Id +E t )−1 =

∞  j =0

(−E t ) j .

(10.70)

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Given η > 0, there is an N so that for any T < T0 , we have that 9 9 9  9 9 ∞ 9 t j9 9 (−E ) 9 ≤ η. 9 9 j =N+1 9

(10.71)

W F,k,γ ,T

The induction hypothesis and the properties of the solution operators to the model prob t ϕ  has the small time localization property. Therefore lems show that the operator ψ  Q the essential point is to see that this is true of a composition At B t . L EMMA 10.3.1. Suppose that for t ∈ [0, T ], the maps k,γ

k,γ

k,γ

k,γ

At :ᏯW F (P × [0, T ]) −→ ᏯW F (P × [0, T ]) B t :ᏯW F (P × [0, T ]) −→ ᏯW F (P × [0, T ])

(10.72)

are bounded, so that if ϕ and ψ are smooth functions with disjoint supports, then ϕ At ψ and ϕ B t ψ have the small time localization property, i.e.,, are compact and tend to zero in norm as T → 0+ . Moreover, the composition k,γ

k,γ

At B t : ᏯW F (P × [0, T ]) −→ ᏯW F (P × [0, T ])

(10.73)

has the same property. P ROOF. Let ϕ, ψ be as above. Choose θ ∈ Ꮿ∞ (P) with the properties: dist(supp ψ, supp θ ) > 0, and dist(supp ϕ, supp(1 − θ )) > 0,

(10.74)

so that θ ≡ 1 on a neighborhood of supp ϕ. We observe that ψ At B t ϕ = [ψ At θ ]B t ϕ + ψ At [(1 − θ )B t ϕ].

(10.75)

The operators [ψ At θ ] and [(1 − θ )B t ϕ] have the small time localization property. Hence ψ At B t ϕ is compact and converges in norm to zero as T → 0+ .  If we let Id +FNt =

N 

t (Id +F t ), (−E t ) j and ᏽtN = Q N

(10.76)

j =0

then this lemma shows that the operator ψ  (Id +FNt )ϕ  is compact as a map from k,γ ᏯW F (P × [0, T ]) to itself and tends to zero in norm, as T → 0+ . Furthermore, the difference k,γ k,2+γ ᏽt − ᏽtN : ᏯW F (P × [0, T ]) −→ ᏯW F (P × [0, T ]) (10.77) tends to zero in the norm topology. With θ as above:

t θ ](Id +F t )ϕ  + ψ  Q

t [(1 − θ )(Id +F t )ϕ  ], ψ  ᏽtN ϕ  = [ψ  Q N N

(10.78)

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which shows, as above, that ψ  ᏽtN ϕ  has the small time localization property, and therefore ψ  ᏽt ϕ  is also compact. Finally for any η > 0, there is an N so that as a map from k,γ k,2+γ ᏯW F (P × [0, T ]) to ᏯW F (P × [0, T ]) for T < T0 we have ψ  (ᏽt − ᏽtN )ϕ   ≤ η.

(10.79)

ψ  ᏽt ϕ  : ᏯW F (P × [0, T ]) −→ ᏯW F (P × [0, T ])

(10.80)

This shows that the norm of k,γ

k,2+γ

k,γ

tends to zero as T → 0+ . This establishes that as an operator from ᏯW F (P × [0, T ]) to k,2+γ ᏯW F (P × [0, T ]), the solution operator ᏽt has the small time localization property. To complete this part of the argument, we need only show that for any k ≥ 0 the Neumann series for (Id +E t )−1 converges in operator norm topology defined by k,γ k,γ k,γ ᏯW F (P × [0, T ]), and that E t : ᏯW F (P × [0, T ]) → ᏯW F (P × [0, T ]) has the small time localization property. The induction hypothesis shows that the interior error term E it has this property, so it only needs to be verified for the boundary contribution to E t . The detailed construction of the boundary parametrix is done in the following section, for k = 0. The argument for k > 0 is presented at the end of the chapter. 10.4 THE BOUNDARY PARAMETRIX CONSTRUCTION In this section we give the details of the argument that if P is a manifold with corners so that the maximal codimension of b P is M + 1, and L is a generalized Kimura diffusion

t and a function ϕ ∈ Ꮿ∞ defined on P, then given δ > 0, there is an operator Q c (P) so b that 1. ϕ equals 1 in a neighborhood of  M+1 . 2. For any 0 < γ < 1 and some T0 > 0,

tb : Ꮿk,γ (P × [0, T0 ]) → Ꮿk,2+γ (P × [0, T0 ]) Q WF WF k,γ

is a bounded operator. As a map from ᏯW F (P × [0, T0 ]) to itself, this operator tends, as T0 → 0+ , to zero in norm. k,γ

3. For g ∈ ᏯW F (P × [0, T0 ]),

tb g = ϕg + (E 0,t g + E 1,t g), (∂t − L) Q b b k,γ

(10.81)

where E b0,t has norm at most δ as an operator on ᏯW F (P × [0, T0 ]) and E b1,t is a compact operator on this space with norm tending to zero as T0 → 0+ . 4. The family of operators E b0,t has the small time localization property. In this section we verify claims 1–4 in the case that k = 0.

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10.4.1 The Codimension N Case The argument is a little simpler if M + 1 = N = dim P, so that the stratum  N consists of a finite number of isolated points. We begin the construction by choosing an 0 < η 0, Proposition 9.3.1 shows that this term converges

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exponentially to zero in the ᏯW F -operator norm for any k ∈ ⺞0 and γ ∈ (0, 1). That is, for any k, γ , T there are positive constants C(k, γ , T ) and μ(k, γ ), so that, with

we have

∞,t b,t g = [ψ , L]K i,q [χ g], E i,q

(10.87)

∞,t E i,q gW F,k,γ ,T ≤ C(T, k, γ ) −μ(k,γ ) gW F,0,γ ,T .

(10.88)

The constant C(T, k, γ ) tends to zero as T → 0+ . This leaves only the last term: 0,t b,t E i,q g = ψ (L i,q − L)K i,q [χ g] = ⎡ ⎤ N  √  √ b,t  x i x j ai j (x) x i x j ∂xi ∂x j ⎦ K i,q [χ g]. (10.89) − ψ ⎣ bi (x)∂xi + i=1

1≤i = j ≤N 0,γ

We need to estimate the ᏯW F -norm of this term, which involves two parts, the supnorm part 9 ⎡ 9 ⎤ 9 9 N  √  9 9 √ b,t  9 ,  ⎣ ⎦ ψ I =9 (x)∂ + x x a (x) x x ∂ ∂ [χ g] b K i xi i j ij i j xi x j  9 i,q 9  9 9 ∞ i=1 1≤i = j ≤N L (10.90) and the (0, γ )-semi-norm part + ⎤ ⎡+ ⎡ ⎤ + + N  √  + + √ b,t  +  ⎦ ⎣ ⎣ ⎦ I I = +ψ x i x j ai j (x) x i x j ∂xi ∂x j K i,q [χ g]++ , bi (x)∂xi + + + i=1 1≤i = j ≤N W F,0,γ ,T

(10.91) which we estimate using Lemma 10.1.2. Since the function ψ is supported in the set where x i ≤ 5 2 , Proposition 7.2.1 implies that the first term is estimated by C 2 χ gW F,0,γ ,T .

(10.92)

Applying Lemmas 10.1.2 and 10.1.3, we see that I ≤ C 2−γ gW F,0,γ ,T ,

(10.93)

where the constant C is independent of . To estimate II, Lemma 10.1.2 shows that we need to consider terms of the forms + + *+ ,+ + + b,t b,t ψ  [χ g]+ , +ψ  [χ g] L ∞ bi (x) L ∞ +∂xi K i,q bi (x)+ W F,0,γ ,T ∂xi K i,q W F,0,γ ,T

(10.94)

and

+√ + √ + + b,t ψ x i x j ai j (x) L ∞ + x i x j ∂xi ∂x j K i,q [χ g]+ , W F,0,γ ,T + + √ √ + + b,t  x i x j ∂xi ∂x j K i,q [χ g] L ∞ . +ψ x i x j ai j (x)+ W F,0,γ ,T

(10.95)

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Lemma 10.1.3 and Proposition 7.2.1 show that the terms where the sup-norm is on the coefficients are estimated by C 2−γ . Applying Lemma 10.1.4 we see that there is a C independent of  so that + √ + *+ ,+ + + +ψ  ≤ C 2−γ . (10.96) bi (x)+ W F,0,γ ,T + +ψ x i x j ai j (x)+ W F,0,γ ,T

We get an additional order of vanishing in the second term because the coefficients √ vanish to second order in the variables { x i }. We again use the estimate from Proposition 7.2.1 to see that √ b,t b,t ∂xi K i,q [χ g] L ∞ +  x i x j ∂xi ∂x j K i,q [χ g] L ∞ ≤ C −γ gW F,0,γ ,T , (10.97) showing that these products in (10.94) and (10.95) are bounded by a constant times  2(1−γ )gW F,0,γ ,T . Altogether the right-hand side of (10.89) contributes M N terms of these types, which allows us to conclude that there is a C independent of , and T ≤ T0 , so that

whence

I + I I ≤ C M N  2(1−γ )gW F,0,γ ,T ,

(10.98)

0,t E i,q gW F,0,γ ,T ≤ C M N  2(1−γ )gW F,0,γ ,T .

(10.99)

These calculations apply at each of the p points in  N . For each  > 0 we let χi,q ( ψi,q , resp.) denote the function χ (ψ , resp.) in the i th-coordinate patch, with this choice of . The contribution of  N to the boundary parametrix is given by p   b,t

t = Q ψi,q K i,q χi,q . (10.100) b j =1 q∈F j

We therefore have

t g = (∂t − L) Q b

p   

0,t ∞,t χi,q g + E i,q g + E i,q g

 (10.101)

j =1 q∈F j

= ϕ g +

E 0,t g

+

E ∞,t g,

where ϕ =

p  i=1

χi,q ,

E 0,t =

p  

0,t E i,q ,

and

j =1 q∈F j

E ∞,t =

p  

∞,t E i,q .

(10.102)

j =1 q∈F j

The local estimate (10.98) shows that there is a constant C so that for any  > 0 we have E 0,t gW F,0,γ ≤ C 2(1−γ )gW F,0,γ ,T . (10.103) We can therefore choose  > 0 so that C 2(1−γ ) = δ.

(10.104)

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With this choice of  we let ϕ=

p 

χi,q .

(10.105)

i=1

For this fixed  > 0, the estimate in (10.88) shows that E ∞,t gW F,k,γ ,T ≤ C(T, k, γ ) −μ(k,γ ) gW F,0,γ ,T ,

(10.106)

where C(T, k, γ ) → 0 as T → 0+ . Thus with E t = E 0,t + E ∞,t ,

(10.107)

E t 0,γ ≤ [δ + C(T, 0, γ ) −μ(0,γ ) ].

(10.108)

we have the norm estimate

The function ϕ equals 1 in a neighborhood of  N , and we have estimate

tb g − ϕgW F,0,γ ,T ≤ [δ + C(T, 0, γ ) −μ(0,γ ) ]gW F,0,γ ,T . (∂t − L) Q

(10.109)

It only remains to verify the small time localization property for the error term. The operator E t is built from a finite combination of terms of the form G K t θ, where G is a differential operator, θ is a smooth function, and K t is the heat kernel of a model operator. If ϕ and ψ are smooth functions with disjoint supports, then we can choose another smooth function ϕ  so that supp ϕ  ∩ supp ψ = ∅ and ϕ  = 1 on supp ϕ.

(10.110)

Since G is a differential operator, it is immediate that ϕG K t θ ψ = ϕGϕ  K t θ ψ.

(10.111)

As the supports of ϕ  and ψ are disjoint, it follows that ϕ  K t θ ψ is a family of smoothing operators tending to zero as T → 0+ as a map from the space Ꮿ0 (P × [0, T ]) to Ꮿk (P × [0, T ]), for any k ∈ ⺞. This completes the construction of the boundary parametrix in this case. 10.4.2 Intermediate Codimension Case Now assume that n = M + 1 < dim P, and that  M+1 is the maximal codimensional stratum of b P. This includes the case that n = 1, which is the base case needed to start the induction. We let N +  M+1 denote the inward pointing normal bundle of  M+1 . Since  M+1 is the maximal codimensional stratum, the tubular neighborhood theorem for manifolds with corners implies that there is a neighborhood W of  M+1 in P that is diffeomorphic to a neighborhood W0 , of the zero section in N +  M+1 . Let  : W0 → W be such a diffeomorphism, which reduces to the inclusion map along the zero section. We let  denote a Ꮿ∞ -function defined on N +  M+1 so that  = 1 in a neighborhood W1

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of zero section and  = 0 outside a somewhat larger neighborhood W2 . We define a family of functions { :  ∈ (0, 1]} in Ꮿ∞ (W ) by setting    (r ) =   −2 ·  −1 (r ) . (10.112) Here  −2 · denotes the usual action of ⺢+ on the fiber of N +  M+1 . Let U = {U j : j = 1, . . . , p} denote a covering of a neighborhood of  M+1 by NCC charts. The fact that  M+1 is the maximum codimensional stratum implies that all of these charts have coordinates lying in ⺢n+ × ⺢m . Let (x; y) = (x 1 , . . . , x n ; y1, . . . , ym ) be the normal cubic coordinates in a subset U j , so that in these coordinates L is given by L=

n  i=1





x i ∂x2i +

ckl (x, y)∂ yk ∂ yl +

1≤k,l≤m

n 

bi (x, y)∂xi +

i=1

x i x j ai j (x, y)∂xi ∂x j +

1≤i = j ≤n

m n  

x i bil (x, y)∂xi ∂ yl +

i=1 l=1

m 

dl (x, y)∂ yl .

(10.113)

l=1

We let L p denote the sum on the first line; this is the principal part of L. There is a positive constant K so that within the coordinate chart the coefficient matrix ckl satisfies K Idm ≤ ckl (x, y) ≤ K −1 Idm . (10.114) For each point in q ∈  M+1, j =  M+1 ∩ U j we could choose an affine change of coordinates in the y-variables, which we denote by (x, y), so that in these variables q ↔ (0; 0) and p

L q =

n 

x i ∂x2i

+

i=1



(δkl + ckl (x, y))∂ y˜k ∂ y˜l +

1≤k,l≤m

where

n 

(bi +  bi (x, y))∂xi , (10.115)

i=1

 ckl (0n , 0m ) =  bi (0n , 0m ) = 0.

(10.116)

In light of the bounds (10.114) these affine changes of variable come from a compact subset of G L m  ⺢m and therefore, under all these changes of variable, the coefficients  ckl , bi ,  bi and ai j , bil , dl remain uniformly bounded in the Ꮿ∞ -topology. In fact we do not use these changes of variables in our construction, but simply note that the constants in the estimates for the model operators at points q = (0; yq ) ∈ Ui , which we can take to be L i,q =

n m   [x i ∂x2i + bi (0, yq )∂xi ] + ckl (0, yq )∂ yl ∂ yk , i=1

k,l=1

(10.117)

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are uniformly bounded. We have L = L i,q + L ri,q +

m 

dl (x, y)∂ yl ,

(10.118)

l=1

where the residual “second order” part at q is L ri,q =

 1≤k,l≤m



√

 ckl,q (x, y)∂ yk ∂ yl +

n 

 bi,q (x, y)∂xi +

i=1

√ √ √ x i x j ai j (x, y) x i x j ∂xi ∂x j + x i bil (x, y) x i ∂xi ∂ yl . n

1≤i = j ≤n

m

(10.119)

i=1 l=1

The coefficients of L ri,q are smooth functions of (x, y), and   ckl,q (x, y) = ckl (x, y) − ckl (0, yq ) and bi,q (x, y) = bi (x, y) − bi (0, yq ), (10.120)

so that

  ckl,q (0, yq ) = bi,q (0, yq ) = 0.

We let χ(x, y) and ψ(x, y) be functions in

n Ꮿ∞ c (⺢ +

× ⺢m )

(10.121) so that

χ ≡ 1 in [0, 4]m × (−2, 2)n supp χ ⊂ [0, 9]m × (−3, 3)n ,

(10.122)

ψ ≡ 1 in [0, 16]m × (−4, 4)n supp ψ ⊂ [0, 25]m × (−5, 5)n .

(10.123)

and

With (0; yq ) the coordinates of q, we define  χ i,q = χ and

x y − yq ,  2

 ψi,q = ψ

x y − yq , 2 

 ,

(10.124)

.

(10.125)



Of course these functions depend on the choice of , but to simplify the notation, we leave this dependence implicit. We let Fi, be the points in Ui ∩ M+1 with coordinates {(0,  j) : j ∈ ⺪m }. The following lemma is immediate from these definitions. L EMMA 10.4.1. Every point lies in the support of at most a fixed finite number of the functions {ψi,q : q ∈ Fi, ; i = 1, . . . , p}, independently of .

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From the definition of the sets Fi, it is clear that there is a constant S, independent of , so that for r ∈ P we have the estimate X  (r ) =

p  

χ i,q (r ) ≤ S.

(10.126)

i=1 q∈Fi,

It is also clear that X  (r ) ≥ 1 for r ∈  M+1 . By choosing the neighborhoods W1 , W2 (independently of ) used in the definition of  (see (10.112)), we can arrange to have  = 1 on the set where X  ≥ 12 , and supp  ⊂ X −1 ([

1 , S]). 16

(10.127)

To get a partition of unity of a neighborhood of  M+1 , we replace the functions { χi,q } with / 0 χ i,q χi,q =   p  . (10.128) i,q q∈Fi, χ i=1 For any choice of  > 0, these functions are smooth and define a partition of unity in a neighborhood of  M+1 . By repeated application of Lemmas 10.1.2 and 10.1.3 and (10.127), it follows that there is a constant C independent of  > 0 so that ψi,q W F,0,γ + χi,q W F,0,γ ≤ C −γ .

(10.129)

For each  > 0 we define a boundary parametrix by setting

t = Q b

p  

b,t ψi,q K i,q χi,q ,

(10.130)

i=1 q∈Fi, b,t denotes the solution operator constructed above for the model problem where K i,q

(∂t − L i,q )u = g

u(r, 0) = 0,

with L i,q defined in (10.117), and with b = (b1 (0, yq ), . . . , bn (0, yq )). We now consider the typical term appearing in the parametrix. If g is a H¨older continuous function defined in a neighborhood of supp χi,q , then b,t u i,q = ψi,q K i,q [χi,q g]

(10.131)

is well defined throughout Ui and can be extended, by zero, to all of P. We apply the operator to u i,q obtaining: q

b,t b,t (∂t − L)u i = ψi,q (∂t − L i,q + L i,q − L)K i,q [χi,q g] + [ψi,q , L]K i,q [χi,q g] q

b,t b,t = χi,q g + ψi,q (L i,q − L)K i,q [χi,q g] + [ψi,q , L]K i,q [χi g].

(10.132)

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The estimates for the sizes of these errors will be in terms of χi,q giW F,0,γ ,T . It follows from Lemmas 10.1.2 and 10.1.3 and (10.129) that there is a constant C, independent of , T so that χi,q giW F,0,γ ,T ≤ C −γ gW F,0,γ ,T . There are three types of error terms: ∞,t b,t E i,q g = [ψi,q , L]K i,q [χi,q g],

1,t E i,q g = ψi,q 0,t E i,q

=

/ m 

(10.133) 0

b,t dl (x, y)∂ yl K i,q [χi,q g],

l=1 b,t r ψi,q L i,q K i,q [χi,q g],

(10.134) where L ri,q is given by (10.119). For each  > 0 and d ∈ {0, 1, ∞} we define E d,t

=

p  

d,t E i,q .

(10.135)

i=1 q∈Fi,

Observe that the support of the coefficients of [ψi,q , L] is disjoint from that of ∞,t is a smoothing operator tending χi,q and therefore Proposition 9.3.1 shows that E i,q + exponentially to zero as T → 0 . As before there are positive constants C(T, k, γ ) and μ(k, γ ) so that, for any  > 0, we have ∞,t E i,q gW F,k,γ ,T ≤ C(T, k, γ ) −μ(k,γ ) gW F,k,γ ,T ,

(10.136)

where C(T, k, γ ) = O(T ) as T → 0+ . 1,t g produced by the tangential first derivatives is of lower order, The error term E i,q but more importantly, equation (9.45) shows that the norm of this term also tends to zero as T → 0+ . Hence there is a positive constant C independent of  so that γ

1,t gW F,0,γ ,T ≤ C −γ T 2 gW F,0,γ ,T . E i,q

(10.137)

Recalling Lemma 10.4.1, each point will lie in the support of at most S of the functions ψi,q and therefore there is a constant C independent of  so that the sum of these terms satisfies an estimate of the form γ

[E 1,t + E ∞,t ]gW F,0,γ ,T ≤ SC −(μ(0,γ )+γ ) T 2 gW F,0,γ ,T .

(10.138)

The remaining error term is 0,t b,t g = ψi,q L ri,q K i,q [χi,q g], E i,q

(10.139)

k,γ

which is a bounded + map + of ᏯW F to itself, for any k ∈ ⺞0 . We need to estimate both + 0,t + 0,t g L ∞ and +E i,q g+ . The vanishing properties of the coefficients of L ri,q , E i,q W F,0,γ

Proposition 9.2.1 and Lemmas 10.1.2 and 10.1.3 imply that the L ∞ -term satisfies 0,t g L ∞ ≤ Cχi,q gW F,0,γ ≤ C 1−γ gW F,0,γ . E i,q

(10.140)

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The second inequality follows from Lemmas 10.1.2 and 10.1.3. To estimate the H¨older semi-norm we need to consider a variety of terms, much like those in (10.94) and (10.95). For the case at hand we have the terms + + + + + + + + b,t b,t   (x, y)∂xi K i,q [χi,q g]+ , +ψi,q ckl,q (x, y)∂ yk ∂ yl K i,q [χi,q g]+ , +ψi,q bi,q W F,0,γ W F,0,γ +  + √ √ + + b,t , +ψi,q x i x j ai j (x, y) x i x j ∂xi ∂x j K i,q [χi,q g]+ W F,0,γ +  + √ √ + + b,t , (10.141) +ψi,q x i bil (x, y) x i ∂xi ∂ yl K i,q [χi,q g]+ W F,0,γ

each of which is estimated by using the Leibniz formula in Lemma 10.1.2. Lemma 10.1.3 shows that there is a constant C so that the terms + + + + b,t  (x, y) L ∞ +∂xi K i,q [χi,q g]+ , ψi,q bi,q W F,0,γ + + + + b,t  (x, y) L ∞ +∂ yk ∂ yl K i,q [χi,q g]+ , ψi,q ckl,q W F,0,γ +  + √ +√ + b,t [χi,q g]+ (10.142) ψi,q x i bil (x, y) L ∞ + x i ∂xi ∂ yl K i,q W F,0,γ

are all bounded by

,+ *+ C +χi,q g + W F,0,γ ≤ C 1−γ gW F,0,γ .

(10.143)

Similarly, we see that ψi,q

√

+√ + + + b,t x i x j ai j (x, y) L ∞ + x i x j ∂xi ∂x j K i,q [χi,q g]+

W F,0,γ

≤ C 2−γ gW F,0,γ . (10.144)

To complete the estimates for the terms in (10.141) we need to bound + + + +  (x, y)+ +ψi,q bi,q

b,t ∂xi K i,q [χi,q g] L ∞ , + + + + b,t  (x, y)+ ∂ yk ∂ yl K i,q [χi,q g] L ∞ , +ψi,q ckl,q W F,0,γ +  + √ √ + + b,t  x i x j ∂xi ∂x j K i,q [χi,q g] L ∞ , +ψi,q x i x j ai j (x, y)+ W F,0,γ +* + , √ b,t +ψi,q √ x i b (x, y)+  x i ∂xi ∂ yl K i,q [χi,q g] L ∞ . il W F,0,γ W F,0,γ

(10.145)

Proposition 9.2.1 shows that for any 0 < γ  ≤ γ < 1 the sup-norms appearing in (10.145) are bounded by 

Cγ  χi,q gW F,0,γ  ≤ Cγ   −γ gW F,0,γ  .

(10.146)

We therefore fix a 0 < γ  ≤ γ so that γ  + γ < 1.

(10.147)

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To complete this estimate we only need to bound the H¨older semi-norms of the coefficients. Lemma 10.1.4 shows that all of these terms are bounded by C 1−γ , for a constant independent of  > 0. Together these estimates show that there is a constant C independent of , i and q so that 

0,t E i,q gW F,0,γ ,T ≤ C 1−γ −γ gW F,0,γ ,T .

(10.148)

Once again we use the fact that for any point in P at most a fixed finite number of terms in the sum defining E 0 is non-zero to conclude that there is a constant S so that 

E 0,t gW F,0,γ ,T ≤ SC 1−γ −γ gW F,0,γ ,T .

(10.149)

We can therefore choose  > 0 so that 

SC 1−γ −γ ≤ δ.

(10.150)

With this fixed choice of  we let ϕ=

p  

χi,q ;

(10.151)

i=1 q∈Fi,

t with this function equals 1 in a neighborhood of  M+1 . Using the definition for Q b this choice of , we see that with

we have that

E t = E 0,t + E 1,t + E ∞,t

(10.152)

t − ϕg = E t g, (∂t − L) Q b

(10.153)

and therefore γ

tb − ϕgW F,0,γ ,T ≤ [δ + C −μ(0,γ ) T 2 ]gW F,0,γ ,T . (∂t − L) Q

(10.154)

This estimate completes the construction of the boundary parametrix for the case of arbitrary maximal codimension between 1 and dim P. It only remains to verify the small time localization property for the error term. As before, the operator E t is built from a finite combination of terms of the form G K t θ, where G is a differential operator, θ is a smooth function, and K t is the heat kernel of a model operator. Precisely the same argument as given in the maximal codimension case shows that if ϕ and ψ are smooth functions with disjoint supports, then ϕG K t θ ψ is a family of smoothing operators tending to zero as T → 0+ as a map from Ꮿ0 (P ×[0, T ]) to Ꮿ j (P × [0, T ]), for any j ∈ ⺞. This in turn completes the proof, in case k = 0, of the existence of a solution to the inhomogeneous problem up to a time T0 > 0. In the next section we show how to use this result to demonstrate the existence of solutions to the Cauchy problem, which in turn allows us to prove a global in time existence result for the inhomogeneous problem.

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10.5 SOLUTION OF THE HOMOGENEOUS PROBLEM Assuming the existence of a solution to the inhomogeneous problem for data belonging k,γ to ᏯW F (P × [0, T0 ]) for a fixed T0 > 0, a very similar parametrix construction is used to show the existence of v, the solution for all time, to the homogeneous Cauchy k,2+γ problem, with initial data f ∈ ᏯW F (P). Assume that ᏽt , the solution operator for the inhomogeneous problem, is defined for t ∈ [0, T0 ]. As above, we use Proposition 9.1.1 to build a boundary parametrix for the homogeneous Cauchy problem, which we then glue to the exact solution operator for PU . This gives an operator

t : Ꮿk,2+γ (P) −→ Ꮿk,2+γ (P × [0, ∞)) Q 0 WF WF t t

t f t =0 = f, (∂t − L) Q 0 f = E 0 f and Q 0 where

k,2+γ

k,γ

E 0t : ᏯW F (P) −→ ᏯW F (P × [0, ∞))

(10.155)

(10.156)

is a bounded map. A slightly stronger statement is true. P ROPOSITION 10.5.1. Given δ > 0, we can make lim E 0t Ꮿk,2+γ ( P)→Ꮿk,γ

T →0+

W F ( P×[0,∞))

WF

≤ δ.

(10.157)

t is a simple consequence of the induction hypothThe existence of the operator Q 0 esis and the properties of the solution operators for the model homogeneous Cauchy problems established in Proposition 9.1.1. Suppose that the maximal codimension of b P is M. Let {W j : j = 1, . . . , J } be an NCC cover of  M , and W0 a relatively compact subset of int P, which covers P \ ∪ Jj=1 W j , and has a smooth boundary. Let {ϕ j } be a partition of unity subordinate to this cover of P, and {ψ j } smooth functions

t of compact support in W j , with ψ j ≡ 1 on supp ϕ j . For each j ∈ {1, . . . , J } let Q j0 be the solution operator for the homogeneous Cauchy problem defined by the model

t be the exact solution operator for the Cauchy operator in W j . As above, we let Q 00 problem (∂t − L)u = 0 on W0 with Dirichlet data on bW0 × [0, ∞). We then define

t = Q 0

J 

t ϕ j . ψj Q j0

(10.158)

j =0

From the mapping properties of the component operators it follows that, for any 0 < γ < 1, and k ∈ ⺞0 , this operator defines bounded maps:

t : Ꮿk,γ (P) −→ Ꮿk,γ (P × [0, ∞)) Q 0 WF WF

t : Ꮿk,2+γ (P) −→ Ꮿk,2+γ (P × [0, ∞)). Q 0 WF WF

(10.159)

t tends strongly to the identity, with respect to the topologies As t → 0+ , the operator Q 0 k, γ k,2+ γ ᏯW F (P), ᏯW F (P), respectively, for any  γ < γ.

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t f = ᏽt E t f, and ᏽt f = ( Q

t − Q t ) f, then If we set Q 0 0 0 0 0 k,2+γ

k,2+γ

ᏽt0 : ᏯW F (P) −→ ᏯW F (P × [0, T0 ]) is bounded

(∂t − L)ᏽt0 f = 0.

(10.160)

k,2+ γ

For any 0 <  γ < γ , the solution ᏽt0 f tends to f in ᏯW F (P). From the induction

t f. To hypothesis and the properties of the boundary terms this is certainly true of Q 0 t treat the correction term we observe that ᏽ defines a bounded map from the space k,γ k,2+γ ᏯW F (P ×[0, T ]) to ᏯW F (P ×[0, T ]). For a fixed δ > 0, by constructing the partition of unity {ϕ j } as in Section 10.4, and choosing  > 0 sufficiently small, we can arrange to have lim E 0t f W F,k,γ ,T ≤ δ f W F,k,2+γ . T →0+

Hence, for any δ > 0, lim ᏽt f − f W F,k,2+ γ ≤ Cδ f W F,k,2+γ .

t →0+

(10.161)

To show that the solution to the homogeneous problem exists for all t > 0, we observe that the time of existence T0 already obtained is independent of the initial data, and there is a constant C so that, with v = ᏽt0 f, v(·, T0 )W F,k,2+γ ≤ C f W F,k,2+γ .

(10.162)

We can therefore apply this argument again, with data v(·, T0 ) specified at t = T0 , to obtain a solution on [0, 2T0 ]. We have the same estimate on [0, 2T0] with C replaced by C 2 . This can be repeated ad libitum to show that there is are constants C1 , M and a k,2+γ solution v to the homogeneous Cauchy problem, belonging to ᏯW F (P × [0, T ]), for any T > 0, which satisfies the estimate vW F,k,2+γ ,T ≤ M exp(C1 T ) f W F,k,2+γ .

(10.163)

To verify that ᏽt0 satisfies the small time localization property (condition (3) in

t − ᏽt E t . The induction hypothesis the induction hypothesis) we recall that ᏽt0 = Q 0 0

t has this property. We have and the properties of the model heat kernels show that Q 0 established this for the operator ᏽt . The error term is again of the form G K 0t θ, where G is a differential operator and K 0t is either a model heat kernel, or the heat kernel from the interior. As before, if ϕ and ψ have disjoint support, then we can choose k,2+γ ϕ  satisfying (10.110). From this it is immediate that, as maps from ᏯW F (P) to k,γ ᏯW F (P), the operators ϕG K 0t θ ψ = ϕGϕ  K 0t θ ψ (10.164) have the small time localization property. Using the arguments in the proof of Lemma k,2+γ 10.3.1 it follows easily that, as maps from ᏯW F (P) to itself, the operator ᏽt E 0t also has the small time localization property. This completes the proof of the following theorem, which is part of Theorem 10.0.2, in the k = 0 case.

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T HEOREM 10.5.2. Let P be a manifold with corners and L a generalized Kimura diffusion operator defined on P. There is an operator k,2+γ

so that

k,2+γ

ᏽt0 : ᏯW F (P) −→ ᏯW F (P × [0, ∞)),

(10.165)

(∂t − L)ᏽt0 f = 0 for all t > 0;

(10.166)

k,2+ γ ᏯW F (P).

There are constants

ᏽt0 f W F,k,2+γ ,T ≤ Mk,γ exp(Ck,γ T ) f W F,k,2+γ .

(10.167)

moreover, for any  γ < γ , ᏽt0 f converges to f in Ck,γ , Mk,γ so that

Contingent upon verification of the convergence of the Neumann series for k ∈ ⺞, and the proof of Theorem 10.2.1, this completes the proof of Theorem 10.5.2. The maximum principle has a useful corollary about the point spectrum of L on the k,2+γ spaces ᏯW F (P). 0,2+γ

C OROLLARY 10.5.3. If there is a non-trivial solution f ∈ ᏯW F (P) to the equation (L − μ) f = 0, then Re μ ≤ 0. P ROOF. Suppose there were a solution f μ = 0, for a complex number μ = μ1 + i μ2 with μ1 > 0. The maximum principle, Proposition 3.3.5, shows that μ2 = 0. The unique solution to the initial value problem (∂t − L)v = 0, with v(x, 0) = f μ (x), is v(x, t) = eμt f μ (x). The Re(v(x, t)) and Im(v(x, t)) are also solutions. The supj norms of these solutions at times of the form 2π μ2 grow exponentially, which contradicts Proposition 3.3.1.  We also observe that the solution of the homogeneous problem can be used to extend the time of existence for the inhomogeneous problem. Contingent upon proving the convergence of the Neumann series for (Id +E t )−1 , we have proved the existence k,2+γ of a solution, u ∈ ᏯW F (P × [0, T0 ]) to k,γ

(∂t − L)u = g ∈ ᏯW F (P × [0, T ]) with u(w, 0) = 0,

(10.168)

where we assume that T > T0 . We now let v 1 denote the solution to the Cauchy k,2+γ problem with initial data u(w, T0 ) ∈ ᏯW F (P), which exists on the interval [0, T0 ], and let u 1 denote the solution to (10.168), with g replaced by g(w, t + T0 ). We see that setting u(w, t) = v 1 (w, t − T0 ) + u 1 (w, t − T0 ) for t ∈ [T0 , 2T0 ]

(10.169)

extends u as a solution of (10.168) to the interval [0, 2T0]. This process is repeated n times until nT0 ≥ T, or infinitely often if T = ∞. It is clear that the solution k,2+γ u ∈ ᏯW F (P × [0, T ]), and its norm grows at most exponentially in T. To complete the proof of Theorem 10.0.2 we need to prove 10.2.1, and the converk,γ gence of the Neumann series for (Id +E t )−1 in the topologies defined by ᏯW F (P × [0, T0 ]), for k > 0. Theorem 10.2.1 is proved in the next section, and the higher regularity is established at the end of this chapter.

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10.6 PROOF OF THE DOUBLING THEOREM Let P be a manifold with corners up to codimension M and L a generalized Kimura diffusion operator on P. Let  =  M denote the corner of maximal codimension M. This is a closed manifold without boundary; for simplicity we assume here that it is connected, although this is not important. We first examine the geometry of P near  and use this to indicate how to perform the doubling construction for P itself. Once we have accomplished this, we show how to extend L to an operator of the same type on the doubled space. A key property of manifolds with corners is that  possesses a neighborhood U˜ which is diffeomorphic in the category of manifolds with corners to a bundle over , where each fiber is the positive unit ball B+M = {x ∈ ⺢ M : x j ≥ 0 ∀ j, ||x|| < 1} in the positive orthant in ⺢ M . Indeed, the existence of this fibration is just the correct global version of the fact that near any point q ∈  there is an adapted coordinate chart (x 1 , . . . , x M , y1 , . . . , y ) for P with each x j ∈ [0, 1) and yi ∈ (−1, 1). The point we do not belabor is that one can choose a coherent set of coordinate charts of this type so that in the overlaps of these charts, the fibers {y = const.} are the same and the transition maps induce diffeomorphisms of the positive orthant fibers. In fact, we need a slightly more refined version of this. Use polar coordinates 0 ≤ r < 1 and ω ∈ S+M = {x ∈ B+M : ||x|| = 1} to identify each fiber with a truncated cone C1 (S+M−1 ). Then it is possible to choose the atlas of coordinate charts so that the transition maps preserve the radial coordinate r . In other words, each hypersurface {r = const.} is globally defined, and is itself a manifold with corners up to codimension M − 1. In particular, set  o = {r = 1}. Note that  o is the total space of a fibration over  with fiber S+M−1 .  Let P o denote the open manifold with corners We next define the doubled space P. P \ . As a set, define $ %  = (−P o ) % P o % (−1, 1) ×  o / ∼, P where −P o denotes P o with the opposite orientation. The identification is the obvious one between P o ∩ ᐁ ∼ = (0, 1) ×  o and the corresponding portion of the cylinder, with the analogous identification between −P o and the other side of the cylinder. This space has the structure of a smooth manifold with corners only up to codimension M − 1. For the second step of the proof, we must define an extension of the operator L to  It is most convenient now to express the restriction of L to the neighborhood ᐁ of P.  in polar coordinate form. For this we recall that in these coordinates, 1 ∂x j = ω j ∂r + V j , r where V j is tangent to each hypersurfaces r = const. and transversal to {ωi = 0}. On

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the other hand, each ∂ yi lifts to a vector field of precisely the same form. Therefore, L=

M   1

1 − ωi2 Vi + ωi Vi (ωi )∂r r





 ckl ∂ yk ∂ yl r i=1    + ai j ωi2 ω2j (r ∂r )2 + ωi ω j Vi V j + ωi Vi (ω2j )r ∂r + ωi Vi (ω j )V j

r ∂r2

+

ωi Vi2

i = j

+



+

   bil r ωi2 ∂r ∂ yl + ωi Vi ∂ yl + dl ∂ yl .

 , d are smooth in (y, r, ω). Notice that the first term, r ∂ 2 , The coefficients ai j , bil , ckl l r and the operator in the first parenthetic expression are both homogeneous of degree −1 and odd in r and are independent of y. All of the other operators are homogeneous of degree 0 and even in r provided we neglect the smooth dependence of their coefficients in r . We can obviously regard this as an operator on the cylinder, at least away from r = 0, so we must simply define a modification of the coefficients which extends smoothly and in the same class of Kimura-type operators across r = 0. Recall that we wish to make this modification in any fixed but arbitrarily small region |r | < η. To this end, choose a smooth non-negative cutoff function χ(r ) which equals 1 in r ≥ η and vanishes when r ≤ η/2. Now replace ai j , for example, by

aij := χ(r )ai j (y, r, ω) + (1 − χ(r ))ai j (y, 0, ω), and similarly for all the other coefficients. These modified terms are now exactly homogeneous of degree 0 in r ≤ η/2 and extend by even reflection across r = 0. It remains only to define the extensions of the first two terms. For this, let ρ(r ) be a smooth function defined when |r | < 1 with the following properties: ρ(r ) = ρ(−r ), ρ(r ) ≥ η/4 for all |r | < 1, ρ(r ) = r when |r | ≥ η and ρ(r ) ≤ η for |r | < η. We then replace these first two terms in L by ρ(r )∂r2 +

 M   1 2 1 ωi Vi2 − ωi Vi + ωi Vi (ωi )∂r . ρ(r ) ρ(r ) i=1

We have now defined the full extension of L to an operator  L of Kimura type on  This completes the proof of Theorem 10.2.1. the doubled space P. 10.7 THE RESOLVENT OPERATOR AND C0 -SEMI-GROUP 0,2+γ

The existence of a solution to the Cauchy problem with initial data in ᏯW F (P) suffices to establish the existence of a contraction semi-group on Ꮿ0 (P), generated by the 0,2+γ Ꮿ0 -graph closure of L acting on ᏯW F (P), for any 0 < γ < 1. Though these results suffice to establish the uniqueness of the solution to the stochastic differential equation associated to L and therefore the existence of a strong Markov process with support in P, they are not optimal as regards the smoothing properties of the resolvent (μ − L)−1 . We revisit the question of the resolvent operator in the next chapter.

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If f ∈ Ꮿ0,2+γ (P) then Theorem 10.5.2 shows that there is a unique solution v ∈ Ꮿ0,2+γ (P × [0, ∞)) to the initial value problem (∂t − L)v = 0 with v(·, 0) = f.

(10.170)

The maximum principle shows that v L ∞ ( P×[0,∞)) ≤  f  L ∞ ( P) ,

(10.171)

and Theorem 10.5.2 gives the estimate vW F,0,2+γ ,T ≤ M exp(C T ) f W F,0,2+γ .

(10.172)

The estimate in (10.171) easily implies that, so long as Re μ > 0, the limit 1

 lim

→0+

v(·, t)e−μt dt

(10.173)

 0,2+γ

exists as a Ꮿ0 (P)-valued integral, and, if Re μ > C, as a ᏯW F (P)-valued integral. We denote this limit by R(μ) f. The estimates on v given above imply that 1  f L ∞ Re μ M  f W F,0,2+γ . ≤ [Re μ − C]

R(μ) f  L ∞ ≤ R(μ) f W F,0,2+γ

(10.174)

Using the same integration by parts argument as was used in Section 4.3 we establish that (μ − L)R(μ) f = f. (10.175) The maximum principle shows that the operator L with domain Ᏸ2W F (P), considered 0,2+γ as an unbounded operator on Ꮿ0 (P), is dissipative (see Lemma 3.2.2). As ᏯW F (P) 0 is a dense subset of Ꮿ (P), we can apply a theorem of Lumer and Phillips [30] to conclude the existence of a Ꮿ0 -semi-group of operators et L : Ꮿ0 (P) → Ꮿ0 (P), with 0,2+γ domain given by the Ꮿ0 -graph closure of (L, ᏯW F (P)). The maximum principle implies that this semi-group is contractive. This establishes, for example, the uniqueness of the solution to the martingale problem, supported on Ꮿ0 ([0, ∞); P) and the uniqueness-in-law for the solution to the stochastic differential equation formally defined by this second order operator. The fact that the paths of this process are confined, almost surely, to P follows using an argument like that in [6, 7, 8]. We will return to these questions in a later publication.

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10.8 HIGHER ORDER REGULARITY In the earlier sections of this chapter we constructed a boundary parametrix with an error term E t defined in (10.107) or (10.152). These operators define bounded maps k,γ from ᏯW F (P ×[0, T ]) to itself for any k ∈ ⺞0 and 0 < γ < 1. To complete the proof of Theorems 10.0.2 and 10.5.2 we need only establish the convergence of the Neumann k,γ series for (Id +E t )−1 in the operator norm topology defined by ᏯW F (P × [0, T0 ]), for some  > 0 and T0 > 0. We accomplish this by using a general result about the convergence of Neumann series in higher norms proved in [14]. We begin by recalling the main result of that paper. Suppose that we have a ladder of Banach spaces X 0 ⊃ X 1 ⊃ X 2 ⊃ · · · , with norms  · k , satisfying xk−1 ≤ xk for all x ∈ X k .

(10.176)

T HEOREM 10.8.1. Fix any K ∈ ⺞. Assume that A is a linear map so that AX k ⊂ X k for every k ∈ ⺞0 , and that there are non-negative constants {α j : j = 0, 1, . . . , K } and {β j : j = 1, . . . , K }, with αj < 1

for 0 ≤ j ≤ K ,

(10.177)

for which we have the estimates Ax0 ≤ α0 x0 for x ∈ X 0 , and

(10.178)

Axk ≤ αk xk + βk xk−1 for x ∈ X k . In this case the Neumann series (Id −A)−1 =

∞ 

Aj

(10.179)

j =0

converges in the operator norm topology defined by (X k ,  · k ) for all k ∈ {0, . . . , K }. To apply this theorem we need to show that for any K ∈ ⺞ and 1 < γ < 0, we can choose 0 < , 0 < T0 so that there are constants {α0 , . . . , α K } and {β0 , . . . , β K } with β0 = 0, α j < 1, for 0 ≤ j ≤ K , and we have the estimates E t gW F,k,γ ,T0 ≤ αk gW F,k,γ ,T0 + βk gW F,k−1,γ ,T0 for 0 ≤ k ≤ K .

(10.180)

k,γ

Recalling the definitions of the norms on the function spaces ᏯW F (P × [0, T ]) and k,2+γ ᏯW F (P × [0, T ]), we see that the proofs of such estimates follow quite easily from what is done in Chapter 10. Equivalent norms can be defined inductively by starting at k = 0 with the definitions in (5.59) and (5.67) and then setting gW F,k,γ ,T = gW F,k−1,γ ,T +

sup

|α|+|β|+2l=k

gW F,k,2+γ ,T = gW F,k−1,2+γ ,T +

sup

β

∂tl ∂ xα ∂ y gW F,0,γ ,T

|α|+|β|+2l=k

β

∂tl ∂ xα ∂ y gW F,0,2+γ ,T .

(10.181)

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The operators appearing in the sum that defines the boundary contributions to E t are of the form b,t G K i,q χ , (10.182) where G is a differential operator. From the form of this operator it is clear that we can regard it as acting on functions with support in a compact subset of the coordinate chart, independent of . This allows the application of the higher order estimates proved in Chapters 6–9 with constants that are independent of . The higher order estimates for the contributions from the interior are covered by the induction hypothesis. The part of the estimate for E t gW F,k,γ ,T which cannot be subsumed into a large multiple of E t gW F,k−1,γ ,T will be called E t gW F,k,γ ,T rel E t gW F,k−1,γ ,T .

(10.183)

This arises only from terms of the form β

∂tl ∂ xα ∂ y E t gW F,0,γ ,T , where 2l + |α| + |β| = k.

(10.184)

The structure of the operators that make up E t shows that the parts of these terms that cannot be estimated by a multiple of E t gW F,k−1,γ ,T arise from one of two sources. The simpler terms to estimate are of the form β

t ∂tl ∂ xα ∂ y gW F,0,γ ,T , E

(10.185)

t is the error term in the parametrix construction for a generalized Kimura where E diffusion operator  L α derived in a straightforward manner from L. The other “new” terms arise from x-derivatives being applied to the coefficients of terms in E t involving x j ∂x j , x i ∂xi ∂ yl and x i x j ∂xi ∂x j . These terms are not of lower order, but applying a derivative to the coefficients of one of these terms leaves one less derivative to apply to g. Terms of the type appearing in (10.185) are controlled by choosing a small  > 0, whereas this latter type of term is controlled by taking T0 sufficiently small. 10.8.1 The 1-Dimensional Case We explain this first in the 1-dimensional case, where P is the interval [0, 1]. The operator takes the form L = x(1 − x)∂x2 + b(x)∂x , with b(x)∂x inward pointing at each boundary component. We can introduce coordinates x 0 , x 1 , respectively, so that j ↔ x j = 0, j = 0, 1 and, in these coordinates b j (x))∂x j , where b j ≥ 0 and  b(0) = 0. L = x j ∂x2j + (b j + 

(10.186)

We let L b = x∂x2 + b∂x denote the model operators, and K tb the solution operators for (∂t − L b )u = g, u(x, 0) = 0. The boundary parametrix has the form

t = Q b

1  j =0

b

ψ (x j )K t j ϕ (y j ).

(10.187)

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Here ϕ(x) is a smooth function equal to 1 in [0, 18 ], and supported in [0, 14 ], and ψ is a smooth function equal to 1 in [0, 12 ], and supported in [0, 34 ]. As usual we let f  (x) = f (x/ 2 ). We observe that for any smooth function θ [L, θ ]u = 2x(1 − x)∂x θ ∂x u + [b(x)∂x θ + x(1 − x)∂x2θ ]u,

(10.188)

which consists entirely of lower order terms, and

tb = (L − ∂t ) Q

1  

 b ϕ j, + ([ψ j, , L] + ψ j, ( b j (x j )∂x j ))K t j ϕ j, .

(10.189)

j =0

The error terms are E ∞,t = [ψ0, , L]K t 0 ϕ0, + [ψ1, , L]K t 1 ϕ1, b

b

(10.190)

E 0,t = [ψ0, ( b0 (x 0 )∂x0 )]K t 0 ϕ0, + [ψ1, ( b1 (x 1 )∂x1 )]K tb1 ϕ1, . b

t = E 0,t + E ∞,t . Together E b t gW F,k,γ ,T of the form We want to give an estimate E b t E b gW F,k,γ ,T ≤ αk gW F,k,γ ,T + βk gW F,k−1,γ ,T ,

(10.191)

where αk < 1. The new terms in going from k − 1 to k are of the form t ∂tm ∂xl E b gW F,0,γ ,T ,

(10.192) b

where 2m + l = k. Any derivatives that fall onto the coefficients of K t j ϕ j, g, other than  b j (x j ), will lead to terms that can be estimated by multiples (possibly depending on ) of gW F,k−1,γ ,T , which are of no consequence. From Lemma 4.1.2 it follows that m−1  q m−q−1 m l b b+l m l ∂t ∂x K t g ≡ K t [L b+l ∂ y g] + L b+l ∂t g. (10.193) q=0

We write that l ∂tm ∂xl K tb g ≡ K tb+l [L m b+l ∂ y g] + ᏻ(k − 2).

(10.194)

Here ᏻ(k −2) denotes terms for which (W F, 0, γ , T )-norms are estimated by multiples of gW F,k−2,γ ,T , which are also of no consequence. The new contributions to E t gW F,k,γ ,T come from terms like b j +l

[ψ j, , L]K t

l Lm b+l ∂ y (ϕ j, g)W F,0,γ ,T ,

b +l l ψ j, ( b j (x 0 )∂x j )K t j L m b+l ∂ y (ϕ j, g)W F,0,γ ,T ,

and

b +l−1 m ψ j, [∂x j  L b+l−1 ∂ yl−1 (ϕ0, g), b j ]∂x j K t j

(10.195)

(10.196)

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for j = 0, 1. The terms in (10.195) are precisely the sorts of terms estimated earlier in the chapter, with exactly the same coefficients. All that has changed is that we have b b +l l replaced K t j with K t j and ϕ j, g with L m b+l ∂ y (ϕ j, g). From the Leibniz formula, it is again clear that the only terms that cannot be subsumed into gW F,k−1,γ ,T are those of the form b j +l

[ψ j, , L]K t

l ϕ j, L m b+l ∂ y (g)W F,0,γ ,T , b +l l ψ j, ( b j (x 0 )∂x j )K t j ϕ j, L m b+l ∂ y (g)W F,0,γ ,T .

These terms can all be estimated by C 2−γ gW F,k,γ ,T ,

(10.197)

where the constant C is uniformly bounded for k ≤ K . To complete this case we need to consider the terms in (10.196); these are not a priori of lower order because ∂x j  b j may not vanish at x j = 0. On the other hand, the estimate given in (6.104) shows that there are constants Ck , μ(k, γ ) so that b j +l−1 m L b+l−1 ∂ yl−1 (ϕ0, g)W F,0,γ ,T = b +l l−1 ψ j, [∂x j  b j ]K t j ∂ y L m b+l−1 ∂ y (ϕ0, g)W F,0,γ ,T

ψ j, [∂x j  b j ]∂x j K t

≤ Ck  −μ(k,γ ) T

1− γ2

(10.198)

gW F,k,γ ,T .

If we fix any δ > 0, then we can choose an  > 0 so that, C 2−γ < δ and therefore for some constants {βk }, the estimates γ

t gW F,k,γ ,T ≤ (δ + Ck  −μ(k,γ ) T 1− 2 )gW F,k,γ ,T + βk gW F,k−1,γ ,T (10.199) E b

hold for k ≤ K . Fix a δ < 1/2, which thereby fixes an  > 0. Let ϕb = ϕ0, +ϕ1, , and choose ψi with compact support [a, b] ⊂ (0, 1) and equal to 1 on a neighborhood of

t be the exact solution operator to the Dirichlet problem supp(1 − ϕb ). Finally we let Q i (∂t − L)u = g on [a, b] with u(x, 0) = u(a, t) = u(b, t) = 0.

(10.200)

With the global parametrix given by

we see that

t + ψi Q t = Q

t (1 − ϕb ), Q b i

(10.201)

t t g = g + E b

ti (1 − ϕb )g. (∂t − L) Q g + [ψi , L] Q

(10.202)

Since the support of 1 − ψi and 1 − ϕb do not overlap, the induction hypothesis shows that there is a constant C(T, k, γ ), which tends to 0 as T → 0+ , so that

t (1 − ϕb )gW F,k,γ ,T ≤ C(T, k, γ )gW F,k,γ ,T . [ψi , L] Q i Note that  > 0 has already been fixed.

(10.203)

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215

t

t (1 − ϕb )g, then for some T0 > 0, there are If we let E t g = E b g + [ψi , L] Q i constants {βk : k = 0, . . . , K } so that we have the estimates

E t gW F,k,γ ,T0 ≤ 2δgW F,k,γ ,T0 + βk gW F,k−1,γ ,T0 , for 0 ≤ k ≤ K .

(10.204)

Theorem 10.8.1 applies to show that the Neumann series for (Id +E t )−1 converges in k,γ the operator norm topologies defined by ᏯW F ([0, 1] × [0, T0 ]) for 0 ≤ k ≤ K . The argument at the end of Section 10.3 applies to show that this operator has the small time k,γ k,2+γ localization property as a map from ᏯW F ([0, 1] × [0, T0 ]) to ᏯW F ([0, 1] × [0, T0 ]). As K is arbitrary we see that this completes the proof, in dimension 1, of the induction step for the inhomogeneous case and any k. 10.8.2 The Higher Dimensional Case The argument in the general case is quite similar to the 1-dimensional case, though there are more terms analogous to those appearing in (10.196). We now briefly describe it. As above the key point is to show that estimates like those in (10.191) and (10.199) hold for the error terms coming from the boundary parametrix. This fixes a choice of  > 0, and then we can apply the induction hypothesis to obtain similar estimates for the contribution of the interior parametrix to the error term, which, along with the contributions of terms like those in (10.196), is made as small as we like by taking 0 < T0 small enough. The boundary contributions to the error term are enumerated in (10.134). It is immediate that the only contributions to [E ∞,t + E 1,t ]gW F,k,γ ,T rel [E ∞,t + E 1,t ]gW F,k−1,γ ,T are of the terms of the types b+α,t ψi,q dl (x, y)∂ yl K i,q [L b+α,m ∂wα ∂ zβ ]χi,q gW F,0,γ ,T

(10.205)

and b+α,t [L b+α,m ∂wα ∂ zβ ]χi,q gW F,0,γ ,T . [ψi,q , L]K i,q

(10.206)

These are types of terms that we have estimated earlier (see (10.138)); hence for 0 ≤ k ≤ K , there are constants C(T, k, γ ), μ(k, γ ) and {βk } so that [E ∞,t + E 1,t ]gW F,k,γ ,T ≤ C(T, k, γ ) −μ(k,γ ) gW F,k,γ ,T + βk gW F,k−1,γ ,T . (10.207) Moreover C(T, k, γ ) tend to zero as T → 0+ . This leaves only terms of the form   β b,t [χi,q g] W F,0,γ ,T , (10.208) ∂tm ∂ y ∂ xα ψi,q L ri,q K i,q

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with 2m + |α| + |β| = k. As in the 1-dimensional case, there are two types of terms that now need to be estimated. The first type arises by passing all derivatives through to χi,q g, which are of the form   β b+α,t ψi,q L ri,q K i,q L b+α,m ∂ y ∂ xα [χi,q g] W F,0,γ ,T . (10.209) These are precisely the sorts of terms estimated in the k = 0 case. As before we choose 0 < γ  so that γ + γ  < 1. For 0 ≤ k ≤ K , there are constants C(k, γ ) for which    β b+α,t L b+α,m ∂ y ∂ xα [χi,q g] W F,0,γ ,T ≤ C(k, γ ) 1−γ −γ gW F,k,γ ,T . ψi,q L ri,q K i,q (10.210) The only terms that remain result from differentiation of the coefficients of terms appearing in L ri,q of the forms x j ∂x j , x j ∂x j ∂ yl , or x i x j ∂xi ∂x j . The parts of terms of these types that cannot be subsumed by a large multiple of gW F,k−1,γ ,T are   β  b+α  ,t ψi,q ∂x j K i,q L b+α  ,m ∂ y ∂ xα [χi,q g] W F,0,γ ,T =   β  b+α,t ψi,q K i,q ∂x j L b+α ,m ∂ y ∂ xα [χi,q g] W F,0,γ ,T , (10.211)   β  b+α  ,t L b+α  ,m ∂ y ∂ xα [χi,q g] W F,0,γ ,T = ψi,q ∂x j ∂ yl K i,q   β  b+α,t ψi,q K i,q ∂x j ∂ yl L b+α ,m ∂ y ∂ xα [χi,q g] W F,0,γ ,T , (10.212) where α  = α − e j ; and 

 β  L b + α  , m∂ y ∂ xα [χi,q g] W F,0,γ ,T =   β  b+α,t ψi,q x i K i,q ∂xi ∂x j L b+α ,m ∂ y ∂ xα [χi,q g] W F,0,γ ,T , (10.213)

b+α ψi,q x i ∂x j ∂xi K i,q

 ,t

where α  = α − ei − e j ; and   β  b+α  ,t ψi,q ∂x j ∂xi K i,q L b+α ,m ∂ y ∂ xα [χi,q g] W F,0,γ ,T =   β  b+α,t ψi,q K i,q ∂x j ∂xi L b+α  ,m ∂ y ∂ xα [χi,q g] W F,0,γ ,T .

(10.214)

It follows from (9.50) and the foregoing argument that for 0 ≤ k ≤ K , there are constants C(k, γ ), μ(k, γ ) so that, for T < 1, each of the terms in (10.211)–(10.214) is bounded by γ C(k, γ ) −μ(k,γ ) T 2 gW F,0,γ ,T . (10.215) Combining (10.207) with (10.210) and (10.215), we see that there are constants {βk } so that t E b gW F,k,γ ,T ≤ 

[C(T, k, γ ) −μ(k,γ ) + C(k, γ ) 1−γ −γ ]gW F,k,γ ,T + βk gW F,k−1,γ ,T ,

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EXISTENCE OF SOLUTIONS

where C(T, k, γ ) → 0 as T → 0+ . If we fix 0 < δ < 1/2, then by first choosing  > 0 and then T0 > 0 we can arrange to have t E b gW F,k,γ ,T ≤ δgW F,k,γ ,T + βk gW F,k−1,γ ,T ,

(10.216)

for 0 ≤ k ≤ K . As in the 1-dimensional case, the argument is finished by augmenting the boundary parametrix with an interior term, obtaining

tb + ψi Q

ti (1 − ϕb ). ᏽt = Q

(10.217)

Possibly decreasing T0 , we obtain an error term E t that satisfies E t gW F,k,γ ,T0 ≤ 2δgW F,k,γ ,T0 + βk gW F,k−1,γ ,T0 ,

(10.218)

for 0 ≤ k ≤ K . This completes the proof of (10.180) for an arbitrary K ∈ ⺞ and 0 < γ < 1.

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Chapter Eleven The Resolvent Operator We have shown that et L , the formal solution operator for the Cauchy problem (∂t − L)v = 0, v( p, 0) = f ( p), 0,2+γ

0,2+γ

makes sense for initial data f ∈ ᏯW F (P), and that the solution belongs to ᏯW F (P× [0, ∞)). Of course, much more is true, but the extension to merely continuous data seems to entail rather different techniques from those employed herein. The Laplace transform of et L is formally the resolvent operator: (μ − L)

−1

∞ =

e−μt et L f dt.

(11.1)

0

Using the Laplace transform of a parametrix for the heat kernel and a perturbative argument, we construct below an operator R(μ), which depends analytically on μ lying in the complement of a set E ⊂ ⺓ which lies in a conic neighborhood of (−∞, 0]. This means that for any α > 0 there exists an 0 < Rα so that E ⊂ {| arg μ| > π − α or |μ| < Rα }.

(11.2)

See Figure 11.1. 0,γ 0,2+γ If f ∈ ᏯW F (P), then R(μ) f ∈ ᏯW F (P), satisfies (μ − L)R(μ) f = f.

(11.3) 0,γ

Hence R(μ) is a right inverse for (μ− L). As a map from ᏯW F (P) to itself the operator R(μ) is compact. In fact for any 0 < γ < 1 and k ∈ ⺞, R(μ) defines a bounded map k,γ k,2+γ from ᏯW F (P) to ᏯW F (P). In H¨older spaces, these are the natural elliptic estimates for generalized Kimura diffusions. Coupling this with Corollary 10.5.3 shows that the k,2+γ operator μ − L, acting on the spaces ᏯW F (P), is injective for μ in the right half plane. Since equation (11.3) shows that for such μ, 0,2+γ

0,γ

(μ − L) : ᏯW F (P) → ᏯW F (P) is surjective, the open mapping theorem implies that R(μ) is also a left inverse, and hence equals the resolvent operator (μ − L)−1 . 218

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THE RESOLVENT OPERATOR

x x x x

x

x x

x

x

x x x

x

π−α Rα

x

Figure 11.1: A discrete set lying in a conic neighborhood of (−∞, 0). 0,2+γ

0,γ

Since the domain ᏯW F (P) is not dense in ᏯW F (P) a few more remarks are in order. Suppose that μ is a value for which (μ − L) is invertible. We can rewrite (μ + ν − L) = [Id +ν(μ − L)−1 ](μ − L). 0,2+γ

(11.4)

0,γ

The map (μ − L) : ᏯW F (P) → ᏯW F (P) is an isomorphism. The maps [Id +ν(μ − L)−1 ] : ᏯW F (P) → ᏯW F (P) 0,γ

0,γ

depend analytically on ν and are Fredholm of index zero. From this we conclude that the set of ν for which (μ + ν − L) fails to be invertible is discrete and coincides with the set {ν : ker(μ + ν − L) = 0}. (11.5) 0,2+γ

0,γ

Thus L : ᏯW F (P) → ᏯW F (P) has a compact resolvent, with discrete spectrum lying in a conic neighborhood of the negative real axis. Moreover, the elliptic estimates show that all eigenfunctions belong to Ꮿ∞ (P), so the spectrum of L acting on the k,2+γ spaces ᏯW F (P) does not depend on k or γ . Using standard functional analytic techniques this allows us to show that the solution to the Cauchy problem (∂t − L)v = 0 with v( p, 0) = f ( p) 0,γ

(11.6)

is defined for f ∈ ᏯW F (P) and, in fact, extends analytically in t to the right half plane. 0,2+γ The solution belongs to ᏯW F (P) for any time with positive real part. Indeed we also

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CHAPTER 11 k,γ

k,2+γ

show that, for any k ∈ ⺞, if f ∈ ᏯW F (P), then the solution belongs to ᏯW F (P), for t in the right half plane. The solution operator, ᏽt0 , defines a semi-group; thus, for any N ∈ ⺞ t

ᏽt0 f = [ᏽ0N ] N f. 1,γ

(11.7)

0,2+γ

We have the obvious inclusions ᏯW F (P) ⊂ ᏯW F (P), and in fact for any k ∈ ⺞, we k+1,γ k,2+γ have ᏯW F (P) ⊂ ᏯW F (P). These inclusions, the semi-group property, and these regularity results show that the solution v to the Cauchy problem, with H¨older initial data, belongs to Ꮿ∞ (P × (0, ∞)). In the next section we construct the resolvent kernel, using an induction over the maximal codimension of b P, similar to that employed in the previous chapter to construct the heat kernel. We also prove various estimates on it and corresponding estimates for the solution operator for the Cauchy problem. 11.1 CONSTRUCTION OF THE RESOLVENT To construct the resolvent operator we proceed very much as for the construction of the heat kernel. We use an induction over the maximal codimension of b P, which allows us to construct an approximate solution operator for the Cauchy problem of the form

t = Q

ti + Q

tb , Q

(11.8)

t the “interior” and Q

t the “boundary” contributions, respectively. We then with Q i b analyze the operator:

R(μ) =

∞

t dt e−t μ Q

0

(11.9)

b (μ) + R

i (μ). =R

t extends analytically to Re t > 0, and from its form we see that R

b (μ) The operator Q b extends analytically to the complement of (−∞, 0]. From the induction hypothesis

i (μ) is analytic in the complement of a discrete set lying in a conic it follows that R neighborhood of (−∞, 0]. We show that

(μ − L) R(μ) = (Id −E μ ), (11.10) k,γ

k,γ

where the operator E μ : ᏯW F (P) → ᏯW F (P) is bounded for arbitrary k ∈ ⺞0 and 0 < γ < 1. We show that for a given k, 0 < γ , and 0 < α there is an Rα so that for μ satisfying | arg μ| ≤ π − α and |μ| > Rα , (11.11) the norm of this operator is less than 1 and therefore, for μ in this domain, we can define the analytic family of operators:

R(μ) = R(μ)(Id −E μ )−1 .

(11.12)

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This operator is a right inverse k,γ

(μ − L)R(μ) f = f for all f ∈ ᏯW F (P).

(11.13)

We then verify the estimates in the induction hypothesis. As noted above this allows us to construct the solution operator for the Cauchy problem for the heat equation via the contour integral:  1 ᏽt0 = R(μ)eμt dμ. (11.14) 2πi α

The contour α is the boundary of the complement of the region described in (11.11). k,γ This defines a semi-group, analytic in Re t > 0, acting on the spaces ᏯW F (P). The theorem we prove is the following: T HEOREM 11.1.1. Let P be a manifold with corners of codimension n and L a generalized Kimura diffusion operator. Fix k ∈ ⺞ and 0 < γ < 1. There is a discrete subset E, independent of (k, γ ), contained in Re μ ≤ 0 and lying in a conic neighk,2+γ borhood of (−∞, 0], such that the spectrum of L acting on ᏯW F (P) is contained in the set E. The resolvent operator R(μ) is analytic in ⺓ \ E. For 0 < α there is an Rα so that for μ satisfying (11.11) there are constants Cα , Ck,α so that R(μ) satisfies the following estimates: Cα  f  L ∞ for μ ∈ (0, ∞) μ Ck,α  f W F,k,γ ≤ |μ| ≤ Ck,α  f W F,k,γ .

R(μ) f  L ∞ ≤ R(μ) f W F,k,γ R(μ) f W F,k,2+γ

(11.15)

Let V be a vector field defined in P so that, in the neighborhood of a boundary point of codimension l, V takes the form V (x, y) =

l 

b j (x, y)x j ∂x j +

j =1

n−l 

dl (x, y)∂ yl .

(11.16)

l=1

For μ satisfying (11.11) there are constants Ck,α so that, if |μ| > 1, then V R(μ) f W F,k,γ ≤

Ck,α γ

|μ| 2

 f W F,k,γ .

(11.17)

P ROOF. The proof is very similar to that of Theorem 10.0.2, and so many details are left to the reader. The construction of the resolvent is done by induction over the maximal codimension of b P. The verification of the induction hypothesis in this proof is actually somewhat simpler, as we do not use the weak localization property. We begin with the case that P is compact manifold without boundary, i.e., the maximum

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CHAPTER 11

codimension of b P is zero, and L is a non-degenerate elliptic operator without constant term. The H¨older spaces are simply the classical H¨older spaces, and the statement of the theorem is more or less contained in [29], though this text does not address the compact manifold case explicitly. As the detailed estimates for R(μ) stated in (11.15) and (11.17) also do not seem to be available in the literature, we start by briefly outlining this case. As before we begin with the k = 0 case. The case of k > 0 follows by applying 10.8.1, very much like in the proof of Theorem 10.0.2. The details of this argument are also left to the reader. We continue to use the notation and constructions from Sections 10.2–10.4. 11.1.1 The Compact Manifold Case For  > 0 we cover P by open balls of radius , {B (x q ) : q = 1, . . . , N }, so that any point lies in at most S of the balls {B3 (x q ) : q = 1, . . . , N }. As noted earlier, S can be taken to be independent of  > 0. Let {ϕq } denote a partition of unity subordinate to this cover and {ψq } smooth functions, such that ψq (x) = 1 in B2 (x q ), and supp ψq ⊂ B3 (x q ).

(11.18)

We let L q denote the constant coefficient operator obtained by freezing the coefficients of the second order part of L at the point x q . If (y1 , . . . , yn ) are local coordinates near to x q , then q

L =

n 

clm (x q )∂ yl ∂ ym .

(11.19)

l,m=1

We let Q tq be the heat kernel defined by L q . This is obtained from the Euclidean heat kernel by a linear change of variables. The parametrix for the heat kernel is defined, for t in the right half plane, by

t = Q

N 

ψq Q tq ϕq ,

(11.20)

q=1

and the resolvent, for μ ∈ ⺓ \ (−∞, 0], by

R(μ) =

∞

e−se

iθ μ

seiθ eiθ ds, Q

(11.21)

0

where Re[eiθ μ] > 0. In the sequel we let θ = {seiθ : s ∈ [0, ∞)}.

(11.22)

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THE RESOLVENT OPERATOR

We now compute the “error term,” E μ 

L R(μ) f =

e−t μ

Lψq Q tq ϕq f dt

q=1

θ



=

N 

e

−t μ

N  : ; [L, ψq ] + ψq (L − L q ) + ψq L q r Q tq ϕq f dt.

(11.23)

q=1

θ

Using that L q Q tq = ∂t Q tq , we integrate by parts in the last term and obtain

(μ − L) R(μ) f = f −



e−t μ

N  :

; [L, ψq ] + ψq (L − L q ) Q tq ϕq f dt

q=1

θ

(11.24)

= f − E μ f. There are two kinds of error terms: those arising from the commutators [L, ψq ], which are lower order, and those arising from freezing coefficients ψq (L − L q ). The differences L − L q are of the form q

L−L = =

n  l,m=1 n 

(c( y) − clm (x q ))∂ yl ∂ ym +

n 

dl ( y)∂ yl

l=1

(11.25)

cq ( y)∂ yl ∂ ym + V q f.

l,m=1

As in the previous case, the second order terms of this type are controlled by taking  sufficiently small. The contribution of each such term is of the form ψq cq ∂ yl ∂ ym R q (μ)ϕq f W F,0,γ , 

where R q (μ) f =

e−μt Q tq f dt.

(11.26)

(11.27)

θ

Arguing as in the proof of Theorem 10.0.2 and using the estimates for R q (μ) given in Proposition 9.4.1, we see that there is a  γ < 1, so that γ ψq cq ∂ yl ∂ ym R q (μ)ϕq f W F,0,γ ≤ Cα  1−  f W F,0,γ .

(11.28)

As before, for each point in P there is at most a fixed finite number of terms, S, independent of , contributing to this error term. This gives the estimate 

N  q=1

ψq

n 

γ (c( y) − clm (x q ))∂ yl ∂ ym R q (μ)ϕq f W F,0,γ ≤ SCα  1−  f W F,0,γ .

l,m=1

(11.29)

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We can now fix  > 0 so that the coefficient γ SCα  1− ≤

1 . 4

The commutators are first order operators: [L, ψq ] f = 2

n 

clm ( y)∂ yl ψq ∂ ym f +

l,m=1

n 

dl ( y)(∂ yl ψq ) f.

(11.30)

l=1

These terms, along with that defined by the vector fields {V q }, are controlled using the estimates in (9.168) and (9.169). These estimates show that, for some positive ν, there is a constant Cα so that [L, ψq ]ϕq f W F,0,γ + V q ϕq f W F,0,γ ≤

 −ν Cα γ

|μ| 2

 f W F,0,γ .

(11.31)

Combining these estimates gives / E μ f W F,0,γ ≤ SCα

 −ν

γ

|μ| 2

0 +

1− γ

 f W F,0,γ .

(11.32)

This shows that there is an R0 so that if |μ| > R0 , then the norm of E μ is less than 0,γ 1 2 , and therefore (Id −E μ ) is well defined as an operator from Ꮿ W F (P) to itself. The analytic dependence on μ follows from the analyticity of Id −E μ and the uniform norm convergence of the Neumann series. If we define

R(μ) f = R(μ)(Id −E μ )−1 f,

(11.33)

0,γ

then we see that, for any f ∈ ᏯW F (P), we have that 0,2+γ

R(μ) f ∈ ᏯW F (P) and (μ − L)R(μ) f = f.

(11.34)

Finally, it is a classical result that for Re μ > 0, on a compact manifold, the only solution of the equation (μ − L) f = 0 (11.35) 0,2+γ

0,γ

is f ≡ 0, and therefore (μ − L) : ᏯW F (P) → ᏯW F (P) is a one-to-one and onto mapping. The open mapping theorem implies that R(μ) is also a left inverse. Hence the identity R(μ)(μ − L) = Id = (μ − L)R(μ) (11.36) holds in the connected component where R(μ) is analytic, which contains the right half plane. We have shown that this set contains the complement of a conic neighborhood of (−∞, 0]. The first estimate in (11.15) follows from the maximum principle. As a map 0,γ from ᏯW F (P) to itself R(μ) is compact, and therefore the spectrum of L acting on

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THE RESOLVENT OPERATOR 0,2+γ

ᏯW F (P) is a discrete set E. We have shown that E is contained in a conic neighbor-

hood of (−∞, 0]. Arguing as in Section 10.8 we can show that, for any 0 < φ, there are constants {αk , βk , Rk }, with αk < 1, so that, if | arg μ| < π − φ, and |μ| > Rk , then E μ f W F,k,γ ≤ αk  f W F,k,γ + βk  f W F,k−1,γ .

(11.37)

Applying Theorem 10.8.1 we see that the Neumann series for (Id −E μ )−1 converges k,γ in the operator norm defined by ᏯW F (P), thus establishing that these results extend to show that, for any k ∈ ⺞, the maps k,γ

k,2+γ

R(μ) : ᏯW F (P) −→ ᏯW F (P)

(11.38)

are also bounded. The estimates in the statement of the theorem, (11.15) and (11.17) for k > 0 follow easily since it is simply a matter of establishing these estimates for



R(μ). For example, using that the second estimate in (11.15) holds for R(μ), we see that

R(μ) f W F,k,γ =  R(μ)(Id −E μ−1 ) f W F,k,γ Cα (Id −E μ−1 ) f W F,k,γ ≤ |μ| C ≤ α  f W F,k,γ . |μ|

(11.39)

As the other estimates hold for R(μ) it follows by the same sort of argument that they also hold for R(μ). 0,2+γ Suppose that μ ∈ E, and f μ ∈ ᏯW F (P) is a non-trivial eigenfunction, with (μ − L) f μ = 0.

(11.40)

If we select ν so that Re(ν + μ) is sufficiently large, then the eigenvalue equation implies that f μ = ν R(μ + ν) f μ . (11.41) k,2+γ

k+1,γ

Since ᏯW F (P) ⊂ ᏯW F (P), we can use (11.38) in a bootstrap argument to conclude that k,γ f μ ∈ ᏯW F (P) for all k. (11.42) k,2+γ

From this we conclude that f μ ∈ Ꮿ∞ (P), and the spectrum of L acting on ᏯW F (P), does not depend on k. 11.1.2 The Induction Argument The proof now proceeds by induction on the maximal codimension of the components    is a of b P. Suppose that the theorem has been proved for all pairs ( P, L) where P  manifold with corners, with the maximal codimension of b P at most M, and  L is a

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 We let P be a manifold with corners where the generalized Kimura diffusion on P. maximal codimension of b P is M + 1 and L be a generalized Kimura diffusion on



P. The parametrix R(μ) for R(μ) is constructed as in Section 10.4.2, with R(μ) =

b (μ) + R

i (μ). As R ∞ 

(11.43) Ri (μ) = e−t μ ψet L (1 − ϕ)dt, 0

i (μ) has an analytic extension to ⺓ \ F, where the induction hypothesis implies that R F is a discrete set lying in a conic neighborhood of (−∞, 0]. The only change is that, instead of (10.130), we let

b (μ) = R

p  

b ψi,q Ri,q (μ)χi,q ,

(11.44)

b,t e−μt ki,q dt,

(11.45)

i=1 q∈Fi,

where

∞ b Ri,q (μ)

= 0

with

b,t ki,q

the solution operator for the model problem (∂t − L i,q )v = 0 with v( p, 0) = f ( p).

(11.46)

The error terms are quite similar to those arising in the previous case. If b (μ)χi,q f, wi,q = ψi,q Ri,q

(11.47)

then q

b b (μ)[χi,q f ] + [ψi,q , L]Ri,q (μ)[χi f ]. (μ − L i,q )wi,q = χi,q f + ψi,q (L i,q − L)Ri,q (11.48) We again use the decomposition of L i,q − L given in (10.118) to write the error terms as 2 b E i,q,μ f = ψi,q L ri,q Ri,q (μ)[χi,q f ], 6 & m  1 b dl (x, y)∂ yl Ri,q (μ)[χi,q f ]. E i,q,μ f = [ψi,q , L] +

(11.49)

l=1

Using the estimates in (9.173) and the argument from Section 10.4.2 we conclude that there is a constant Cα so that if | arg μ| < π − α, then 

2 E i,q,μ f W F,0,γ ≤ Cα  1−γ −γ  f W F,0,γ ,

(11.50)

where γ + γ  < 1. Once again there is an S independent of  > 0, so that 

2 E ,μ f W F,0,γ ≤ SCα  1−γ −γ  f W F,0,γ ,

(11.51)

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THE RESOLVENT OPERATOR

with 2 E ,μ =

p  

2 E i,q,μ .

(11.52)

i=1 q∈Fi, 

We can now fix  > 0 so that SCα  1−γ −γ < 14 . The commutators are of the form [ψi,q , L] =

M+1 

bi (x, y)x i ∂xi +

n−(M+1) 

i=1

dl (x, y)∂ yl .

(11.53)

l=1

The estimates in (9.169) and (9.170), along with the argument in Section 10.4.2, show that there are constants Cα and ν so that 1 f W F,0,γ ≤ SCα E ,μ

 −ν

γ

|μ| 2

 f W F,0,γ .

(11.54)

With the given choice of  > 0, we again define ϕ as in (10.151). With this choice we have the estimate

b (μ) f − ϕ f W F,0,γ = E b,μ f W F,0γ ≤ (μ − L) R / 0  −ν 1−γ −γ  SCα  f W F,0,γ . γ + |μ| 2

(11.55)

We now proceed exactly as in Section 10.3: let U be a neighborhood of  M+1   with U ⊂⊂ int ϕ −1 (1). As before we apply Theorem 10.2.1 to find ( P, L) so that    the maximal codimension of b P is M and (PU , L U ) is embedded into ( P, L). We let  R(μ) be the resolvent operator for L U , whose existence and properties follow from the induction hypothesis. Finally we choose ψ, a smooth function equal to 1 on the support of (1 − ϕ), compactly supported in U c , and let

and



i (μ) f = ψ R(μ)[(1 − ϕ) f ], R

(11.56)

b (μ) f.

i (μ) f + R R(μ) f =R

(11.57)

 (μ − L) R(μ) f = f − [L, ψ] R(μ)[(1 − ϕ) f ] + E b,μ f.

(11.58)

We see that

The commutator [L, ψ] is a vector field of the form (11.16) in each adapted coordinate frame. Hence the induction hypothesis implies that there is a constant Cα so that Cα  −ν  [L, ψ] R(μ)[(1 − ϕ) f ]W F,0,γ ≤ γ  f  W F,0,γ . |μ| 2

(11.59)

Altogether this shows that with

f − f −E μ f = (μ − L) R(μ)

(11.60)

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CHAPTER 11

/

we have E μ f W F,0,γ ≤ SCα

 −ν

γ

|μ| 2

0 +

1−γ −γ 

 f W F,0,γ .

Thus, we can choose Rα so that if |μ| > Rα then / 0 1  −ν  1−γ −γ < , SCα γ + 2 |μ| 2

(11.61)

(11.62)

so that the Neumann series for (Id −E μ )−1 converges in the operator norm topology 0,γ defined by ᏯW F (P) in the set | arg μ| < π − α and |μ| > Rα .

(11.63)

It is clear that the family of operators (Id −E μ )−1 is analytic in a conic neighborhood of (−∞, 0]. If we let

R(μ) = R(μ)(Id −E μ )−1 , (11.64) 0,γ

0,2+γ

then this is an analytic family of operators, mapping ᏯW F (P) to ᏯW F (P), which satisfies 0,γ (μ − L)R(μ) f = f for f ∈ ᏯW F (P). (11.65) As before, Corollary 10.5.3 shows that (μ − L) is injective for μ in the right half plane. The open mapping theorem then implies that (μ − L) is actually invertible for Re μ > 0, and therefore 0,2+γ

R(μ)(μ − L) f = f for f ∈ ᏯW F (P),

(11.66)

as well. As noted in the compact manifold case, the fact that R(μ) satisfies all the estimates in (11.15) and (11.17) follows immediately from the boundedness of (Id −E μ )−1 : ᏯW F (P) −→ ᏯW F (P), 0,γ

0,γ

(11.67)

and the fact that R(μ) satisfies these estimates. This latter claim follows from the fact that the model operators satisfy these estimates, and, by the induction hypothesis, so  does R(μ). This completes the induction step in the k = 0 case. The cases where k > 0 are quite similar to that treated in Section 10.8. This case is somewhat simpler, as we do not need to estimate time derivatives. This means that we only need to use the formulæ in (4.27) with j = 0. In this case powers of L b,m do not appear on the right-hand side, and no hypothesis is required on the support of the data. The only other significant difference concerns the higher order estimates in (11.17). The contributions of the interior terms are estimated, for all k, by using the induction hypothesis and the fact that the commutator [ψ, L] is of the form given in (11.16). The estimates in (9.170) give / 0 1 1

b (μ) f W F,k,γ + ∇ y R

b (μ) f W F,k,γ ≤ Cα  −ν x · ∇ x R  f W F,k,γ . γ + |μ| |μ| 2 (11.68)

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THE RESOLVENT OPERATOR

For |μ| > 1, this therefore gives the desired estimate, and completes the verification of the induction hypothesis for M + 1. This completes the proof of the theorem.  11.2 HOLOMORPHIC SEMI-GROUPS Now that we have constructed the resolvent operator for L and demonstrated that it is analytic in the complement of a conic neighborhood of (−∞, 0], we can use contour integration to construct the solution to the heat equation. Our second pass through this problem represents a distinct improvement over our previous result for several reasons: 0,γ

0,2+γ

1. This time we can work with data belonging to ᏯW F (P), rather than ᏯW F (P). 0,2+γ

2. For such data the solution is shown to belong to ᏯW F (P) for positive times. If k,γ k,2+γ the data is in ᏯW F (P), then the solution belongs to ᏯW F (P). 3. A bootstrapping argument, using the inclusion k,2+γ

k+1,γ

ᏯW F (P) ⊂ ᏯW F (P)

and the semi-group property, gives that the solution belongs to Ꮿ∞ (P), for positive times. 4. The solution extends analytically to t in the right half plane, H+ . For any α > 0, 0 < γ < 1, and k ∈ ⺞, there is an 0 < Rα so that as a map from R(μ), constructed in Theorem 11.1.1, is analytic in a domain containing the set | arg μ| ≤ π − α, and |μ| ≥ Rα , and satisfies the estimates in the theorem. We let α denote the boundary of the complement of this region. From these observations, the following theorem follows from standard results in semi-group theory. See, for example, the proof of Theorem 8.2.1 in [29], or that of Theorem 2.34 in [13]. k,γ k,2+γ ᏯW F (P) to ᏯW F (P) the operator

k,γ

T HEOREM 11.2.1. For f ∈ ᏯW F (P) and t with | arg t| < 1 v(t, p) = T f ( p) = 2πi



t

π 2

− α, define

et μ R(μ) f ( p)dμ.

(11.69)

α

Then: 1. For any p ∈ P the function t → v(t, p) is analytic in the right half plane, and, for s, t in the right half plane: T t T s f = T t +s f. k,2+γ

2. For any t ∈ H+ , v(t, ·) ∈ ᏯW F (P).

(11.70)

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CHAPTER 11

3. For any 0 < α, there is a Cα so that for t with | arg t| < π2 − α, we have the estimates  1 T t f W F,k,2+γ ≤ Cα e Rα |t | +  f W F,k,γ . (11.71) |t| 4. v satisfies the heat equation in {t : Re t > 0} × P : ∂t v = Lv.

(11.72)

5. For t real, we have T t f  L ∞ ≤  f  L ∞ and lim T t f − f  L ∞ = 0.

(11.73)

t →0+

6. For  γ < γ , we have

lim T t f − f W F,k, γ = 0.

(11.74)

t →0+

Remark. From the higher order regularity results and a simple integration by parts 2(k+1),γ (P), for a k ∈ ⺞ and 0 < γ < 1, then argument, it follows that if f ∈ ᏯW F t L v( p, t) = e f ( p) is given by the Taylor series, with remainder, for the exponential:  k  t j L j f ( p) (t − s) j +1 j +1 + ∂ v( p, s)ds. v( p, t) = j! ( j + 1)! t t

j =0

(11.75)

0

k+1,γ

As noted earlier, the regularity statement in this theorem and the fact that ᏯW F (P) ⊂ have an important corollary:

k,2+γ ᏯW F (P)

0,γ

C OROLLARY 11.2.2. If, for some 0 < γ , f ∈ ᏯW F (P), then v = T t f belongs to + × P).

Ꮿ∞ (H

11.3 DIFFUSIONS WHERE ALL COEFFICIENTS HAVE THE SAME LEADING HOMOGENEITY In this brief section we return to an issue which we have alluded to earlier. Namely, from the points of view of both probability and analysis, it is quite natural to study a slightly broader class of Kimura diffusions with generators of the form L=

n  i, j =1

√ √ ai j (x, y) x i x j ∂x2i x j + x i bik (x, y)∂x2i yk + n

m

i=1 k=1 m 

ckl (x,

k,l=1

(11.76) y)∂ y2k yl

+ V,

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THE RESOLVENT OPERATOR

where V is an inward pointing vector field, and assuming a degenerate ellipticity condition on the coefficients. The difference between this expression and the one in the √ original Definition 2.2.1 is that we allow cross-terms which vanish at the rates x i x j instead of x i x j . This is not an innocuous change, since as we show below, the crossterms are no longer negligible error terms, which can be easily absorbed. It is possible to adapt the methods developed in this monograph to study operators of this more general type. The strategy is essentially the same: one first analyzes the model heat problem and establishes appropriate H¨older estimates, and then uses a similar parametrix construction and perturbation argument to obtain the corresponding estimates for solutions of the actual heat operator ∂t − L. The important difference between this and our earlier work is that the model operator is no longer “diagonal,” and hence may be substantially more difficult to analyze. We do not claim to have a general method to carry this out at this time, though we hope to return to this elsewhere. If we assume that the model operator is “nearly diagonal,” then one can obtain sufficient information using perturbation arguments. This is similar to what was done by Bass and Perkins [4], who treated a more restricted class of operators on the orthant, ⺢n+ , instead of a manifold with corners. In particular these operators do not involve tangential, or y-variables. For simplicity, we assume that the coefficients ai j , bik and ckl are smooth functions on P. This hypothesis can be relaxed substantially, but, at present, we are concerned with issues connected to the rate of vanishing of the off-diagonal terms. First let us comment on the ellipticity of L. Under the change of variables ξi = √ 2 x i , we have √

x i ∂xi = ∂ξi ,

√

& x i x j ∂x2i x j

=

∂ξ2i ξ j ∂ξ2i

+

i = j 1 2ξi ∂ξi

otherwise,

hence the second order part of L becomes n  i, j =1

ai j ∂ξ2i ξ j +

m n  

bik ∂ξ2i yk +

i=1 k=1

m 

ckl ∂ y2k yl .

k,l=1

Thus ellipticity corresponds to the positive definiteness of the matrix ⎛

a11 ⎜ .. ⎜ . ⎜ ⎜ an1 A=⎜ ⎜ 1 b11 ⎜2 ⎜ . ⎝ .. 1 2 b1m

... ... ... ... ... ...

a1n .. . ann 1 2 bn1

.. . 1 b 2 nm

1 2 b11

.. . 1 b 2 n1 c11 .. . cm1

... ... ... ... ... ...

⎞ 1 2 b1m

.. ⎟ . ⎟ ⎟ 1 ⎟ b 2 nm ⎟ . c1m ⎟ ⎟ .. ⎟ . ⎠ cmm

The fact that the x and y directions are no longer orthogonal when x = 0 means that any linear change of coordinates which diagonalizes this matrix will not respect the corner product structure.

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We must also consider the first order terms. If V = these new coordinates, the first order part of L becomes n  j =1



bi ∂ x i +



dk ∂ yk , then in

 1 1 (b j − ) ∂ξ j + dk ∂ yk . 2 ξj

Thus the singularity of the full equation cannot be transformed away except in the very special case where b j ≡ 12 for every j = 1, . . . , n. Let us now revert to the original coordinate system (x, y). The model operator for L at the origin in these coordinates is defined to be LA = +

n  i, j =1 m  k,l=1

√ √ ai j (0, 0) x i x j ∂x2i x j + x i bik (0, 0)∂x2i yk n

m

i=1 k=1

ckl (0, 0)∂ y2k yl

+

n 

bi (0, 0)∂xi +

m 

(11.77) ek (0, 0)∂ yk .

j =1

i=1

Here we no longer think of x and y as local coordinates, but rather as global linear coordinates on ⺢n+ × ⺢m . The key fact which should make it possible to obtain detailed information about L A is that it has a number of symmetries. First, it is translation invariant in y. In addition, provided the coefficients ek (0, 0) all vanish, it is also homo√ geneous of degree −1 with respect to the dilation (x, y) → (λx, λ y). However, even reducing by these symmetries, we are typically left with a multidimensional problem (at least provided n > 1), which may be quite difficult to solve. The one case where the solution is explicit is precisely the case we have already considered, where the bik all vanish and (ai j ) is diagonal. For in this case, the operator is a sum of 1-dimensional operators and hence its heat kernel is a product of the corresponding 1-dimensional heat kernels. We say that the coefficient matrix A is near-diagonal if there is a linear change of coordinates of Sn,m which preserves the orthant structure in ⺢n+ such that the transformation A of A in these new coordinates satisfies ||A − Id|| ≤ η

(11.78)

for some constant η ' 1. For simplicity we call this transformed matrix simply A again. We now explain how it is possible, under this hypothesis, to find a solution of the model heat equation (∂t − L A )w = g,

w|t =0 = f

(11.79)

which satisfies the same H¨older estimates as in the diagonal case. Again to simplify notation, the coefficients in the first order vector field are still denoted bi after this linear change of coordinates, and we also only consider the case that the coefficients ek all vanish. This last hypothesis is less important since if some bi = 0, then the affine change of coordinates y˜ = y −(x i /bi )e removes these tangential coefficients.

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THE RESOLVENT OPERATOR

P ROPOSITION 11.3.1. Let k ∈ ⺞0 , 0 < R, (b1 , . . . , bn ) ∈ ⺢n+ , and 0 < γ < 1. k,γ Suppose that (11.78) holds. If the initial data f lies in ᏯW F (⺢n+ × ⺢m ), and is supk,γ ported in B R+ (0) × B R (0), and similarly the inhomogeneous term g lies in ᏯW F (⺢n+ × ⺢m × ⺢+ ) and is also supported in the same product of balls for all t ≤ T , then the k,γ solution w to (11.79) can be written as a sum v + u with v ∈ ᏯW F (⺢n+ × ⺢m × [0, T ]) k,2+γ and u ∈ ᏯW F (⺢n+ × ⺢m × [0, T ]). Moreover, there are constants C b,γ ,R so that * , vW F,k,γ + uW F,k,2+γ ≤ C b,γ ,R (1 + T )gW F,k,2+γ +  f W F,k,γ . (11.80) k,2+γ

k,2+γ

If f ∈ ᏯW F (⺢n+ × ⺢m ), then v belongs to ᏯW F (⺢n+ × ⺢m × ⺢+ ), and * , vW F,k,2+γ + uW F,k,2+γ ≤ C b,γ ,R (1 + T )gW F,k,2+γ +  f W F,k,2+γ (11.81) for some constant C b,γ ,R . These constants are uniformly bounded for b ≤ B1. P ROOF. We consider only the case f = 0 and k = 0 for simplicity. Write L A = L b,m + Q A , where Q A is the second order operator with off-diagonal components, and then define  u j +1 =

t

0

e(t −s)L b,m (g(s, ·) + Q A u j (s, ·)) ds. 0,2+γ

We claim that this sequence of functions converges in ᏯW F . This is accomplished by a standard contraction mapping argument. If we let  t uˆ j (t, ·) = e(t −s)L b,m Q A u j (s, ·) ds, (11.82) 0

then uˆ j (0) = 0 and (∂t − L b,m )uˆ j = Q A u j . We use the estimates in Proposition 9.2.1 and the fact that √ || x i x j ∂x2i x j u||W F,0,γ ≤ CuW F,0,2+γ , where C depends on only the constants above, to obtain u j +1 − u j W F,0,2+γ ≤ Cηu j − u j −1 W F,0,2+γ , where η = max{|ai j | : i = j } and the constant C is independent of these coefficients. We may then choose η so that Cη < 12 , which gives convergence of the sequence, and hence a solution to the problem. The estimates for the problem when f ≡ 0 and when k > 0 are all handled in a similar manner and we leave details to the reader.  Once we have established these estimates for the model problem, it is then possible to follow the arguments from the previous two chapters to establish the existence of the heat kernel for the generalized Kimura diffusion problem with generator L having sufficiently small off-diagonal terms near each point of b P. The exact smallness condition depends on the geometry of P and the “size” of the diagonal terms in normalized coordinates. The full details needed to verify all these facts are very lengthy. As before,

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CHAPTER 11

the main point, which we leave the reader to verify, is that one may use perturbation arguments in the WF-H¨older spaces to pass from the solution operator for the model problem to the solution operator for the general problem on manifold with corners. This in turn only requires the estimates for the model problems that we have just generalized to the near-diagonal case, and general facts above these function spaces established in Section 10.1.1. As before, we may also use the parametrix for the heat kernel and the Laplace transform to perturbatively construct the resolvent for L. The reader may well wonder if there is an essential difference between the theory for the operator L in this more general setting and the simpler version we have been studying in the rest of this monograph. The answer is not obvious, but it is likely that the key new phenomena involve the regularity of solutions of Lu = 0 (or (∂t −L)u = 0) near the corners where some combination of the x i vanish. We expect that the regularity of solutions near the boundary hypersurfaces where only one of the x i vanish does not change much.

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Chapter Twelve The Semi-group on Ꮿ0 (P) In the previous chapters we have dealt almost exclusively with solutions to (10.1) with inhomogeneous terms f and g in the WF-H¨older spaces. As explained early in this monograph, the reason for working in H¨older spaces in the first place is to handle the perturbation theory in passing from the model operator to the actual one. The original problem, suggested by applications to population genetics, is to study (10.1) with g = 0 and f ∈ Ꮿ0 (P). As noted earlier, the existence theory we have developed 0,2+γ suffices to prove that the Ꮿ0 (P)-graph closure of L with domain ᏯW F (P), for any 0 < γ , is the generator of a C0 -semi-group on Ꮿ0 (P). We let L denote this operator. As noted earlier this suffices to establish the uniqueness of the solution to the martingale problem, and the weak uniqueness of the solution to associated stochastic differential equation, which leads to the existence of a strong Markov process, whose paths are confined to P, almost surely. Perhaps surprisingly, the refined regularity of solutions with initial data in Ꮿ0 (P) does not seem to follow easily from all that we have accomplished thus far. In fact, if the Cauchy data f is continuous but has no better regularity, then it is not clear that the solution u gains any smoothness along b P at times t > 0. Of course, we do know that 0,γ u becomes smooth if f ∈ ᏯW F (P); we also know that solutions to the model problem n m (∂t − L b,m ) on ⺢+ × ⺢ with continuous initial data become smooth. While it seems quite likely that this also holds for continuous initial data, it does not seem easy to prove this for general Kimura diffusions using the present methods. There are related difficulties concerning the graph closure L of L on Ꮿ0 . For example, it is not clear that the resolvent of L is compact. We will return to these questions in a later publication. In this chapter we establish several properties of the elements of Dom(L) and features of the adjoint operator that can be deduced from the analysis above. We also consider the existence of irregular solutions to the equations Lu = f, for functions that do not belong to the range of L. The graph closure L on Ꮿ0 (P) is the unique linear operator defined on the dense subspace Dom(L) ⊂ Ꮿ0 (P) characterized by the condition that u ∈ Dom(L) and 0,2+γ Lu = f ∈ Ꮿ0 (P) if there exists a sequence < u j >⊂ ᏯW F (P) such that Lu j = f j and u j → u and f j → f in Ꮿ0 (P). (12.1) Since L is a non-degenerate elliptic operator away from b P, it is standard that any u ∈ Dom(L) is “almost” twice differentiable in int P in the sense that  2, p Dom(L) ⊂ Wloc (int P) (12.2) 1< p 0,2+γ contained in ᏯW F (P) such that w − wn Ꮿ0 ( P) + Lw − Lwn Ꮿ0 ( P) → 0.

(12.4)

L  [wn  ] = [Lwn ]  .

(12.5)

Clearly, for each n, By assumption, the sequence on the right converges to [Lw]  , and hence the sequence on the left also converges. This shows that w  ∈ Dom(L  ), and L  [w  ] = [Lw]  .  As already observed by Shimakura, this result implies that [et L f ]  = et L  [ f  ],

(12.6)

so that these boundary components are effectively “decoupled” from the rest of P. This result gives some information about the behavior of w ∈ Dom(L) in directions transverse to hypersurfaces to which L is tangent. Let  be such a hypersurface again, then, in adapted coordinates near an interior point of , L takes the form L=

x∂x2

+ b(x, y)x∂x +

N−1  l=1

xa1l (x, y)∂x ∂ yl + (1 + O(x))L  .

(12.7)

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THE SEMI-GROUP ON Ꮿ0 (P)

237

Proposition 12.0.2 implies that if w ∈ Dom(L), then / 0 N−1  2  xa1l (x, y)∂x ∂ yl w(x, y) = 0. lim x∂x + b(x, y)x∂x + x→0+

(12.8)

l=1

A similar result holds at strata of codimension greater than 1 to which L is tangent. 12.1 THE DOMAIN OF THE ADJOINT ∗

The space Ꮿ0 (P) is non-reflexive, which means that the semi-group defined by L on ᏹ(P) (the Borel measures of finite total variation) may not be strongly continuous at ∗ t = 0. This is a reflection of the fact that Dom(L ) may fail to be dense in ᏹ(P). A solution to this problem was introduced by Lumer and Phillips, whereby we consider ∗ et L acting on a smaller space: ∗

ᏹ (P) = Dom(L ).

(12.9)

∗

The semi-group et L , acting on this space, is strongly continuous at t = 0. We denote   its generator by L . The domain of L is then given by 

∗

∗

Dom(L ) = {ν ∈ Dom(L ) : L ν ∈ ᏹ (P)},

(12.10)

which Lumer and Phillips show is a dense subset of ᏹ (P). See Chapter 14 of [26]. Let d V be a smooth non-degenerate density on P. It is clear that if g ∈ Ꮿ∞ c (int P), ∗ then gd V ∈ Dom(L ). The closure of this subspace in the strong topology on ᏹ(P)  equals L 1 (P)d V, which is therefore a subspace of ᏹ (P). Because L is formally a non-degenerate elliptic operator in the interior of P it is clear that at positive times  et L ν is represented at interior points of P by a measure with a smooth density. From  this it is apparent that all elements of Dom(L ) are represented by absolutely continuous measures in the interior of P. As indicated by Proposition 12.0.2, the behavior  of elements of Dom(L ) along a hypersurface in b P depends crucially upon the first order part of L at that hypersurface. Let H be a boundary hypersurface. In adapted coordinates (x, y), defined near an interior point of p0 of H, where p0 ↔ (0, 0), a generalized Kimura diffusion takes the form: L = x∂x2 + b(x, y)∂x +

N−1 

xa1l (x, y)∂x ∂ yl + (1 + O(x))L H ,

(12.11)

l=1

where L H is a generalized Kimura diffusion tangent to H. Integrating by parts, we see  that an element ν = gd xd y ∈ Dom(L ) that is smooth in int P, and has no support, as a measure, on b P, must satisfy the boundary condition: lim [b(x, y)g(x, y) − ∂x (xg(x, y))] = 0.

x→0+

(12.12)

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This is called the no-flux boundary condition. Generally, this condition forces g to have a complicated singularity along {x = 0}, i.e., g(x, y) ∼ x b(0, y)−1 . Let P = { p ∈ P : dist( p, b P) ≥ }. Because ν ∈ ᏹ (P) is represented by an absolutely continuous measure in the interior of P, we can define the interior measure νi via the equation  f, νi  = lim χ P f, ν. (12.13) →0+

The interior measure is represented as gd V, for a function g ∈ L 1loc (P). In fact it is easy to see that g L 1 ( P) ≤ ν, so that the boundary measure, νb = ν − νi , is also in ᏹ(P). From the construction it is clear that the support of νb is contained in b P. If ν is a non-negative measure, then evidently νb is as well. We also observe that there must exist a sequence of {n } tending to zero for which  n |g( p)|d Sn ( p) = 0, (12.14) lim n→∞ b Pn

for otherwise the L 1 -norm of g would be infinite.  Suppose that ν ∈ Dom(L ), then L ∗ ν = μ ∈ ᏹ (P), and therefore L ∗ ν = μ i + μb .

(12.15)

It is easy to see, in the sense of distributions in the interior of P, L ∗ νi = μi . We  now study the boundary behavior of elements of Dom(L ) at points in the interiors of hypersurface boundary components. Let νi = gd V and μi = hd V. For this calculation we assume that h ∈ L p (P), for a p > 1. In this case a distributional solution to 2, p L ∗ (gd V ) = hd V belongs to Wloc (int P), which allows us to integrate by parts. For f ∈ Ꮿ3 (P) and ν as above we have that L f, ν =  f, μi  +  f, μb .

(12.16)

With P as above, we can also do this calculation by observing that L f, ν = lim L f, νi  P + L f, νb . →0+

(12.17)

Using the coordinates, (x, y), introduced in (12.11), we let f (x, y) = ψ(x)ϕ( y).

(12.18)

Here ψ is a smooth function equal to 1 at x = 0, and zero for x > δ > 0, and ϕ is a smooth function compactly supported in int H near to p0 . We can assume that d V = d x d y; using the sequence {n } from (12.14), we integrate by parts to see that   lim L f, νi  Pn = lim  f, μi  Pn + lim ∂x ψ(x)ϕ( y)xg(x, y)+ n→∞

n→∞

n→∞ b Pn

ϕ( y)ψ(x) (b(x, y)g(x, y) − ∂x (xg(x, y))) −



al (x, y)∂ yl ϕ( y)xψ(x)g(x, y) d y.

l

(12.19)

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It follows from (12.19) that the contributions of the first and last terms in the integral over b Pn tend to zero. If we suppose that ∂x ψ(0) = 0, then, for test functions of the type under consideration, L f {x=0} = L H ϕ. If B (g)d y denotes the boundary operator applied to g in (12.19), then we see that ϕ, μb  = lim ϕ, Bn (g)d yb Pn + L H ϕ, νb .

(12.20)

n→∞

This shows that, in the sense of distributions on H, we have the equation: lim [Bn (g)d y + L ∗H νb ] = μb .

(12.21)

n→∞ 



Suppose that L ν = hd V, for an h ∈ L p (P) : that is, in int P, L νi = hd V, in the classical sense, and μb = 0. For f ∈ Ꮿ∞ (P) with compact support in int P we have that  L f, νi  =  f, L νi  =  f, hd V . (12.22) Let f ∈ Ꮿ∞ (P) have support near a point p0 ∈ H, and suppose that f vanishes on b P. The functions f (, ·) ∈ Ꮿ∞ (H ) tend to zero in the Ꮿ∞ -topology, as  → 0+ . Therefore, a limiting argument, using (12.21) and (12.14), shows that L f, νi  =  f, hd V . If f ∈ Ꮿ∞ (P), with f b P = 0, and ∂x f = 0 outside a small neighborhood, U ⊂⊂ int H of p, then it follows from these observations that L f, ν = L f U , νb  +  f, hd V  = b(0, ·)∂x f, νb  +  f, hd V .

(12.23)



In order for ν to belong to Dom(L ), there must be a constant C so that |L f, ν| ≤ C f  L ∞ .

(12.24)

If L meets b P cleanly, then this along with (12.23) implies that supp νb is disjoint from the interior of any face of b P to which L is transverse. We summarize these calculations in the following proposition: 

P ROPOSITION 12.1.1. Suppose that L meets b P cleanly, and ν ∈ Dom(L ) solves  L ν = hd V, with h ∈ L p (P), for a p > 1. The support of νb is disjoint from the interior of faces in b P (L). If νi = gd V , then along faces to which L is transverse, g satisfies the boundary condition in (12.12). Remark. Thus far we have only shown that the support of νb is disjoint from the interiors of faces to which L is transverse. It seems quite likely that supp νb ⊂ b P T . To prove this statement requires a more refined analysis of the regularity of νi and a stronger sense of convergence in (12.21) than that of a sequence of distributions. Below we show that if L is everywhere transverse to b P, then there is a unique   solution ν ∈ Dom(L ) to L ν = 0, which is a probability measure. As explained in [27], Section 15.2, there are circumstances where there may be multiple solutions to this equation, which are non-negative and normalizable. Evidently our method picks out the solution that satisfies the boundary condition in (12.12). A more detailed analysis of this and related questions will need to wait for a later publication.

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12.2 THE NULL-SPACE OF L



As noted above, we are, at present, missing the compactness of the resolvent of L. We  can nonetheless give a precise description of the null-space of the adjoint, L , under the hypothesis that L meets b P cleanly. For the following results it suffices to consider 0,2+γ the operator acting on ᏯW F (P), for a 0 < γ < 1. P ROPOSITION 12.2.1. Suppose that L meets b P cleanly. To each element of b Pter (L)  there is an element of the null-space of L . Each is represented by a non-negative measure supported on a component of  ∈ b Pter (L). P ROOF. For any 0 < γ < 1, denote by L γ the operator 0,2+γ

0,γ

L γ : ᏯW F (P) −→ ᏯW F (P).

(12.25)

We have established that this map is Fredholm; in fact, this map has index zero since it can be deformed amongst Fredholm operators to L γ − 1, which is invertible. Thus dim ker L γ = dim ker L ∗γ .

(12.26)

For the remainder of the argument we fix a 0 < γ < 1. Consider first the extreme case that L is transverse to every boundary hypersurface. It then follows from Lemma 3.2.7 that ker(L γ ) consists of constant functions. Moreover, using this same lemma, if f ≡ 0 is continuous and non-negative, then the equation Lγ w = f (12.27) is not solvable, since any solution would be a subsolution of L γ . 0,γ 0,2+γ The adjoint operator L ∗γ acts canonically as a map from [ᏯW F (P)]∗ to [ᏯW F (P)]∗ . Since we are still assuming that b Pter (L) = P, ker(L γ ) contains only the constant 0,γ functions, so there is precisely one non-trivial element  ∈ [ᏯW F (P)]∗ , unique up to ∗ scaling, which satisfies L γ  = 0. By the Fredholm alternative, the equation L γ w = f 0,γ

is solvable for f ∈ ᏯW F (P) if and only if ( f ) = 0. 0,γ

This means that if f ∈ ᏯW F (P) is non-negative (and non-zero), then L γ w = f is not solvable, so that ( f ) = 0. We may as well assume that ( f ) > 0

(12.28)

on the set of non-negative functions; we further normalize so that (1) = 1. A priori, we only know that  lies in the dual of a H¨older space, and thus could be 0,γ a distribution of negative order. If f ∈ ᏯW F (P), then f + ( p) = max{ f ( p), 0} and f − ( p) = min{ f ( p), 0},

(12.29)

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both lie in this same function space, and therefore (12.28) and our normalization imply that min f ≤ ( f ) ≤ max f, (12.30) 0,γ

and therefore, for f ∈ ᏯW F (P), we have |( f )| ≤  f  L ∞ .

(12.31)

The WF-H¨older spaces are dense in Ꮿ0 (P), so  has a unique extension as an element of [Ꮿ0 (P)] . By the Riesz-Markov theorem, there is a non-negative Borel measure ν so that  ( f ) = f ( p)dν( p). (12.32) P

As



0,2+γ ᏯW F (P)

is dense in Dom(L) this shows that L ν = 0.  The adjoint L is elliptic in int P, so by standard elliptic regularity, νi = v 0 d V,

(12.33)

for some smooth, non-negative function v 0 on int P; here d V is a smooth, nowhere zero density on P. Using (12.28) again, we see that the support of v 0 is all of P. Note,  however, that since L can have a zero order part, there is no obvious reason that v 0 is strictly positive. Proposition 12.1.1 shows that supp νb is disjoint from hypersurfaces in b P (L) and, along such faces, νi satisfies the boundary condition in (12.12). T (L). If dim  = 0, so  = p is a single Let us now turn to components  ∈ b Pmin  point, then the fact that L is tangent to  simply means that the restriction L  ≡ 0. Hence if δ denotes the functional w, δ  = w( p ),

(12.34)

0,2+γ

then clearly, for w ∈ ᏯW F (P), we have Lw, δ  = 0,

(12.35) 

and this equation remains true for w ∈ Dom(L). Hence δ ∈ Dom(L ), and 

L δ = 0.

(12.36)

Suppose, on the other hand, that dim  > 0, i.e.,,  is a compact manifold without boundary, and L  is a non-degenerate elliptic operator, without constant term, acting on Ꮿ2 (). Clearly L  1 = 0. On the other hand, the strong maximum principle shows that all solutions to L  w = 0 are constant. Also from the strong maximum principle, the equation L  w = f is not solvable whenever f ∈ Ꮿ0 () is non-negative and not identically zero. Arguing as above for the case that b Pter (L) = P we conclude that there is a non-negative measure with smooth density, ν = v 0 d V that spans the  null-space of L  . The functional  (w) = wdν (12.37) 

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defines an element of ker L . As before, the support of v 0 is all of .  To complete the construction of ker L we need only consider boundary compoT (L). In this case L is a generalized Kimura diffusion on nents  ∈ b Pter (L) \ b Pmin  , and bter (L  ) = . The argument above produces a measure ν with support equal  to  and such that L  ν = 0. If we define  (w) = wdν, (12.38)  

then L  = 0. This completes the proof of the proposition.



D EFINITION 12.2.2. We denote by {δ :  ∈ b Pter (L)}, the measures, belonging  to ker L , constructed in the proof of Proposition 12.2.1. 0,γ

These measures define non-trivial functionals on Ꮿ0 (P) ⊃ ᏯW F (P), and are certainly linearly independent. This argument shows that dim ker L ∗γ ≥ |b Pter (L)|. On the other hand, by Corollary 3.2.10, dim ker L γ ≤ |b Pter (L)|. We summarize all of this in a proposition: P ROPOSITION 12.2.3. If L meets b P cleanly, then, for any 0 < γ < 1, dim ker L ∗γ = dim ker L γ = |b Pter (L)|.

(12.39)

The ker L γ is contained in Ꮿ∞ (P); on the other hand, ker L ∗γ is spanned by a finite collection of non-negative Borel measures, each of which has a smooth non-negative density supported on one of the terminal boundary components of P. The operator L 0,2+γ has no generalized eigenvectors at 0, i.e.,, functions w ∈ ᏯW F (P) with Lw ∈ ker L. T (L) = ∅, and b P (L) = P, then Remark. If b Pmin ter

dim ker L ∗γ = dim ker L γ = 1.

(12.40)

The ker L ∗γ is spanned by a non-negative measure with support all of P. This is the equilibrium measure. If |b Pter (L)| > 1, then instead of a single equilibrium measure, there is a collection of such measures, each supported on one of the terminal components of b P. Zero-dimensional components of b Pter (L) are classical absorbing states of the underlying Markov process. Higher dimensional components correspond to generalized absorbing states; these are again characterized by an equilibrium measure. P ROOF. Only the last statement still requires proof. If b Pter (L) = P, then ker L γ consists of constant functions. We observe that L γ w = 1 is not solvable, for otherwise w would be a non-trivial subsolution. 0,2+γ Suppose that b Pter (L) = P and that w ∈ ᏯW F (P) satisfies Lw = f, where L f = 0. Then necessarily  f, δ  = 0 for all  ∈ b Pter (L).

(12.41)

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However, any f ∈ ker L ∩ ᏯW F (P) is constant on each component of b Pter (L). Since each of the measures {δ } is non-negative and non-trivial, Proposition 3.2.8 shows that f ≡ 0.  We have not proved that all elements of ker L belong to Ᏸ2W F (P), nor have we established the Hopf maximum principle for elements of Dom(L), hence we cannot  presently conclude that dim ker L = dim ker L . On the other hand, elements of   ker L are represented by Borel measures, and furthermore ker L ⊂ ker L ∗γ . Since ∗ we have shown that ker L γ is also spanned by Borel measures, we obtain: P ROPOSITION 12.2.4. If L meets b P cleanly, then dim ker L



= |b Pter (L)|.

(12.42)

The null-space is spanned by Borel measures with support on the components of b Pter (L). 12.3 LONG TIME ASYMPTOTICS These observations have several interesting consequences. P ROPOSITION 12.3.1. If f in Ꮿ0 (P), and et L f denotes the action of the semigroup, then the functions t → et L f, δ  (12.43) are constant for every  ∈ b Pter (L). P ROOF. Indeed, this is clear when f ∈ Dom(L) since ∂t et L f, δ  = Let L f, δ  ∗

= et L f, L δ  = 0.

(12.44)

However, the domain Dom(L) is dense in Ꮿ0 (P), so for any f ∈ Ꮿ0 (P) we can choose a sequence < f n > in Dom(L) which converges to f in Ꮿ0 (P). Then et L f, δ  = lim et L fn , δ . n→∞

(12.45)

The right-hand side is independent of t for each n, hence so is the limit. If b Pter (L) = P, then we can also conclude that et L f, δ P  is constant.

(12.46) 

Remark. A similar observation, for a special case, appears in [9]. We now show that 0 is the only element in the spectrum of L γ on the imaginary axis, Re μ = 0.

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L EMMA 12.3.2. Let P be a compact manifold with corners, and L a generalized 0,2+γ Kimura diffusion on P. If ϕ ∈ ᏯW F (P) is a non-trivial solution to Lϕ = i αϕ, for α ∈ ⺢, then α = 0. P ROOF. Let qt (x, d y) denote the Schwartz kernel for et L . Then for each x ∈ P,  qt (x, d y) = 1, (12.47) P

and qt (x, d y) is a non-negative measure. By the strong maximum principle, et L is strictly positivity improving within int P. Hence if U ⊂ int P is any open subset, then  qt (x, d y) > 0 (12.48) U

for each x ∈ int P. Now, e

it α

 tL

ϕ(x) = e ϕ =

qt (x, d y)ϕ(y),

(12.49)

P

so by the non-negativity of qt (x, d y),  |ϕ(x)| ≤ qt (x, d y)|ϕ(y)| = [et L |ϕ|](x).

(12.50)

P 0,γ

Note also that |ϕ| lies in ᏯW F (P), so et L |ϕ| ∈ Ꮿ∞ (P) for t > 0. The estimate in (12.50) implies that for any s > 0, 0≤

[e(s+t )L |ϕ|](x) − [es L |ϕ|](x) . t

(12.51)

Since es L |ϕ| ∈ Dom(L γ ), we can let t → 0+ to conclude that Les L |ϕ|(x) ≥ 0 for all x ∈ P.

(12.52)

Choose any non-negative ψ ∈ Ꮿ∞ c (int P). Integrating by parts with respect to some smooth non-degenerate density on P gives 0 ≤ es L |ϕ|, L ∗ ψ,

(12.53)

so letting s → 0+ , we obtain that 0 ≤ |ϕ|, L ∗ ψ.

(12.54)

If the support of ψ is further constrained to lie in a set where |ϕ| is smooth, then we can integrate by parts again to conclude that 0 ≤ L|ϕ|, ψ.

(12.55)

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In particular, L|ϕ| ≥ 0 in the open subset of int P where |ϕ| > 0. There are now several cases to consider. If b P T (L) = ∅, then Lemma 3.2.6 shows that |ϕ(x)| ≡ 1. We have + +   + + + 1 = |ϕ(x)| = + qt (x, d y)ϕ(y)++ ≤ qt (x, d y)|ϕ(y)| = qt (x, d y) = 1, (12.56) < so the inequality in the middle is an equality, and since qt (x, d y) = 1, this can only happen if ϕ has constant phase. This shows that α = 0 in this case. If b P T (L) = ∅, then we use an induction on the dimension of P. The result has been proved when dim P = 1 in [20] and [15], so we now assume that it is true whenever dim P ≤ N − 1. Let P have dimension N and assume that b P T (L) = ∅. Suppose that ϕ is a non-trivial solution, as above, and that ϕᏯ0 ( P) = 1. For each  ∈ b P T (L) we know that L  ϕ = i αϕ . (12.57) By induction, either α = 0 or ϕ  = 0. In the former case we are done, so we can reduce to the case that ϕ = 0 for every  ∈ b P T (L). If L is tangent to every face of b P, then |ϕ| attains its maximal value 1 at some point x 0 ∈ int P. It follows directly from (12.47) and (12.48) that |ϕ(x)| ≡ 1 in int P. For if this were false, then the fact that et L is strictly positivity improving in int P would show that  qt (x 0 , d y)|ϕ(y)| < 1, (12.58) P

which contradicts (12.50). By induction ϕ vanishes on b P, which is clearly impossible, as ϕ ∈ Ꮿ0 (P). We are left to consider the case where b P T (L) = b P. If |ϕ(x)| assumes its maximum in the interior of P then we conclude as above that |ϕ(x)| ≡ 1 in P, which leads to the same contradiction as before. Thus |ϕ(x)| must assume the value 1 at x 0 ∈ b P \ b P T (L). Indeed x 0 ∈ b P (L), for otherwise x 0 would belong to the closure of b P T (L) and hence would vanish. Applying Lemma 3.2.6 gives that |ϕ| is identically equal to 1 in a neighborhood U of x 0 . Using the previous argument at x 1 ∈ U ∩ int P gives |ϕ(x)| ≡ 1 in P, which contradicts that ϕ b P T (L)= 0. Thus the only tenable case is that α = 0, as claimed.  Combining this lemma with Theorem 11.1.1 and Lemma 12.3.2 gives the following corollary: C OROLLARY 12.3.3. For any 0 < γ < 1, the spec(L γ ) \ {0} lies in a half plane Re μ < η < 0. Write N0 = |b Pter (L)| and let { j : j = 1, . . . , N0 } enumerate the components  of b Pter (L). Also, denote by  j = δ j the probability measures which span ker L , constructed above. From the support properties of these measures, we can choose smooth functions { f j : j = 1, . . . , N0 } so that k ( f j ) = δ j k .

(12.59)

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Choose a smooth basis {w j : j = 1, . . . , N0 } for ker L γ , for any 0 < γ < 1. Corollary 12.3.3 and the fact that L γ has no generalized eigenvectors at 0, shows that spec L γ \ {0} lies in Re μ < η, for an η < 0. Thus, as the spectrum lies in a conic neighborhood of the negative real axis, we can deform the contour in (11.69) to show 0,γ there exist continuous linear functionals {a j } such that, for any f ∈ ᏯW F (P), we have e

tL

f =

N0 

a j ( f )w j + O(eηt ).

(12.60)

j =1

Proposition 12.3.1 shows that the quantities {k (et L f )} are independent of t, so letting t → ∞, we conclude that k ( f ) =

N0 

a j ( f )k (w j ).

(12.61)

j =1

In light of (12.59), k (w j ) is an invertible matrix, so we can find a new basis {w˜ j } for ker L γ so that k (w˜ j ) = δ j k and therefore et L f =

N0 

 j ( f )w˜ j + O(eηt ).

(12.62)

j =1

By duality we can conclude that if ν is a Borel measure belonging to ᏹ (P), then ∗

et L ν =

N0 

w˜ j , ν j + O(eηt ).

(12.63)

j =1

The { j } are non-negative measures with disjoint supports. Since the forward Kolmogorov equation maps non-negative measures to non-negative measures, it follows that the eigenfunctions {w˜ j } must be non-negative. As a special case of (12.62) note that N0  1=  j (1)w˜ j . (12.64) j =1

We summarize these results in a proposition. P ROPOSITION 12.3.4. For 0 < γ < 1, there is a basis for ker L γ consisting of non-negative smooth functions {w˜ j }. There is an η < 0, so that for initial data f ∈ Ꮿ0,γ (P), the asymptotic formula (12.62) holds. For initial data ν ∈ ᏹ (P), the asymptotic formula (12.63) holds. Remark. In the classical case, with L = L Kim and P = ᏿ N , a basis for ker L Kim is given by the functions {1, x 1, . . . , x N }. It is very likely that (12.62) also holds for data in Ꮿ0 (P).

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12.4 IRREGULAR SOLUTIONS OF THE INHOMOGENEOUS EQUATION In many applications to probability one would like to solve equations of the form Lw = f,

(12.65)

where f is a non-negative function. From the analysis in Section 12.2 it is clear that there is often no regular solution to this equation, for  f, δ  will, in general, not vanish for all  ∈ b Pter (L). Nonetheless, probabilistic considerations show that this equation should sometimes have a solution. For example, if f = 1, then −w( p) is the expected time for a path of the process defined by L, starting at p, to arrive at b P. In this section we show that, under certain hypotheses on L, there is an irregular solution to this equation (12.12), by which we mean a solution that does not belong to Dom(L). We begin by noting that, since L defines a contraction semi-group on Ꮿ0 (P), it automatically satisfies the maximum principle: L EMMA 12.4.1. If w ∈ Dom(L) assumes its maximum at p ∈ P, then Lw( p) ≤ 0. See Theorem 7.22 in [13]. By choosing a smooth non-degenerate density d V on P we can represent the formal adjoint operator L ∗ to L as a sum L∗ =  L + c,

(12.66)

where c = L ∗ 1 is a smooth function, and L −  L = X is a vector field. Denote by {ρ j : j = 1, . . . , K } non-negative defining functions for the hypersurface components of b P. If L ∗ ρ j ρ j =0 ≥ 0 for j = 1, . . . , K , (12.67) then  L is also a generalized Kimura diffusion operator. We call a generalized Kimura diffusion operator with this property adversible. We use this slightly odd neologism as the more natural term “reversible” already has a well established meaning in the context of Markov processes. For a adversible Kimura diffusion we can apply the analysis above to study the k,2+γ operator  L on the spaces ᏯW F , and then consider its Ꮿ0 (P)-graph closure. We denote this closed operator by  L r ; it generates a contraction semi-group of operators on Ꮿ0 (P). If the scalar function c is non-positive, then the closed operator L r∗ =  L r + c also generates a contraction semi-group on Ꮿ0 (P), with the same domain as  L r . Using Lemma 12.4.1 we can easily establish the following form of the maximum principle: P ROPOSITION 12.4.2. If L is an adversible generalized Kimura diffusion, with c non-positive in P and strictly negative on b P, then, for w ∈ Dom(L r∗ ), the equation L r∗ w = 0

(12.68)

has only the trivial solution. P ROOF. Suppose that there is a non-trivial solution. Evidently w is non-constant, and we can therefore assume that w has a positive maximum. The strong maximum principle implies that the maximum must occur at a point p0 ∈ b P. Lemma 12.4.1 shows that  L r w( p0 ) ≤ 0, and therefore L r∗ w( p0 ) < 0, which is a contradiction. 

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If we assume that c( p) < −λ < 0 throughout P, then L r∗ =  L r + (c + λ) − λ.

(12.69)

The operator  L r + (c + λ) generates a contraction semi-group, and therefore L r∗ is invertible. The formal symbol of the operator (L r∗ )∗ agrees with L. Because Ꮿ0 (P) is not reflexive, we need to use the operator (L r∗ ) on the space ᏹ (P) introduced above. With these preliminaries we can show the existence of irregular solutions. P ROPOSITION 12.4.3. Suppose that P is a manifold with corners and L is an adversible generalized Kimura diffusion on P such that L ∗ 1 < 0 throughout P. If g ∈ L 1 (P) then there is a unique solution w ∈ Dom((L r∗ ) ) to the equation (L r∗ ) w = g.

(12.70)

Remark. In the interior equation (12.70) is equivalent to the classical equation Lw = g, which explains why we say that this equation has an irregular solution for any g ∈ L 1 (P). P ROOF. Under the hypotheses of the proposition the operator L r∗ is invertible. Theorem 14.3.4 of [26] shows that (L r∗ ) is also invertible. As noted above, L 1 (P) is a subspace of ᏹ (P), from which it follows that (12.70) has a unique solution, for any g ∈ L 1 (P).  Remark. As the operator L is non-degenerate elliptic in the interior of P, the solution w has roughly two more derivatives than g in this subset. On the other hand, the best one can hope for is that w is an absolutely continuous measure, satisfying the no-flux boundary conditions (12.12), along b P. Let {ρi } be defining functions for the faces of β( y) b P. These conditions force w to have leading singularities of the form ρi along the sets {ρi = 0}. In the classical population genetic case, P is a simplex N ⊂ ⺢ N , and L = L Kim +

N 

b j (x)∂x j .

(12.71)

j =1

A simple calculation shows that, with respect to Lebesgue measure, L ∗ = L Kim +

N 

[2 − bi (x) − 2(N + 1)x j ]∂x j − [N(N + 1) +

j =1

N 

∂x j b j (x)]. (12.72)

j =1

The conditions required for L to be adversible are: the functions b j (x) ≤ 2, where x j = 0, for j = 1, . . . , N and ⎤ ⎡ N  ⎣ b j (x)⎦ ≥ −2. (12.73) j =1

{x 1 +···+x N =1}

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That is we are in a “weak mutation” regime. The hypothesis that L ∗ 1 < 0 becomes the requirement: N  ∂x j b j (x) > −N(N + 1). (12.74) j =1

This inequality involves the strength of both mutation and selection, but holds when both are sufficiently weak. Along {x j = 0} the no-flux condition would force a solution 1−b (x)

to behave like x j j . Along faces where 0 < b j (x) < 1, the solution will tend to zero. Where 1 < b j (x) < 2, the solution will tend to infinity. Where b j = 0, we expect the solution to behave like x j log x j , as it is precluded from being regular.

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Appendix A Proofs of Estimates for the Degenerate 1-d Model This Appendix contains proofs of estimates, used throughout the paper, of the 1-dimensional solution operator ktb (x, y), which we recall is given by ktb (x, y) =

y b−1 − (x+y)  x y  e t ψb 2 , t t

where ψb (z) =

∞  j =0

zj . j !( j + b)

This function has the following asymptotic development, as z → ∞ : ⎡ ⎤ √ 1 b ∞  cb, j z 4 − 2 e2 z ⎣ ⎦. 1+ ψb (z) ∼ √ j 4π 2 z j =1

(A.1)

(A.2)

(A.3)

In several of the arguments below we need a suitable replacement for the mean value theorem that is valid for complex valued functions. L EMMA A.0.4. Let f be a continuously differentiable, complex valued function defined on the interval [a, b]. There is a point c ∈ (a, b) such that | f (b) − f (a)| ≤ (b − a)| f  (c)|.

(A.4)

P ROOF. As an immediate consequence of the fundamental theorem of calculus and the triangle inequality we see that b | f (b) − f (a)| ≤

| f  (y)|d y.

(A.5)

a

The estimate in the lemma now follows from the standard mean value theorem applied to the differentiable function x F(x) =

| f  (y)|d y.

(A.6)

a

 251

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APPENDIX A

The kernel functions ktb (x, y) extend to be analytic for Re t > 0, and we prove estimates for the spatial derivatives of this analytic continuation. These are needed to study the resolvent kernel, which for the 1-dimensional model problem is defined in the right half plane by ∞ R(μ) = e−μt et L b dt. (A.7) 0

The contour of integration can be deformed to lie along any ray arg t = θ with |θ | < π2 . This provides an analytic continuation of R(μ) to ⺓ \ (−∞, 0]. In these arguments we let t = τ eiθ , where τ = |t|. Notational Convention. To simplify the notation in the ensuing arguments we let d

eφ = e−iφ .

(A.8)

A.1 BASIC KERNEL ESTIMATES L EMMA A.1.1. [L EMMA 6.1.4] For b > 0, 0 < γ < 1, and 0 < φ < constants Cb,φ uniformly bounded with b, so that for t ∈ Sφ ∞

γ

γ

|ktb (x, y) − ktb (0, y)|y 2 d y ≤ Cb,φ x 2 .

π 2,

there are

(A.9)

0

P ROOF. We let t = τ eiθ , where |θ | < ∞ 0

γ |ktb (x, y) − ktb (0, y)|y 2 d y

π 2

− φ. Using the formula for ktb we see that

∞   + γ dy  x ye  y b − cos θ y ++ −eθ x 2θ + 2 τ +e τ ψb − ψ = e (0) +y b τ y τ2 0

∞ ≤ 0

+ + γ dw . wb e− cos θw +e−λeθ ψb (λwe2θ ) − ψb (0)+ (wτ ) 2 w (A.10)

On the second line we let w = y/τ and λ = x/τ. We split the integral into a part, I1 (t, λ), from 0 to 1/λ, and the rest, I2 (t, λ). We estimate the compact part first; using the FTC we see that 1 ψb (λwe2θ ) − ψb (0) =λwe2θ 0

=Mλwe2θ ,

ψb (sλwe2θ )ds

(A.11)

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PROOFS OF ESTIMATES FOR THE DEGENERATE 1-D MODEL

where M is a complex number satisfying: |M| ≤ sup |ψb (z)|.

(A.12)

z: |z|≤1

This gives 1

γ

λ

I1 (t, λ) ≤ Cb τ 2

* , γ wb+ 2 −1 e− cos θw e− cos θλ λw + |1 − e−eθ λ | dw.

(A.13)

0

The constant Cb is uniformly bounded for 0 < b < B. For λ bounded, and |θ | < π2 we can estimate |1 − e−eθ λ | by λ and therefore the integral can also be estimated by a constant times λ. Altogether we get γ

γ

γ

I1 (t, λ) ≤ Cb,θ τ 2 λ = Cb,θ x 2 λ1− 2 .

(A.14)

As λ is bounded, this is the desired estimate. Now we turn to λ → ∞. In this case it is easy to see that the integral tends to zero. As λ > 1, this implies that γ

I1 (t, λ) ≤ Cb,θ x 2 .

(A.15)

We are left to estimate I2 . In this case there is no cancellation between the terms on the right-hand side of (A.10). As ψb (0) = 1/ (b), it is elementary to see that, in all cases, ∞ γ γ dw γ 2 ≤ Cb,θ x 2 . τ wb e− cos θw |ψb (0)|w 2 (A.16) w 1 λ

To complete the proof, for this case, we need to estimate the other term, which we denote I2 (t, λ). To that end we use the asymptotic expansion to estimate ψb (wλ) : 1

b

|ψb (wλe2θ )| ≤ Cb (wλ) 4 − 2 e2 cos θ

√ wλ

.

(A.17)

Inserting this estimate gives: I2 (t, λ)

≤ Cb τ

γ 2

∞  1 λ

w  b2 − 14 − cos θ(√w−√λ)2 γ dw e w2 √ . λ w

(A.18)

Applying Lemma 6.1.23 it is a simple matter to see that this is uniformly bounded by γ Cb,θ x 2  f W F,0,γ , for a constant bounded when b is bounded, and therefore γ

I2 (t, λ) ≤ Cb,θ x 2 .

(A.19)

Combining (A.14), (A.15), (A.16) and (A.19) completes the proof of the lemma.



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APPENDIX A

L EMMA A.1.2. [L EMMA 6.1.5] For b > 0, there is a constant Cb,φ so that, for t ∈ Sφ , ∞ x/|t| . (A.20) |ktb (x, z) − ktb (0, z)|dz ≤ Cb,φ 1 + x/|t| 0

For 0 < c < 1, there is a constant Cb,c,φ so that if cx 2 < x 1 < x 2 and t ∈ Sφ , then ⎛

∞ |ktb (x 2 , z) − ktb (x 1 , z)|dz ≤ 0

√ √ | x2 − x1 | √ |t | √ √ Cb,c,φ ⎝ | x2 − x1 | √ 1+ |t |

⎞ ⎠.

(A.21)

P ROOF. First observe that Lemma 6.1.3 implies that, for t ∈ Sφ , the integrals in (A.20) and (A.21) are always bounded by a constant Cφ . We start with the proof of (A.20). We let t = τ eiθ , with |θ | < π2 , and set w = z/τ, and λ = x/τ, then we see that the expression on the right-hand side of (A.20) equals ∞

wb−1 e− cos θw |e−eθ λ ψb (λwe2θ ) − ψb (0)|dw.

(A.22)

0

We split this into an integral over [0, λ1 ] and the rest. As λ → ∞ it is clear that the compact part remains bounded, and as λ → 0, it is O(λ). In the non-compact part we use the trivial bound when λ → ∞, and the asymptotic expansion when λ → 0. This cos θ latter term is easily seen to be bounded by O(e− 2λ ), completing the proof in this case. To prove (A.21), we assume that cx 2 < x 1 < x 2 , and use the formula for ktb ; setting w = z/τ, λ = x 1 /τ, and μ = x 2 /x 1 , we obtain: ∞ |ktb (x 2 , z) − ktb (x 1 , z)|dz = 0

∞

wb−1 e− cos θw |e−eθ λ ψb (λwe2θ ) − e−eθ μλ ψb (μλwe2θ )|dw.

(A.23)

0

We let F(μ) = e−eθ μλ ψb (μλwe2θ ); from Lemma A.0.4 it follows that |F(μ) − F(1)| ≤ (μ − 1)λe− cos θξ λ |ψb (ξ λwe2θ ) − eθ wψb (ξ λwe2θ )|,

(A.24)

for a ξ ∈ (1, μ) ⊂ (1, 1c ). We split the integral into the part from 0 to 1/λ, I− , and the rest, I+ . Using the Taylor expansion we see that 1

λ I− ≤ C(μ − 1)λ

w 0

b−1 − cos θ(w+λ)

e



1 + w(1 + λ) dw. (b)

(A.25)

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255

As λ → ∞ this is bounded by a constant times λ1−b (μ − 1)e−λ . This in turn satisfies √ √  λ x2 − x1 I− ≤ e − 2 . (A.26) √ |t| As λ → 0 the integral in (A.25) remains bounded and therefore   √ √  √ √  x2 − x1 x2 + x1 x2 − x1 I− ≤ Cb,θ = Cb,θ , √ √ |t| |t| |t| which shows that

√ √ I− ≤ Cb,θ λ( μ + 1)

√

√  x2 − x1 , √ |t|

(A.27)

(A.28)

thus completing this case. Using the asymptotic expansions for ψb and ψb we see that ∞  I+ ≤ Cb (μ − 1)λ 1 λ

w ξλ

b−1 2

4

e

−λ cos θ(

√w

√ 2 λ − ξ)

+ + 2 + + dw +1 − w + O( √1 + 1 )+ √ . + ξλ wλ λ + w (A.29)

As λ → 0 this satisfies an estimate of the form   x2 − x1 − 1 I+ ≤ Cb,θ e 2λ . |t|

(A.30)

To analyze the non-compact part as λ → ∞, we first note that we are only interested in the case that √ √ x2 − x1 ≤ 1, (A.31) √ |t| √ for otherwise we use the trivial estimate. Dividing by λ we see that this constraint is equivalent to 1 √ μ−1≤ √ , (A.32) λ which clearly implies that μ → 1 as λ → ∞. √ We change variables in (A.29) letting z = w/λ, to obtain: ∞ I+ ≤ Cb (μ − 1)λ

√ 2 ξ)

1

z b− 2 e−λ cos θ(z−

1 λ

+ + + +√ +1 − √z + O( 1 + 1 )+ λdz. + zλ λ + ξ

(A.33)

√ To estimate this integral, we split the domain into three pieces: [ λ1 , 1], [1, μ], and √ √ [ μ, ∞]. Recall that ξ ∈ [1, μ], and therefore, in the first segment we see that |z −

 ξ | > |z − 1|,

(A.34)

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APPENDIX A

and in the third segment, |z −



ξ | > |z −

√ μ|.

(A.35)

With these observations we see that   1 1 √ 1 √ b− 12 −λ cos θ(z−1)2 |1 − z| + | μ − 1| + O( + ) z e λdz+ I+ ≤ Cb (μ−1)λ zλ λ √

1 λ

+ + μ √ 2+ 1 z 1 +√ 1 z b− 2 e−λ cos θ(z− ξ ) ++1 − √ + O( + )++ λdz+ zλ λ ξ 1

∞

1

√ 2 μ)

z b− 2 e−λ cos θ(z−



√ μ

1 √ 1 √ √ | μ − z| + | μ − 1| + O( + ) λdz . zλ λ

(A.36)

Using Laplace’s method to estimate the first and third terms, as well √ as (A.32), we easily show that the sum of the three integrals is bounded by Cb,c,θ / λ, which implies that √ √  x2 − x1 I+ ≤ Cb,c,θ . (A.37) √ |t| This completes the proof of the lemma.



L EMMA A.1.3. [L EMMA 6.1.6] For b > 0, 0 < γ < 1 and t ∈ Sφ , 0 < φ < there is a Cb,φ so that

π 2,

∞

√ γ √ |ktb (x, y)|| x − y|γ d y ≤ Cb,φ |t| 2 .

(A.38)

0

For fixed 0 < φ, and B, these constants are uniformly bounded for 0 < b < B. P ROOF. We let t = τ eiθ , with |θ | < ∞

π 2

−φ, and set w = y/τ, λ = x/τ, obtaining:

√ √ |ktb (x, y)|| x − y|γ d y =

0

|t|

γ 2

∞ 0

√ √ dw . wb e− cos θ(w+λ) |ψb (wλe2θ )|| w − λ|γ w

(A.39)

We split this integral into the part from 0 to 1/λ, I1 , and the rest, I2 . Using the estimate   1 |ψb (z)| ≤ Cb +z , (A.40) (b)

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we easily show that the compact part is uniformly bounded by Cb,θ |t| 2 , for a constant Cb,θ uniformly bounded for b < B and |θ | ≤ π2 − φ. In the non-compact part use the asymptotic expansion to obtain that γ 2

I2 ≤ Cb |t| λ

1 b 4−2

∞

1

w2

  √ b− 12 − cos θ(√w− λ)2

e

1 λ

√ √ dw | w − λ|γ √ . w γ

(A.41)

1

1

Lemma 6.1.23 shows that as λ → 0 this is bounded by Cb,θ |t| 2 λ−(b+γ − 2 ) e− λ , showing that in this regime (6.34) holds. γ As λ → ∞, Lemma 6.1.23 shows that this is bounded by Cb,θ |t| 2 , thus completing the argument to show that (6.34) holds for all x, and t ∈ Sφ .  L EMMA A.1.4. [L EMMA 6.1.7] We assume that x 1 /x 2 > 1/9 and J = [α, β], as defined in (6.35). For b > 0, 0 < γ < 1 and 0 < φ < π2 , there is a Cb,φ so that, for t ∈ Sφ ,  √ √ √ √ |ktb (x 2 , y) − ktb (x 1 , y)|| y − x 1 |γ d y ≤ Cb,φ | x 2 − x 1 |γ . (A.42) Jc

P ROOF. We let t = τ eiθ , with |θ | < α I− =

π 2

− φ, and set

√ √ |ktb (x 2 , y) − ktb (x 1 , y)|| y − x 1 |γ d y

0

(A.43)

∞

√

|ktb (x 2 , y) − ktb (x 1 , y)|| y −

I+ =

√

x 1 |γ d y.

β

Since x 1 /x 2 > 1/9, we know that α > 0. α   x y  x y + √ x dy y b − cos θ y ++ − x2 √ 2 1 + − t1 τ +e t ψb − e e ψ I− ≤ + | x 1 − y|γ . (A.44) b 2 2 τ y t t 0

As usual we let y/τ = w and x 1 /τ = λ, obtaining α

I− ≤ |t|

γ 2

|t|

b − cos θw

w e 0

+ x +   + − 2 eθ λ + √ √ +e x1 ψb x 2 wλe2θ − e−eθ λ ψb (wλe2θ )+ | λ− w|γ dw . + + x w 1

The upper limit of integration can be re-expressed as   2 x2 3 − x1 α =λ . |t| 4

(A.45)

(A.46)

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APPENDIX A

As before we use Lemma A.0.4 to obtain: + + x   + + − 2 eθ λ +e x1 ψb x 2 wλe2θ − e−eθ λ ψb (wλe2θ )+ ≤ + + x1   + + x2 λe−λξ cos θ +weθ ψb (ξ λwe2θ ) − ψb (ξ λwe2θ )+ −1 , x1

(A.47)

where ξ ∈ (1, xx21 ) ⊂ (1, 9). As in an earlier estimate we need to split this integral into the part from 0 to 1/λ and the rest. In the first part, I−1 , we estimate |ψb (ξ λwe2θ )| ≤

1 + Cb (ξ λw) (b)

(A.48)

and |ψb (ξ λwe2θ )| by a constant; in the second part, I−2 , we will use the asymptotic expansions. The term in (A.45) coming from 1/ (b) is estimated by   γ e− cos θλ x 2  I−1 ≤ Cθ |t| 2 λ1+b+γ /2 −1 . (A.49) b(b) x 1 √ √ We observe that this is bounded by Cθ  f W F,0,γ | x 2 − x 1 |γ provided that λb+γ /2 e− cos θλ

2

x2 − |t|

2

x1 |t|

1−γ 2

x2 + |t|

2

x1 |t|

 ≤ C.

(A.50)

As c < x 1 /x 2 , we see that the quantity on the left is bounded by a multiple of λb+1 e− cos θλ , which remains bounded as λ → ∞. Thus, there is a constant Cθ , independent of b, so that √ √  I−1 ≤ C θ | x 2 − x 1 |γ . (A.51) The other part of I−1 (coming from the Cb [ξ λw + w]-terms) is easily seen to satisfy an estimate of the form   γ x2  I−1 ≤ Cθ |t| 2 λ2+b+γ /2 e− cos θλ −1 , (A.52) x1 for a constant independent of b. Arguing as before shows that this also satisfies (A.51), so that I−1 satisfies the desired estimate. Using the asymptotic expansions we see that the other part, I−2 , satisfies γ 2

I−2 ≤ C|t| λ

3 b 4−2



x2 −1 x1



α

|t| 1 λ

e− cos θ(

√ √ w− λξ )2

 √ √ b 1 √ 1 dw w 2 − 4 | w − ξ λ + O((wλ)− 2 )|| w − λ|γ √ . w

(A.53)

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PROOFS OF ESTIMATES FOR THE DEGENERATE 1-D MODEL

As ξ > 1, for w ∈ [0, |tα| ] the exponential is only increased if we replace λξ with λ. For a large enough C it is also the case that for w in the domain of integration:  √ √ √ ξ λ − w ≤ C( λ − w). (A.54)  Letting z = wλ − 1, we obtain γ

3

γ

I−2 ≤ Cb |t| 2 λ 2 + 2



 x2 −1 × x1 1 2



  x 1− x 2



1

1

e− cos θλz (1 + z)b− 2 |z|1+γ dz. (A.55) 2

1 λ −1

We are interested in the case x 2 /x 1 approaches 1, and λ → ∞. Even if b < 12 , then we see that the part of the integral near to z = −1 contributes a term much like I−1 . Lemma 6.1.24 shows that γ

3

γ

I−2 ≤ Cb,θ I−1 + Cb,θ |t| 2 λ 2 + 2



x2 −1 x1

 e−

cos θ

(√x1 −√x2 )2  √x2 −√x1 γ 4|t|

2|t |

1+ γ2

λ

The complicated expression on the right-hand side can be rewritten as √ √  √ √  √ √ 2 2 − x1 ) √ √ x2 − x1 x 2 + x 1 − cos θ ( x4|t| e √ , Cb,θ ( x 2 − x 1 )γ √ x1 |t| showing that

√ √ I−2 ≤ Cb,θ I−1 + Cb,θ | x 2 − x 1 |γ ,

. (A.56)

(A.57)

(A.58)

which is precisely the bound that we need. The error term contributes a term of this size times λ−1 , completing the analysis of this term. We now turn to I+ ; in this part the lower limit of integration is  2 2 λ β x2 = 3 w= − 1 ≥ λ. |t| 4 x1 If λ < 1, we need to split the integral into the part from β/t to 1/λ, and use the Taylor expansion at zero to estimate the ψb - and ψb -terms. If λ > 1, then we only need to use the asymptotic expansions of ψb and ψb . For λ < 1, we have to estimate 1

λ β t

+ x +   + − 2 λe + √ √ dw x2 . wb e− cos θw ++e x1 θ ψb wλe2θ − e−λeθ ψb (wλe2θ )++ | λ − w|γ x1 w (A.59)

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APPENDIX A

Using (A.47) and arguing as before we can show that the contribution of this term is γ estimated by Cb,θ |t| 2 (x 2 − x 1 )/|t|. This, in turn, is estimated by √ √ Cb,θ | x 2 − x 1 |γ . e−

If λ < 1, then the contribution of the integral from 1/λ to infinity is of the form √ √ Cb,θ | x 2 − x 1 |γ . Assuming now that λ > 1, we change variables as before, to see that

cos θ 2λ

γ 2

I+ ≤ Cb |t| λ ∞ λ 4

3 b 4−2



 x2 −1 × x1

w 2 − 4 e− cos θ ( b

1

√

√ 2 w− λξ )

2   x 3 x 2 −1

 √ √ √ 1 dw | ξ λ − w + O((wλ)− 2 )|| λ − w|γ √ . w

1

(A.60)  In this case ξ ≤ xx21 , and so changing variables again, as above, we see that the leading term is bounded by γ

3

γ

I+ ≤ Cb |t| 2 λ 2 + 2



 x2 −1 × x1 ∞ e 3 2

  2 x −λ cos θ z− x 2 b− 21 1

z

|z − 1|γ +1dz.

(A.61)

x

2 1 x1 − 2

This term, as well as the error term, satisfy the same estimates as those satisfied by I2− , thereby completing the proof of Lemma 6.1.7.  L EMMA A.1.5. [L EMMA 6.1.8] For b > 0, 0 < γ < 1 and c < 1 there is a Cb such that if c < s/t < 1, then ∞ + + √ γ √ + b + +kt (x, y) − ksb (x, y)+ | x − y|γ d y ≤ Cb |t − s| 2 .

(A.62)

0

P ROOF. If we let w = y/t, λ = x/t and μ = t/s, then this becomes: ∞ + + √ √ + b + +kt (x, y) − ksb (x, y)+ | x − y|γ d y ≤ 0

∞ + + √ √ dw + b −(w+λ) + t . ψb (wλ) − (μw)b e−μ(w+λ) ψb (μ2 wλ)+ | w − λ|γ +w e w γ 2

0

(A.63)

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261

We denote the quantity on the left by K (x, s, t). If F(μ) = (μw)b e−μ(w+λ) ψb (μ2 wλ), then the difference in the integral can be written: F(μ) − F(1) = F  (ξ )(μ − 1) for a ξ ∈ (1, μ).

(A.64)

Computing the derivative, we see that ∞ K (x, s, t) ≤ t (μ − 1) (ξ w)b e−ξ(w+λ) × γ 2

0

+ +  √ + b + √ 2  2 + + | w − λ|γ dw . (A.65) − (w + λ) ψ (ξ wλ) + 2ξ wλψ (ξ wλ) b b + ξ + w As usual, we split this into a part, I1 from 0 to 1/λ, and the rest, which we denote by I2 . To bound I1 we estimate ψb by a constant and use the estimate of ψb in (A.48). Arguing exactly as before we see that   γ t −s I1 ≤ C b t 2 . (A.66) s The fact that t/s < 1/c, implies that there is a constant C so that   γ γ t −s 2 t ≤ C(t − s) 2 , s showing that

γ

I1 ≤ Cb (t − s) 2 ,

(A.67)

(A.68)

for constants Cb that are uniformly bounded for 0 < λ < B. To estimate I2 we use the asymptotic formulæ for ψb and ψb . It is straightforward to estimate the bξ -term. To estimate the other two terms we need to take advantage of cancellations that occur, to leading order, and then use the error terms in the asymptotic expansions to estimate the remainder. The ξb -term is estimated by γ 2

∞

Cbt (μ − 1)

wb e−ξ(

1 λ

√ √ w− λ)2

√ 1 b √ dw . (wλ) 4 − 2 | w − λ|γ w

(A.69)

As before we apply Lemma 6.1.23 to show that this integral is uniformly bounded for λ ∈ (0, ∞). This term is again bounded by   γ t −s 2 Cb t , (A.70) s which is handled exactly like I1 .

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APPENDIX A

This leaves only γ

I2 = t 2 (μ − 1)× ∞ + + √ √ dw + + . wb e−(w+λ) +2ξ wλψb (ξ 2 wλ) − (w + λ)ψb (ξ 2 wλ)+ | w − λ|γ w

(A.71)

1 λ

Using the asymptotic expansions for ψb and ψb this is bounded by I2

∞

γ 2

≤ Cb t (μ − 1)

wb e−ξ(

1 λ

√ √ w− λ)2

1

b

(ξ 2 wλ) 4 − 2 ×

+√ + √ √ 1 + √ dw + . +( w − λ)2 + O((wλ)− 2 )+ | w − λ|γ w

(A.72)

This is negligible as λ → 0; Lemma 6.1.23 implies that the leading term is bounded % γ $ by Cb t 2 t −s , as before, and that the error term is bounded by s   γ 1 t −s λ− 2 Cb t 2 . s 

This completes the proof of the lemma. The proof of Lemma 6.1.8 also establishes the following simpler result: L EMMA A.1.6. [L EMMA 6.1.9] For b > 0, there is a Cb such that if s < t, then  ∞ + + + b + +kt (x, y) − ksb (x, y)+ d y ≤ Cb 0

 t/s − 1 . 1 + [t/s − 1]

(A.73)

We now consider the effects of scaling these kernels by powers of x/y. L EMMA A.1.7. [L EMMA 7.1.2] If 0 ≤ γ ≤ 1, and b > ν − γ2 > 0, then there is a constant Cb,φ , bounded for b ≤ B, and B −1 < b + γ2 − ν, so that, for t ∈ Sφ , where 0 < φ < π2 , we have the estimate ∞  ν γ γ x |ktb (x, y)|y 2 d y ≤ Cb,φ x 2 . y

(A.74)

0

P ROOF. We let t = τ eiθ , with |θ | < shows that we need to bound |t|

γ 2

∞  0

λ w

ν

π 2

− φ. Using w = y/|t| and λ = x/|t|, it

γ

wb e− cos θ(w+λ) |ψb (wλe2θ )|w 2

dw . w

(A.75)

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As usual we split this into an integral from 0 to 1/λ and the rest. The compact part we can estimate by 1

γ 2

Cb |t| λν e− cos θλ

λ

γ

wb+ 2 −ν−1 e− cos θw dw.

(A.76)

0 γ 2

As b + ν − > 0, the integral is clearly bounded uniformly in λ. Because ν − γ2 ≥ 0, the contribution of this term is bounded by γ

γ

γ

 Cb,θ x 2 λν− 2 e− cos θλ ≤ Cb,θ x2.

(A.77)

We use the asymptotic expansion of ψb to see that the non-compact part is bounded by γ 2

Cb |t|

∞  1 λ

w  b2 − 14 −ν − cos θ(√w−√λ)2 γ dw e w2 √ . λ w

(A.78)

cos θ

The integral tends to zero like e− 2λ as λ → 0, showing that again this term is bounded √ γ √ by C x 2 . We let w − λ = z, to obtain that, as λ → ∞, this is bounded by γ 2

∞

ν+ 14 − b2

Cb |t| λ

(z + √1 λ

√ b− 1 −2ν+γ − cos θ z 2 λ) 2 e dz.

(A.79)

√ − λ γ

It is again not difficult to see that, as λ → ∞, this is bounded by Cb,θ x 2 .



L EMMA A.1.8. [L EMMA 7.1.3] Let J = [α, β], where α and β are given in (7.61). Assuming that x 1 and x 1 satisfy (7.60), for b > 0, 0 < γ ≤ 1, and 0 < φ < π2 , there is a Cb,φ so that if t ∈ Sφ , then, + 3 ++ 2  +   + + b+1 +k (x 1 , z 1 ) x 1 − k b+1 (x  , z 1 ) x 1 + |√z 1 − x  |γ dz 1 ≤ Cb,φ |√x 1 − x  |γ . t 1 1 1 + t + z1 z1 + + Jc

(A.80)

P ROOF. We let t = τ eiθ , with |θ | < π2 − φ. Observe that it suffices to show that  + +√ + x − x +    1 + 1+ √ + | z 1 − x  |γ dz 1 ≤ Cb,φ |√x 1 − x  |γ , (A.81) I = |ktb+1 (x 1 , z 1 )| ++ √ 1 1 + z1 + + c J

and  II = Jc

3

  + √ x 1 ++ b+1 √ + +kt (x 1 , z 1 ) − ktb+1 (x 1 , z 1 )+ | z 1 − x 1 |γ dz 1 ≤ Cb,φ | x 1 − x 1 |γ . z1 (A.82)

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APPENDIX A

The integral in I is relatively simple to bound, and we can extend the integral over [0, ∞), rather than just over J c . Before switching the domain of integration we observe that there is a constant C so that if z 1 ∈ J c , then  √ √ √ √ √ C −1 | z 1 − x 1 | ≤ | z 1 − x 1 | ≤ C| z 1 − x 1 |. (A.83) It therefore suffices to show that  + +√ + x − x + ∞  1 + 1+ √ + | z 1 − √x 1 |γ dz 1 ≤ Cb,φ |√x 1 − x  |γ . (A.84) I  = |ktb+1 (x 1 , z 1 )| ++ √ 1 + z1 + + 0

To estimate this integral we let w = z 1 /|t| and λ = x 1 /|t|, to obtain that 

I = |t|

γ −1 2

√

| x1 −



x 1 |

∞ 0

√ √ dw wb e− cos θ(w+λ) |ψb+1 (wλe2θ )|| w − λ|γ √ . (A.85) w

We observe that |t|

γ −1 2

√ | x1 −



√ x 1 | = | x 1 −



⎤ 1−γ ⎡+ 3 ++ 2 +  √ + + x1 + ⎦ x 1 |γ ⎣++1 − λ . x 1 ++ +

(A.86)

For bounded λ it is easy to see that the integral in (A.85) is bounded, so we need to consider what happens as λ → ∞. As usual we split the integral into the part over [0, 1/λ] and the rest. The compact part is estimated by √

I− ≤ Cb | x 1 −



1

x 1 |γ λ

1−γ 2

e− cos θλ

λ 0

√ √ dw wb e− cos θw | w − λ|γ √ . w

(A.87)

Whether λ is going to zero  or infinity, we see that the contribution of this term is √ bounded by Cb,θ | x 1 − x 1 |γ . The non-compact is estimated using the asymptotic expansion for ψb+1 as I+

√

≤ Cb | x 1 −



x 1 |γ λ

1−γ 2

∞  1 λ

√ dw w  b2 − 14 − cos θ(√w−√λ)2 √ e | w − λ|γ √ . (A.88) λ wλ

1

As λ → 0, the integral is O(e− 2λ ). We let z = I+

√

≤ C| x 1 −



γ x 1 |γ λ− 2

√

w−

√ λ, to obtain that

 1 ∞  z b− 2 − cos θ z 2 γ 1+ √ e |z| dz, λ √

√1 − λ

λ

(A.89)

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PROOFS OF ESTIMATES FOR THE DEGENERATE 1-D MODEL

from which it follows easily that  √ I ≤ I  ≤ Cb,θ | x 1 − x 1 |γ .

(A.90)

We now turn to I I. With y/|t| = w, λ = x 1 /|t|, and μ = x 1 /x 1 , we see that

II ≤



x 1 |t|

γ −1 2

⎡ α ⎤ |t| ∞ ⎢ ⎥ ⎢ + ⎥ wb e− cos θw |e−μλeθ ψb+1 (μλwe2θ )−e−λeθ ψb+1 (λwe2θ )|× ⎣ ⎦ β |t|

0

√ √ dw | w − λ|γ √ . (A.91) w

Note that 1 ≤ μ ≤ 4. To estimate this term we use the Lemma A.0.4 to obtain: |e−μλeθ ψb+1 (μλwe2θ ) − e−λeθ ψb+1 (λwe2θ )| ≤ (μ − 1)λe− cos θξ λ × |wψb+2 (ξ λwe2θ ) − ψb+1 (ξ λwe2θ )|, where ξ ∈ (1, μ). (A.92) The limits of integration in (A.91) can be re-expressed as  √  3− μ 2 α =λ |t| 2

 √  3 μ−1 2 β =λ . |t| 2

(A.93)

When we use the expression in (A.92) in (A.91), we see that the integral is multiplied by 

x 1 |t|

γ −1 2

 √ √ √ √ (μ − 1) = | x 1 − x 1 |γ [ λ( μ − 1)]1−γ ( μ + 1).

(A.94)

We therefore need to show that ⎡ √ √ ⎢ λ[ λ( μ − 1)]1−γ ⎢ ⎣

α

|t| 0

⎤ ∞ ⎥ b − cos θ(w+ξ λ) + ⎥ × ⎦w e β |t|

√ √ dw |weθ ψb+2 (ξ λwe2θ ) − ψb+1 (ξ λwe2θ )|| w − λ|γ √ w

(A.95)

is uniformly bounded. It is clear that the contribution of the integral from 0 to |tα| is bounded for λ bounded, so we only need to evaluate the behavior of this term as λ → ∞. For this purpose we need to split the integral into a part from 0 to 1/λ and the rest. The part from 0 to 1/λ is bounded by a constant times λ2+b e− cos θλ , and is therefore controlled. The remaining

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APPENDIX A

contribution is bounded by α

|t| √ √ 1−γ λ[ λ( μ − 1)] wb e− cos θ(w+ξ λ) |weθ ψb+2 (ξ λwe2θ ) − ψb+1 (ξ λwe2θ )|× 1 λ

√ √ dw | w − λ|γ √ . w

(A.96)

Using the asymptotic expansions for ψb+1 and ψb+2 we see that |weθ ψb+2 (ξ λwe2θ ) − ψb+1 (ξ λwe2θ )| ≤    √  √ √ b 3 1 1 C we2 cos θ ξ λw (ξ λw)− 2 − 4 | w − λξ | + O √ + √ . w λ

(A.97)

The integral in (A.96) is bounded by α

C λ

|t|  1 λ

w ξλ

b−1 2

4

e− cos θ(

√ √ w− ξ λ)2

 √ | w − λξ |γ ×



√ √ | w − λ| + O



1 1 √ +√ w λ



dw √ . w

√ √ √ √ In the interval of integration λξ − w > λ − w, and  √ √ √ λξ − w ≤ C( λ − w),

(A.98)

(A.99)

provided that C > 3. We can therefore estimate the leading term in (A.98) by α

C λ We let z =

√

w−

|t|  1 λ

√ w  b2 − 14 − cos θ(√w−√λ)2 √ dw e | w − λ|1+γ √ . λ w

(A.100)

√ λ, to obtain that this is bounded by 

C λ

√ λ

α |t| −



√ λ

√1 − λ

  1 z b− 2 − cos θ z 2 1+γ 1+ √ e |z| dz, λ

with the upper limit of integration given by 2 √ √ √μ − 1 α − λ=− λ . |t| 2

(A.101)

(A.102)

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PROOFS OF ESTIMATES FOR THE DEGENERATE 1-D MODEL

When the upper limit of integration is bounded, then the  integral is bounded, and the √ contribution of this term is again bounded by C| x 1 − x 1 |γ . If the upper limit tends to −∞, then we easily show that this term is bounded by λ( Cb,θ √ √ [ λ( μ − 1)]γ e− cos θ λ

√ μ−1)2 4

,

(A.103)

 √ and therefore the contribution of this term is again bounded by C| x 1 − x 1 |γ . To complete the analysis of (A.98) we need to estimate the contribution of the error terms. Using the same change of variables we see that these terms are bounded by 

Cb λ

√ λ

α |t| −



√

√1 − λ

λ

   1  z b− 2 − cos θ z 2 γ 1 1 1+ √ e |z| O √ +√ dz. λ z+ λ λ

(A.104)

√ √ These contributions are bounded as before if λ( μ − 1) remains bounded. If the upper limit tends to −∞, then this expression is bounded by √ λ( μ−1)2 Cb,θ √ √ γ −1 − cos θ 4 [ λ( μ − 1)] e , 3 λ2

(A.105)

completing the  proof that the contribution from 0 to α/|t| is altogether bounded by √ Cb,θ | x 1 − x 1 |γ . We turn now to the part of (A.95) from β/|t| to ∞. We need to split this integral into two parts only for λ < 1. In this case we get a term bounded by 1

λ √ √ √ √ dw 1−γ λ[ λ( μ − 1)] wb e− cos θw | w − λ|γ √ , w

(A.106)

β |t|

which is clearly negligible as λ → 0. The other term takes the form ∞

√ √ λ[ λ( μ − 1)]1−γ

max

1 β λ , |t|

wb e− cos θ(w+ξ λ) × .

√ √ dw |weθ ψb+2 (ξ λwe2θ ) − ψb+1 (ξ λwe2θ )|| w − λ|γ √ . w

(A.107)

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APPENDIX A

The integral is bounded by 

∞

Cb λ max

-

1 β λ , |t|

.

w ξλ

b−1 2

4

e− cos θ(

√ √ w− ξ λ)2

 √ | w − λξ |γ ×

   √ √ 1 1 dw | w − λ| + O √ + √ √ . w w λ

(A.108)

For w ∈ [ |tβ| , ∞) we have the inequalities  √ √ √ √ 1 √ ( w − λ) ≤ ( w − λξ ) ≤ ( w − λ), 3

(A.109)

and therefore the expression in (A.108) is bounded by ∞

Cb λ

max

1 β λ , |t|

wb−1 2

.

λ

As before we let z = Cb λ

4

e− cos θ

√ √ ( w− λ)2 9

√ √ | w − λ|γ ×

   √ √ 1 1 dw | w − λ| + O √ + √ √ . w w λ √

w−

(A.110)

√ λ. If 1/λ > β/|t|, then we obtain:

   1  ∞  z b− 2 − cos θ z2 γ 1 1 9 1+ √ e |z| |z| + O √ +√ dz. (A.111) λ z+ λ λ √

√1 − λ

λ

cos θ

This is bounded by Cb,θ e− 20λ /λ as λ → 0, so in thiscase the contribution of the √ integral from β/|t| to infinity is bounded by Cb,θ | x 1 − x 1 |γ . The final case to consider is when 1/λ < β/|t|, so that the lower limit of integration in (A.111) would be: 3 √ √ 3(√μ − 1) β − λ= λ . (A.112) |t| 2 An analysis, essentially identical to that above, shows that this term is bounded by λ( Cb √ √ [ λ( μ − 1)]γ e− cos θ λ

√ μ−1)2 4

.

(A.113)

 √ As before we conclude that the contribution of this term is bounded by C| x 1 − x 1 |γ , which completes the proof of the lemma. 

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269

L EMMA A.1.9. [L EMMA 7.1.4] For b > 0, γ ≥ 0, there is a constant Cb,φ , bounded for b ≤ B, so that for t ∈ Sφ , ∞

√ γ −1 γ √ |ktb+1 (x, y)|| y − x|y 2 d y ≤ Cb,φ |t| 2 .

(A.114)

0

P ROOF. We let t = τ eiθ , with |θ | < and λ = x/|t| to obtain: |t|

γ 2

∞ 0

π 2

− φ, and change variables with w = y/|t|,

√ dw √ γ wb+ 2 e− cos θ(w+λ) |ψb+1 (wλe2θ )|| w − λ| √ . w

(A.115)

In the part of the integral from 0 to 1/λ, we estimate ψb+1 by a constant, obtaining 1

γ

λ

Cb |t| 2 0

√ dw √ γ wb+ 2 e− cos θ(w+λ) | w − λ| √ , w

(A.116)

γ

which is easily seen to be bounded by Cb,θ |t| 2 . In the non-compact part we use the asymptotic expansion of ψb+1 to see that this contribution is bounded by Cb |t|

γ 2

∞  1 λ

√ √ w  b2 − 14 − cos θ(√w−√λ)2 γ | w − λ| dw 2 e w √ √ . λ w λ γ

− As λ√→ 0 this √ is bounded by Cb,θ |t| 2 e z = w − λ to obtain:

Cb |t|

γ 2

cos θ 2λ

. To estimate this term as λ → ∞, we let

 1 ∞  z b− 2 − cos θ z 2 √ |z| 1+ √ e ( λ + z)γ √ dz. λ λ √

√1 − λ

(A.118)

λ

γ

This term is bounded by Cb,θ |t| 2 λ

γ −1 2

, thereby completing the proof of the lemma. 

L EMMA A.1.10. If b > ν > 0, and 0 < φ < bounded for b ≤ B, so that if t ∈ Sφ , we have

π 2,

then there is a constant Cb,φ ,

∞  ν γ √ x √ |ktb (x, y)|| y − x|γ d y ≤ Cb,φ |t| 2 . y 0

(A.117)

(A.119)

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APPENDIX A

P ROOF. We let t = τ eiθ , with |θ | < see that the integral in the lemma equals

π 2

− φ. Using w = y/|t|, and λ = x/|t|, we

∞  ν √ √ dw λ |t| . wb e− cos θ(w+λ) ψb (wλe2θ )| w − λ|γ w w γ 2

(A.120)

0

It therefore suffices to show that this integral is uniformly bounded for λ ∈ [0, ∞). The contribution from [0, 1/λ] is bounded by 1

Cb λν e− cos θλ

λ

√ √ wb−ν−1 e− cos θw | w − λ|γ dw.

(A.121)

0

Since b − ν > 0 this is uniformly bounded for all λ. The remaining contribution comes from ∞  1 λ

λ w

ν

√ √ dw ≤ wb e− cos θ(w+λ) ψb (wλe2θ )| w − λ|γ w

Cb

∞   b √ dw w 2 −ν− 14 − cos θ(√w−√λ)2 √ e | w − λ|γ √ . λ w

(A.122)

1 λ

As λ → 0, this is bounded by Cb e− Cb

cos θ 2λ

. Letting z =

√ √ w − λ, we obtain

1  ∞  z b−2ν− 2 − cos θ z 2 γ 1+ √ e |z| dz. λ √

√1 − λ

(A.123)

λ

It is again straightforward to see that this remains bounded as λ → ∞, thus completing the proof of the lemma.  L EMMA A.1.11. [L EMMA 7.1.5] For 0 ≤ γ < 1, 1 ≤ b, and 0 < c < 1, there is a constant C, so that if ct < s < t, then ∞ + + √ γ √ γ −1 + b + +kt (x, z) − ksb (x, z)+ | x − z|z 2 dz ≤ C|t − s| 2 .

(A.124)

0

P ROOF. The proof of this lemma is very similar to that of Lemma 6.1.8. If we let w = z/t, λ = x/t, and μ = t/s, then the integral we need to estimate becomes t

γ 2

∞ 0

√ dw √ γ wb+ 2 −1 |e−(w+λ)ψb (wλ) − μb e−μ(w+λ) ψb (μ2 wλ)|| w − λ| √ . (A.125) w

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Proceeding as in the proof of Lemma 6.1.8 we see that |e−(w+λ) ψb (wλ) − μb e−μ(w+λ) ψb (μ2 wλ)| = + +  + + b −ξ(w+λ) + b 2 2 + (μ − 1)ξ e + ξ − (w + λ) ψb (ξ wλ) + 2ξ wλψb+1 (ξ wλ)+ ,

(A.126)

where ξ ∈ (1, μ) ⊂ (1, 1c ). Since γ

t2



t −s s



γ

≤

|t − s| 2 , c

(A.127)

it again suffices to show that the integral ∞ 0

+  + b γ − (w + λ) ψb (ξ 2 wλ)+ wb+ 2 −1 e−ξ(w+λ) ++ ξ + √ dw +√ 2ξ wλψb+1 (ξ 2 wλ)++| w − λ| √ w

(A.128)

is uniformly bounded for λ ∈ (0, ∞). We split the integral into a part from 0 to 1/λ and the rest. The compact part is bounded by 1

λ C 0

√ dw √ γ wb+ 2 −1 e−(w+λ) |b + w + λ + 2wλ|| w − λ| √ . w

(A.129)

As b ≥ 1, it is not difficult to see that this integral is uniformly bounded for λ ∈ (0, ∞). For the non-compact part, we first estimate the contribution from the b/ξ -term. Using the asymptotic expansion we see that this part is bounded by ∞   b 1 √ dw w 2 − 4 γ −1 −ξ(√w−√λ)2 √ C w 2 e | w − λ| √ . λ w

(A.130)

1 λ

1

As λ → 0 this is O(e− 2λ ). To estimate this expression as λ → ∞, we let z; this integral is then bounded by  1 ∞  z b− 2 √ 2 1+ √ ( λ + z)γ −1 e−z |z|dz, λ √

√1 − λ

λ

which, as λ → ∞, is bounded by Cλ

γ −1 2

.

√ √ w− λ =

(A.131)

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APPENDIX A

The other term is estimated by + + + + +(w + λ)ψb (ξ 2 wλ) − 2ξ wλψb+1 (ξ 2 wλ)+ = √ / ( 2 )0 2 2 ξ 2 wλ √ √ 1 b e w λ ( w − λ)2 + O 1 + + . (ξ 2 wλ) 4 − 2 √ λ w 4π

(A.132) 1

It is again easy to see that, as λ → 0, the contribution of this term is O(e− 2λ ). To bound term as λ → ∞, we use the estimate from (A.132) in (A.128), and let √ this √ z = w − λ to obtain  1 ∞  z b− 2 √ 2 1+ √ ( λ + z)γ −1 e−z |z|× λ √

√1 − λ

λ

/

)0 √ z λ |z| + O 1 + √ + dz. (A.133) √ λ z+ λ (

2

It is again straightforward to see that all contributions in this integral are O(λ which completes the proof of the lemma.

γ −1 2

), 

A.2 FIRST DERIVATIVE ESTIMATES L EMMA A.2.1. [L EMMA 6.1.10] For b > 0, 0 ≤ γ < 1, and 0 < φ < a Cb,φ so that for t ∈ Sφ we have ∞

√ |∂x ktb (x, y)|| y

−

√

π 2,

there is

γ

γ

x| d y ≤ Cb,φ

0

|t| 2 −1 1

1 + λ2

,

(A.134)

where λ = x/|t|. P ROOF. We let t = τ eiθ , where |θ | < 8.1 in [15] we see that ∞

π 2

− φ. Arguing as in the proof of Lemma

√ √ |∂x ktb (x, y)|| y − x|γ d y ≤

0

|t|

γ 2 −1

∞

√ + + √ wb−1 e− cos θ(w+λ) +weθ ψb (wλe2θ ) − ψb (wλe2θ )+ | w − λ|γ dw.

0

(A.135)

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Here λ = x/|t| and w = y/|t|. We now estimate the quantity: ∞ I (λ) =

√ + + √ wb−1 e− cos θ(w+λ) +weθ ψb (wλe2θ ) − ψb (wλe2θ )+ | w − λ|γ dw.

0

(A.136) We divide this integral into a part from 0 to 1/λ and the rest. We write I (λ) = I0 (λ) + I1 (λ). In the first part we can estimate the integral using the Taylor expansions for ψb and ψb as  1/λ I0 (λ) ≤ Cb wb−1 e− cos θ(w+λ) w + 0

 √ √ 1 | w − λ|γ dw. (b)

(A.137)

When λ remains bounded, the only difficulty that arises is that as b → 0, the wb−1 term introduces a 1/b, but this is compensated for by the 1/ (b), showing that this expression remains bounded as b → 0. This term is bounded by a constant times ( γ ) 2 − cos θλ 1 + λ e . (A.138) b 1+λ To estimate the other part of the integral, we use the asymptotic expansions for ψb and ψb , giving I1 (λ) ≤ C

∞  1 λ

w  b2 − 14 λ

+√ √ ++1+γ − cos θ(√λ−√w)2 dw + e √ . + λ − w+ wλ

(A.139)

1

Applying Lemma 6.1.23, this is O(e− λ ), as λ → 0, and, as λ → ∞, 1

I1 (λ) ≤ Cb,φ λ− 2 .

(A.140)

Combining this with the estimates above, we can show that I (λ) ≤

C √ . 1+ λ

(A.141) 

This proves Lemma 6.1.10. L EMMA A.2.2. [L EMMA 6.1.11] For b > 0, 0 < γ < 1, 0 < φ < 0 < c < 1, there is a constant Cb,c,φ so that for cx 2 < x 1 < x 2 , t ∈ Sφ , ∞

π 2,

and

√ √ √ √ | x 1 ∂x ktb (x 1 , y) − x 2 ∂x ktb (x 2 , y)|| x 1 − y|γ d y ≤

0



Cb,c,φ |t|

γ −1 2

√ √  | x2 − x1 | √ |t |  √ √ . | x2 − x1 | √ 1+ |t |

(A.142)

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APPENDIX A

P ROOF. We let t = τ eiθ , where |θ | < a “trivial” estimate, ∞

π 2

−φ, and note that Lemma 6.1.10 provides

√ √ √ √ | x 1 ∂x ktb (x 1 , y) − x 2 ∂x ktb (x 2 , y)|| x 1 − y|γ d y ≤

0

γ −1 √ γ −1 x 1 |t| 2 Cb,φ √ √ ≤ Cb,φ |t| 2 , (A.143) x 1 + |t|

√ √ √ which is the desired estimate when | x 2 − x 1 |/ |t| >> 1. Recalling that ∂x ktb (x, y) =

 x y  1  y b − x+y  y    x y  ψb 2 − ψb 2 e t yt t t t t

(A.144)

and setting w = y/|t|, λ = x 1 /|t|, and μ = x 2 /x 1 , we see that ∞

√ √ √ √ | x 1 ∂x ktb (x 1 , y) − x 2 ∂x ktb (x 2 , y)|| x 1 − y|γ d y =

0

∞ √ √ √ γ −1 2 λ|t| wb−1 e− cos θw |F(μ, λ, w) − F(1, λ, w)|| λ − w|γ dw, (A.145) 0

where we let F(μ, λ, w) =

, √ −μλeθ * weθ ψb (μλwe2θ ) − ψb (μλwe2θ ) . μe

(A.146)

Using Lemma A.0.4 we see that |F(μ, λ, w) − F(1, λ, w)| ≤ (μ − 1)|∂μ F(ξ, λ, w)|

for a

ξ ∈ (1, μ), (A.147)

where ξ ∈ (1, 1c ). We therefore need to estimate  √ √ √ γ −1 2 λ|t| (μ − 1) wb−1 e− cos θw |∂μ F(ξ, λ, w)|| λ − w|γ dw. ∞

(A.148)

0

Using the equation satisfied by ψb , we see that ∂μ F(ξ, λ, w) =    e−θ  1 e−ξ λeθ  ψb (ξ λwe2θ ) − 2ξ λweθ + (b − )w ψb (ξ λwe2θ ) . ξλ + w − eθ √ 2 2 ξ (A.149)

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As usual we split the integral in (A.148) into the part from 0 to 1/λ and the rest. In the compact part we use the estimate  w 1+λ+w |∂μ F(ξ, λ, w)| ≤ Cb e− cos θξ λ + + λwO(1 + λ + w) . (b) (b + 1) (A.150) Using this estimate we see that 1

√

λ|t|

γ −1 2

λ (μ − 1)

√ √ wb−1 e− cos θw |∂μ F(ξ, λ, w)|| λ − w|γ dw ≤

0

√ γ −1 cos θ λ Cb,θ λ|t| 2 e− 2 |μ − 1| (A.151)

where the constant is uniformly bounded for b ∈ (0, B], and |θ | ≤ π2 − φ. Now we turn to the non-compact part where we use the asymptotic expansions of ψb and ψb to obtain: |∂μ F(ξ, λ, w)| = (ξ λw)

1 b 4−2

e

√ 2 cos θ ξ λw − cos θξ λ

e

/  ( 2 )0 2 2 2 w w λ λ 1− + . +O 1+ ξ ξλ λ w (A.152)

Using this expansion in the integral and setting z = bounded by

Cb λ

γ 2 +1

|t|

γ −1 2

∞ (μ − 1)

√ 2 ξ)

1

z b− 2 e− cos θλ(z−

1 λ



λ(z −

√ w/λ, we see that this term is

|z − 1|γ ×



 ξ )2 + O(1 + z + 1/z) dz. (A.153) γ −1

cos θ

As λ → 0 this term is easily seen to be bounded by Cb,θ |t| 2 (μ − 1)e− 2λ . √ √ √ 1/4, we use the “trivial” estimate in (A.143). In this case, when | x 2 − x 1 |/ |t| >√ √ √ We henceforth assume that | x 2 − x 1 |/ |t| ≤ 1/4, which implies that √

1 μ−1 ≤ √ . 4 λ

(A.154)

In order to estimate the integral, we need to split it into three parts, with z lying in √ √ [1/λ, 1], [1, μ], and [ μ, ∞), respectively. Using the assumption in (A.154) we √ γ −1 √ easily show that the integral over [1, μ] is bounded by Cb,θ λ|t| 2 (μ − 1), as desired. To treat the other two terms we use Laplace’s method. The integral over

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APPENDIX A

[1/λ, 1] is bounded by Cb λ

γ 2 +1

|t|

γ −1 2

1 (μ − 1)

1

2

z b− 2 e− cos θλ(z−1) |z − 1|γ ×

1 λ

  √ λ(z − μ)2 + O(1 + z + 1/z) dz. (A.155)

Laplace’s method, using (A.154), shows that this term is also bounded by √ γ −1 Cb,θ λ|t| 2 (μ − 1). √ Finally the integral over [ μ, ∞) is bounded by Cb λ

γ 2 +1

|t|

γ −1 2

∞ (μ − 1)

√ 2 μ)

1

z b− 2 e− cos θλ(z−

√ μ



|z − 1|γ ×

 λ(z − 1)2 + O(1 + z + 1/z) dz. (A.156)

Applying Laplace’s method to this integral shows that it is bounded by √ γ −1 Cb,θ λ|t| 2 (μ − 1), thereby completing the estimate of the non-compact term. The proof of the lemma is completed by noting that √ √ √ √ √ x2 − x1 x2 + x1 λ(μ − 1) = √ , (A.157) √ x1 |t| and therefore the lemma follows from the assumption that 0 < c < x 1 /x 2 < 1.



L EMMA A.2.3. [L EMMA 6.1.12] For b > 0, 0 < γ < 1, there is a constant Cb so that for t1 < t2 < 2t1 , we have: t1 ∞

√ γ √ |∂x ktb2 −t1 +s (x, y) − ∂x ksb (x, y)|| x − y|γ d yds < Cb |t2 − t1 | 2 . (A.158)

t2 −t1 0

This result follows from the more basic: L EMMA A.2.4. [L EMMA 6.1.13] For b > 0, 0 ≥ γ < 1, and 0 < t1 < t2 < 2t1 , we have for s ∈ [t2 − t1 , t1 ] that there is a constant C so that ∞ 0

√ |∂x ktb2 −t1 +s (x, y) − ∂x ksb (x, y)|| x

−

√

γ

(t2 − t1 )s 2 −1 . y| d y < C √ (t2 − t1 + s)(1 + x/s) γ

(A.159)

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P ROOF OF L EMMA 6.1.12. The estimate in (A.158) follows by integrating the estimate in (6.42): t1 ∞

√ |∂x ktb2 −t1 +s (x, y) − ∂x ksb (x, y)|| x

−

√

t1

γ

y| d yds ≤ C

t2 −t1 0

t2 −t1

∞ ≤C

γ

(t2 − t1 )s 2 −2 ds =

t2 −t1

γ

(t2 − t1 )s 2 −1 ds (t2 − t1 ) + s

γ 2C |t2 − t1 | 2 . 2−γ

(A.160) 

We now give the proof of Lemma 6.1.13. P ROOF. This argument is very similar to the proof of Lemma 6.1.11. Set τ = t2 − t1 , and define   G(μ, λ, w) = μb+1 e−μ(w+λ) μwψb (μ2 wλ) − ψb (μ2 wλ) . (A.161) Setting w = y/s, λ = x/s, and μ = s/(τ + s), we see that ∞

√ √ |∂x ktb2 −t1 +s (x, y) − ∂x ksb (x, y)|| x − y|γ d y =

0

s

γ 2 −1

∞

√ √ wb−1 |G(μ, λ, w) − G(1, λ, w)|| λ − w|γ dw =

0

s

γ 2 −1

∞ (μ − 1)

√ √ wb−1 |∂μ G(ξ, λ, w)|| λ − w|γ dw;

(A.162)

0

here ξ ∈ [μ, 1]. In the last line we use the mean value theorem. The assumption t1 < t2 < 2t1 shows that μ ∈ [ 12 , 1). A calculation, using the equation satisfied by ψb shows that ∂μ G(ξ, λ, w) = ξ b+1 e−ξ(w+λ) ×  1+b 2  2 ) − wψb (ξ wλ)(b − 2 + ξ(w + 3λ)) . ψb (ξ wλ)(3w + λ − ξ

(A.163)

As usual, we split the integral into a part from 0 to 1/λ and the rest. In the compact part we observe that +  ++ +w + λ + 1 − w+λ 2 + + 2 |∂μ G(ξ, λ, w)| ≤ Ce (A.164) + (b) + O w(1 + w + λ) + .

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APPENDIX A

The compact part is therefore bounded by 1

s

γ 2 −1

λ (μ − 1)

wb−1 e−

w+λ 2

0

+  ++ √ +w + λ + 1 √ γ 2 + + + O w(1 + w + λ) + (b) + | λ − w| dw.

(A.165) As λ → ∞ this is bounded by Cs (μ − 1)e , and as λ → 0, by Cs (μ − 1). For the non-compact part we use the asymptotic expansions of ψb and ψb of order 2, given in (A.221), to obtain: γ 2 −1

√

√

γ 2 −1

− λ4

1

2

b

G(ξ, λ, w) ≤ Cξ b+1 e−ξ( w− λ) (ξ 2 wλ) 4 − 2 × 2 3 2  2  +  + +λ 1 − w − a1 (b) w 1 − w + + λ λ λ ( 2 ) 2    + + 1 1 1 w λ +; (A.166) + a3 (b) 1 − +O 1+ + + √ a2 (b) 1 − λ w λ w wλ + here a j (b), j = 1, 2, 3 are polynomials in b. Using this expression in the integral and √ √ letting z = λ( w/λ − 1), we see this is bounded by Cs

γ 2 −1

∞  (μ − 1) √1 λ

√ − λ

z √ +1 λ

b− 1 2

/

2

e

− z2

z

γ

γ

|z|3 + |z| +O √ λ

 0 1 dz. (A.167) λ

1

As λ → 0 this is bounded by Cs 2 −1 (μ − 1)e− 4λ . When λ → ∞, Laplace’s method √ γ applies to show that it is bounded by Cs 2 −1 (μ − 1)/ λ. This completes the proof of the lemma.  A.3 SECOND DERIVATIVE ESTIMATES L EMMA A.3.1. [L EMMA 6.1.14] For b > 0, 0 < γ < 1, and 0 < φ < a Cb,φ so that for t = |t|eiθ with |θ | < π2 − φ, |t | ∞ 0

0

there is

√ γ γ √ b |x∂x2kse x| d yds ≤ Cb,φ x 2 and iθ (x, y)|| y −

0

 t ∞

π 2,

(A.168) √ γ γ √ b |x∂x2kse x| d yds ≤ Cb,φ |t| 2 . iθ (x, y)|| y −

0

We deduce this lemma from the following result, of interest in its own right:

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PROOFS OF ESTIMATES FOR THE DEGENERATE 1-D MODEL

L EMMA A.3.2. [L EMMA 6.1.15] For b > 0, 0 ≤ γ < 1, and 0 < φ < a Cb,φ so that if t ∈ Sφ , then ∞

√ |x∂x2 ktb (x, y)|| x

−

0

√

π 2,

there is

γ

λ|t| 2 −1 , y| d y ≤ Cb,φ 1+λ γ

(A.169)

where λ = x/|t|. We let t = τ eiθ , where |θ | < from (A.169).

π 2

− φ, and first show how to deduce 6.1.14

P ROOF OF L EMMA 6.1.14. To prove the first estimate in (A.168), using (A.169), we see that |t | ∞ 0

√ b |x∂x2 kse iθ (x, y)|| y

−

√

γ

|t |

x| d yds ≤ Cb,φ

0

0

γ

s 2 −1

x/s ds. 1 + x/s

(A.170)

Splitting this into an integral from 0 to x and the rest (if needed), we easily see that the first estimate in (A.168) holds. The second estimate follows from (A.170) and the observation that x/(x + s) ≤ 1.  Now we prove Lemma 6.1.15. P ROOF OF L EMMA 6.1.15. We denote the left-hand side of (A.169) by I. The formula (4.5) for ktb , and the second order equation satisfied by ψb (z) : zψb + bψb − ψb = 0,

(A.171)

imply that x∂x2 ktb (x, y) =             1  y b − (x+y) x+y xy 2x y xy by xy t ψb 2 − ψb ψb − . e yt t t t t t2 t2 t2 (A.172) We let w = y/|t| λ = x/|t| to obtain I =

1 |t|

∞

1− γ2

wb e− cos θ(w+λ) ×

0

√ + √ + +(w + λ)ψb (wλe2θ ) − (2wλeθ + bw)ψ  (wλe2θ )+ | w − λ|γ dw , b w

(A.173)

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APPENDIX A

which we split into a part from [0, λ1 ] and the rest. In the compact part, we use the estimate  + + 1 +(w + λ)ψb (wλe2θ ) − (2wλeθ + bw)ψ  (wλe2θ )+ ≤ Cb λ + w(1 + w + λ) . b (b) (A.174) Applying Lemma 6.1.22 shows that these parts of the w-integral are bounded by λ

Cb,θ e− cos θ 2 as λ → ∞ and Cb,θ λ as λ → 0.

(A.175)

In the non-compact part of the w-integral, we use the asymptotic expansion to obtain + + +(w + λ)ψb (wλe2θ ) − (2wλeθ + bw)ψ  (wλe2θ )+ = b    √ √ 2 1 b w+λ − 2 2 cos θ wλ √ 4 (wλ) ( w − λ) + O √ e + 1 . (A.176) wλ Applying Lemma 6.1.23 shows that the principal terms of the non-compact part of the w-integral are bounded by 1 Cb,θ e− 2λ . (A.177) This leaves only the error term in (A.176). Again applying Lemma 6.1.23 shows that these terms are also bounded by the expression in (A.177).  L EMMA A.3.3. [L EMMA 6.1.16] For b > 0, 0 < γ < 1, 0 < φ < 0 < x 2 /3 < x 1 < x 2 , there is a constant Cb,φ so that, for t ∈ Sφ , we have |t | + + + + b b +(∂ y y − b)kse iθ (x 2 , α) − (∂ y y − b)k seiθ (x 2 , β)+ ds ≤ C b,φ ,

π 2,

and

(A.178)

0

where α and β are defined in (6.35). P ROOF. We let t = |t|eiθ , where |θ | < π2 − φ, and use I to denote the quantity on the left in (A.178). Using (4.5), we see that, for t in the right half plane,  x y  1  y b − (x+y)  x    x y  (∂ y y − b)ktb (x, y) = ψb 2 − ψb 2 , (A.179) e t t t t t t and therefore: t |I | ≤ 0

x2

e− cos θ s s +   x αe  + α b − αeθ  αeθ    x 2 αe2θ  2 2θ + ψb − ψb − + s e s 2 s s s2  b       + + βe β βeθ x 2 βe2θ x 2 βe2θ − sθ  +ds. ψb − ψb e + 2 2 s s s s

(A.180)

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As

x1 1 ≤ < 1, 3 x2

(A.181)

the numbers

x1 x2 β α < < < s s s s are all comparable. If s < x 1 , then we can use the asymptotic expansion to estimate the integrand by 2   √ √ √ √ Cb − cos θ ( x2 − x1 )2 ( x 2 − x 1 ) s 4s e . (A.182) √ +O s α 2 s Changing variables with

√ √ ( x 2 − x 1 )2 , σ = s the principal term in the integral from 0 to x 1 becomes: ∞

x dx e− cos θ 4 √ . x

Cb √

(A.183)

√

(A.184)

( x 2 − x 1 )2 x1

This is uniformly bounded. The integral of the error term is bounded by x1 0

1 ds √ = 2 αs

2

x1 . α

(A.185)

As x 1 /x 2 > 1/3 this is bounded by 1, completing the estimate of this part of the sintegral. If |t| > x 1 , then we also need to estimate the s-integral over [x 1 , t]. If we let Fμ (z) = z b e−zeθ [zeθ ψb (μze2θ ) − ψb (μze2θ )],

(A.186)

then the remaining part of the s-integral can be written: t x1

e− cos θ s

x2 s

+  +   + + + F x2 α − F x2 β + ds. + s s s s +

(A.187)

Lemma A.0.4 shows that this is estimated by t C x1

e− cos θ s

x2 s

|F x 2 (ξ )| s



β −α s

 ds.

(A.188)

Here ξ ∈ [ αs , βs ] ⊂ (0, xβ1 ] and μ ∈ [ xt2 , xx21 ]. The z b -term in Fμ (z) is the only term which may contribute something unbounded to Fμ (z), and this occurs only if b < 1.

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APPENDIX A

The remaining terms are easily seen to contribute a term bounded by Cb (1 − x 1 /x 2 ). The z b−1 -term is bounded by t K = bCb x1

e− cos θ s

x2 s

 α b−1  β − α  s

s

ds.

(A.189)

We let w = x 2 /s to obtain  K = bCb

β −α x2



x2

α x2

b−1 x1 w

b−1 − cos θw

x2 |t|

e

  x1 . dw ≤ Cb,θ b(b) 1 − x2

(A.190) This completes the proof that there is a constant Cb,φ uniformly bounded with b, so that I ≤ Cb,φ . (A.191)  L EMMA A.3.4. [L EMMA 6.1.17] For b > 0, 0 < γ < 1, φ < π2 , and 0 < x 2 /3 < x 1 < x 2 , if J = [α, β], with the endpoints given by (6.35), there is a constant Cb,φ so that if |θ | < π2 − φ, then |t | β I1 =

√ √ √ √ b |L b kse y − x 2 |γ d yds ≤ Cb,φ | x 2 − x 1 |γ iθ (x 2 , y)||

0 α

|t | β I2 =

(A.192) √ b |L b kse y− iθ (x 1 , y)||

√

√

x 1 |γ d yds ≤ Cb,φ | x 2 −

√

x 1 |γ .

0 α

P ROOF. Throughout these calculations we use the formula, valid for t in the right half plane: L b ktb (x, y) = ∂t ktb (x, y) =         1  y b − (x+y) x+y xy 2x y xy − b ψb 2 − ψb . e t 2 yt t t t t t2 We give the argument for I1 ; the argument for I2 is essentially identical. If we let x2 y λ= , w= s s and Rα,β,t = {(w, λ) :

α β x2 ≤ λ}, λ ≤ w ≤ λ and x2 x2 |t|

(A.193)

(A.194)

(A.195)

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283

then I1 becomes:  γ 2 wb−1 e− cos θ(w+λ) |[(w + λ)eθ − b]ψb (wλe2θ ) − 2wλe2θ ψb (wλe2θ ) | |I1 | ≤ x 2 Rα,β,t

2 +γ + + w ++ dwdλ . × ++1 − λ+ λ

(A.196)

As in the previous cases we estimate ψb and ψb using the Taylor expansion where wλ < 1 and using the asymptotic expansion where wλ ≥ 1. In the present instance this divides the argument into two cases: (1) |tx1| ≥ 1 and (2) |tx1| < 1. In case 1 we only need to use the asymptotic expansions, whereas in case 2 we also have to consider another term, where we estimate ψb and ψb using the Taylor expansion. We begin with case 1. The asymptotic expansion gives the estimate  √ 2 γ √ 1 b 2 |I1 | ≤ Cb x 2 wb−1 (wλ) 4 − 2 e− cos θ( w− λ) × Rα,β,t

2 +γ +    + + √ +√ + ++ +( w − λ)2 + b 1 + O √1 + +1 − w + dwdλ . + + + λ+ λ wλ

(A.197)

As w/λ is bounded above and below, this satisfies  √ 2   b 1 w 2 − 4 −λ cos θ 1− wλ |I1 | ≤ Cb x 2 e × λ γ 2

Rα,β,t

We let z =

/  0+ 2 2 2 +γ + w w ++ dwdλ + λ 1− . + 1 +1 − √ λ λ+ wλ

(A.198)

√ w/λ − 1; taking into account that z is bounded we obtain:  γ 2

β

∞ x2

|I1 | ≤ Cb x 2

x2 |t|



−1

e−λ cos θ z

α x 2 −1

2

  dzdλ λz 2 + 1 |z|γ √ . λ

(A.199)

We interchange the order of the integrations and set x = λz 2 , in the λ-integral, to see that  β

x2

γ 2

|I1 | ≤ Cb x 2

−1

∞ e



α x 2 −1

− cos θ x



 √ 1 √ + x d x|z|γ −1dz. x

(A.200)

x2 z2 |t|

The x-integral is bounded by a constant depending only on θ, and this shows that there is a constant Cb,θ , which is bounded for 0 < b ≤ B, so that √ √ (A.201) |I1 | ≤ Cb,θ gW F,0,γ | x 2 − x 1 |γ .

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APPENDIX A

Now we turn to case 2. The foregoing analysis is used to estimate the part of the integral where wλ > 1, by using 1 as the lower limit of integration in (A.199) instead of x 2 /|t|. This leaves the part of the integral in (A.196) over the set Rα,β,t ∩ {(w, λ) : wλ < 1}.

(A.202)

We replace this set, with the slightly larger set  = {(w, λ) : Rα,β,t

x2 α β x2 ≤ λ ≤ }. λ < w < λ and x2 x2 |t| α

(A.203)

Using the Taylor series, we see that this term is bounded by x2

γ

λ

β

α  x 2

C x 22 x2 |t|

λ xα

2

   1 + wλ + wλ × wb−1 (w + λ + b) (b) 2 +γ + + + +1 − w + dwdλ . + λ+ λ

(A.204)

In the w-integral we let σ = w/λ to see that this is bounded by x2

β

   α x2 1 b−1 2 2 + σλ + σλ × (σ λ + λ + b) (σ λ) C x2 (b) γ 2

0

α x2

+ √ + +1 − σ +γ dσ dλ. (A.205)

As c in (A.181) is at least 1/3, we know that range of the σ -integral satisfies √ 3−1 √ ≤ σ ≤ 1. (A.206) 2 γ + √ +γ In the domain of the σ -integral, the quantity x 22 +1 − σ + is bounded by a constant √ √ multiple of | x 2 − x 1 |γ . As σ is bounded above and below, all that remains is the λ-integral. An elementary calculation shows that it remains bounded, even as b → 0. This completes the proof, in all cases, that there is a constant Cb , bounded with b, so that (A.192) holds for I1 . The estimate for I2 is essentially the same.  L EMMA A.3.5. [L EMMA 6.1.18] For b > 0, 0 < γ < 1, 0 < φ < π2 , and 0 < x 2 /3 < x 1 < x 2 , if J = [α, β], with the endpoints given by (6.35), there is a constant Cb,φ so that if |θ | < π2 − φ, then t  0 Jc

√ √ √ √ b b |L b kse y − x 1 |γ d yds ≤ Cb,φ | x 2 − x 1 |γ . iθ (x 2 , y) − L b k seiθ (x 1 , y)|| (A.207)

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P ROOF. We use the formula for L b ktb , given in (A.193), hence:

I

+

|t | ∞   y b−1 − cos θ y s× = e s 0 β

+     x ye   x ye  x 1 eθ + − x2 eθ (x 2 + y)eθ x 2 ye2θ  2 2θ 2 2θ +e s − b ψ − 2 ψb − e− s × b + 2 2 2 s s s s    +      + x 1 ye2θ x 1 ye2θ x 1 ye2θ + √ √ γ dwds (x 1 + y)eθ − b ψb −2 ψb +| y− x 1 | s 2 , s s2 s2 s2 (A.208) and

I

−

|t | α   y b−1 − cos θ y s× = e s 0

0

+     x ye   x ye  + − x2 eθ x e x 2 ye2θ  (x 2 + y)eθ 2 2θ 2 2θ  − 1s θ +e s − b ψb − 2 ψ − e × b + s s2 s2 s2 +     x ye   x ye  + √ √ (x 1 + y)eθ x 1 ye2θ  1 2θ 1 2θ +| y− x 1 |γ dwds . − b ψb − 2 ψb + 2 2 2 s s s s s2 (A.209) For this case we give the details for I − , and leave I + to the interested reader. We change variables, setting w=

y s

λ=

x1 d yds dwdλ so that 2 = ; s λ s

(A.210)

we also let μ = x 2 /x 1 . The integral now satisfies: αλ

|I − | ≤

∞ x1 x1 |t|

wb−1 e− cos θw ×

0

+ + −μλe * ,  −λeθ θ [(μλ + w)e − b]ψ +e × θ b (μλwe2θ ) − 2 (μλwe2θ ) ψb (μλwe2θ ) − e + γ + 2 √ + , * √ x dwdλ [(λ + w)eθ − b] ψb (λwe2θ ) − 2 (λwe2θ ) ψb (λwe2θ ) ++| λ − w|γ 1 γ . λ1+ 2 (A.211)

We split the w-integral into the part, I −− (λ) with w ∈ [0, λ1 ], and the rest, I −+ (λ), which only arises when λ2 > x 1 /α.

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APPENDIX A

To estimate I −− (λ) we let * , F(μ, λ, w) = e−μλeθ [(μλ + w)eθ − b]ψb (μλwe2θ ) − 2 (μλwe2θ ) ψb (μλwe2θ ) . (A.212) It follows from Lemma A.0.4 that for some ξ ∈ (1, μ) we have that + + + + + F(μ, λ, w) − F(1, λ, w)+ ≤ (μ − 1)|∂μ F(ξ, λ, w)|. (A.213) + + In the set {wλ < 1}, we have the bound (see (A.220)):  1 |∂μ F(ξ, λ, w)| ≤ Cb e− cos θξ λ λ(1 + λ + w) + (1 + λ)w . (b)

(A.214)

In this case the w-integral is bounded by 1

λ

γ 2

Cb x 1 (μ − 1)

w

b−1 − cos θ(w+ξ λ)



e

0

(1 + λ) 2 2 + w(1 + λ) + w (1 + λ) × (b) √ √ dwdλ , (A.215) | λ − w|γ γ λ2

and therefore I

−−

γ

γ 2

(λ) ≤ Cb,θ gW F,0,γ x 1 (μ − 1)

e− cos θλ (1 + λ1+ 2 ) γ

λ 2 (1 + λb )

.

(A.216)

As this is integrable from 0 to ∞, we see that I −− ≤ Cb,θ

x2 − x1 1− γ2

.

(A.217)

x1

As x 2 /x 1 is bounded from above, it follows immediately that √ √ I −− ≤ Cb,θ | x 2 − x 1 |γ .

(A.218)

This leaves only I −+ , which is estimated by ∞

γ 2

|I −+ | ≤ x 1 (μ − 1) x

max{ |t|1 ,



αλ

x1

√ √ wb−1 e− cos θw | λ − w|γ ×

1 x1 α } λ

|Fμ (ξ, λ, w)|

dwdλ γ

λ1+ 2

.

(A.219)

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287

As before we apply Lemma A.0.4 as in (A.213) to see that we need to estimate: + |∂μ F(ξ, λ, w)| = e− cos θξ λ +λeθ ψb (ξ λwe2θ )(1 − (ξ λ + w)eθ + b)+ + x2 λwe2θ ψb (ξ λwe2θ )[(3ξ λ+w)eθ +b −2]−2ξ(λwe2θ )2 ψb (ξ λwe2θ )+, ξ ∈ [1, ]. x1 (A.220) To get a controllable error term, we must use the asymptotic expansions for ψb , ψb = ψb+1 through second order: √  1 b 1 z 4 − 2 e2 z (2b − 1)(2b − 3) + O( ) ψb (z) = √ 1− √ 16 z z 4π (A.221) √  1 b 1 z − 4 − 2 e2 z (2b + 1)(2b − 1)  + O( ) . ψb (z) = 1− √ √ 16 z z 4π Using the equation zψb = ψb − bψb , and inserting these relations into (A.220), gives 1

|∂μ F(ξ, λ, w)| ≤ λecos θ(2

b

√ 4−2 ξ λw−ξ λ) (ξ λw)

√

×

 w −1 + ξλ + a1 (b) λξ ) ( 2    2 2 2 1 1 w w w λξ 1 − 1 + a3 (b) − +O + +√ a2 (b) . λξ λξ λξ w λ w λw (A.222) 

2

w −1 λξ

3

4π

2

Here a1 (b), a2 (b) and a3 (b) are polynomials in b. We denote the contributions of these terms by M0 , M1 , M2 , M3 , Me . We first consider M0 : ∞

γ 2

M0 ≤ Cb x 1 (μ − 1) x

max{ |t|1 ,



αλ

x1  1 x1 α } λ

w  b2 − 14 − cos θ(√w−√ξ λ)2 e × λ

 √ √ √ dwdλ | λ − w|γ | ξ λ − w|3 √ 1+γ . wλ 2

(A.223)

In this integral w < λ and 1 ≤ ξ ≤ x 2 /x 1 , and therefore this is bounded by ∞

γ 2

M0 ≤ Cb x 1 (μ − 1) x

max{ |t|1 ,



αλ

x1  1 x1 α } λ

w  b2 − 14 − cos θ(√w−√ξ λ)2 e × λ  √ dwdλ | ξ λ − w|3+γ √ 1+γ . wλ 2

(A.224)

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APPENDIX A

We now let z =

√ √ w/λ − ξ to obtain that 

∞

γ 2

M0 ≤ Cb x 1 (μ − 1) x

max{ |t|1 ,

α √ x1 − ξ





x1 α }

 1 2 ( ξ + z)b− 2 e− cos θλz ×

1 √ λ− ξ 3

|z|3+γ λ 2 dzdλ. (A.225)

√ As λ is bounded from below by x 1 /α, we need to estimate the z-integral as λ → ∞. We apply Lemma 6.1.24 to see that 

α √ x1 − ξ



1 √ λ− ξ

 1 2 ( ξ + z)b− 2 e− cos θλz |z|3+γ dz ≤

⎧ ⎪ ⎨ ⎪ ⎩

C b,θ − cos θ λ e √ C b,θ 4+γ λ 2

√ √ λ( x 2 − x 1 )2 4x 1

λ

if



x2 x1

√

√  x 2 − x 1 2+γ √ x1

 − 1 ≤ 1.

if

 √  x 2 λ − 1 >1 x1

(A.226)

The large λ contribution (the first estimate in (A.226)) leads to terms of the form √ √ √ x1 x2 − x1 Cb,θ | x 2 − x 1 |γ √ √ x1 x2 − x1 √ √  √ x2 + x1 √ | x 2 − x 1 |γ (A.227) ≤ Cb,θ √ x1 as above (see (A.217)–(A.218)). Integrating the second estimate in (A.226) over 2    x1 x1 x1 , , √ , λ ∈ max √ |t| α ( x 2 − x 1 )2 gives a term bounded by

√ Cb,θ

√  √ x2 + x1 √ | x 2 − x 1 |γ . √ x1

(A.228)

This completes the proof that M0 satisfies the desired bound. Using the same change of variables, we see√that M1 and M2 are also bounded by the quantity in (A.228). To treat M3 we let z = w/λ; this gives the bound: 

∞

γ

|M3 | ≤ Cb x 12 (μ − 1) x

max{ |t|1 ,



α

 x1 x1 α }

1

z b− 2 e− cos θλ(

√ ξ −z)2

×

1 λ

 |z 2 − ξ | √ λdzdλ. (A.229) | ξ − z|γ √ z ξ

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As λ → ∞, the part of the z-integral from 1/λ to 1/2 (e.g.,) is bounded by a constant λ multiple of λ1−b e− cos θ 4 , and so contributes a term to M3 that satisfies the desired estimate. √ are left to estimate the contribution from near the diagonal, i.e., for λ ∈ [1/2, α/x 1 ]. √We√ If λ( x 2 /x 1 − 1) > 1/2, then the z-integral is bounded by √

√

x2 − x1 +√ √ √ + + x 2 − x 1 +γ e− cos θλ 2 x1 + + Cb,θ + √ 2 x1 + λ

2

;

(A.230)

√ √ the contribution of this term satisfies the desired bound. If λ( x 2 /x 1 − 1) < 1/2, C b,θ then the z-integral is bounded by 2+γ . Integrating in λ completes the proof that λ

2

√

|M3 | ≤ Cb,θ

√  √ x2 + x1 √ | x 2 − x 1 |γ . √ x1

(A.231)

To complete the estimate of I −+ , and thereby of I − , we only √ need to show that the error terms satisfy the desired bound. To that end we let z = w/λ; the contribution of the error terms is bounded by 

∞

γ

|Me | ≤ Cb x 12 (μ − 1) x

max{ |t|1 ,



α

 x1 x1 α }

1

z b− 2 e− cos θλ(

1 λ



| ξ − z|

γ



√ ξ −z)2

×

1 1 1 √ + +√ z λ λz

 dzdλ. (A.232)

Arguing as above, we see that these terms all satisfy the desired bound. As noted, the estimate of I + is quite similar and is left to the reader.  L EMMA A.3.6. [L EMMA 6.1.19] For b > 0, 0 < γ < 1, and t1 < t2 < 2t1 there is a constant Cb so that t1 ∞

√ γ √ |L b ktb2 −t1 +s (x, y) − L b ksb (x, y)|| x − y|γ d yds ≤ Cb |t2 − t1 | 2 . (A.233)

t2 −t1 0

This lemma follows from the more basic: L EMMA A.3.7. [L EMMA 6.1.20] For b > 0, 0 < γ < 1, and t1 < t2 < 2t1 and s > t2 − t1 , there is a constant Cb so that ∞ 0

γ √ √ |L b ktb2 −t1 +s (x, y) − L b ksb (x, y)|| x − y|γ d y ≤ Cb (t2 − t1 )s 2 −2 .

(A.234)

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APPENDIX A

P ROOF OF L EMMA 6.1.19. The derivation of (A.233) from (A.234) is quite easy: t1 ∞

√ L b ksb (x, y)|| x

|L b ktb2 −t1 +s (x, y) −

−

√

t1

γ

y| d yds ≤ Cb (t2 − t1 )

t2 −t1 0

γ

s 2 −2 ds

t2 −t1 γ

≤ Cb |t2 − t1 | 2 . (A.235)  P ROOF OF L EMMA 6.1.20. To prove (6.49) we need to apply Taylor’s formula to estimate the difference L b ktb2 −t1 +s (x, y)−L b ksb (x, y). To that end, we let F(τ, s, x, y) = L b kτb+s (x, y); we denote the left-hand side in (6.49) as I, which we can rewrite as ∞ √ √ I = [F(τ, s, x, y) − F(0, s, x, y)]| x − y|γ d y;

(A.236)

0

here τ = t2 − t1 . From the mean value theorem, we get the estimate ∞ |I | ≤ τ

√ √ |∂τ F(ξ, s, x, y)|| y − x|γ d y.

(A.237)

0

Using the differential equation satisfied by ψb we can show that ∂τ F(ξ, s, x, y) =

   − x+y b−1 s+ξ y e

  x+y x+y −b − (b + 1) ψb (s + ξ )b+2 s+ξ s+ξ        4x y 2x y x+y xy x+y  − + . (A.238) + 3 − 2 ψ s+ξ s+ξ (s + ξ )2 (s + ξ )2 b (s + ξ )2 

xy (s + ξ )2



Since s ∈ [t2 − t1 , t1 ] and ξ ∈ [0, t2 − t1 ], we see that s < s + ξ < 2s, and therefore xy xy xy ≤ ≤ 2. (A.239) 4s 2 (s + ξ )2 s 2

We can therefore split the y-integral into a compact part with y ∈ [0, 4sx ], I − and the remaining non-compact part I + . In the compact part we use the usual estimates for ψb and ψb . Setting w = y/s, λ = x/s, we obtain that 4

γ

|I − | ≤ s 2 −2 τ

λ

1

wb−1 e− 2 (w+λ) ×

0

 +   √ ++γ 1 +√ + wλ + wλ + w + λ + λ − w+ dw. (λ + w + 1)2 (b)

(A.240)

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If λ is bounded then we easily see that this satisfies: γ

|I − | ≤ Cb τ s 2 −2 .

(A.241)

In the case that λ is large, then we see that γ

λ

|I − | ≤ Cs 2 −2 τ e− 2 ,

(A.242)

which therefore applies for λ ∈ [0, ∞). To estimate I + , we use the second order asymptotic expansions for ψb and ψb to see that   b − 1 −(√μ−√w)2  √ w 2 4 e √ (b − 1)2 ( w − μ)4 + |∂τ F(ξ, s, x, y)| = √ 3 μ w(s + ξ )    9 √ √ ( w − μ)2 + O(1 + w/μ + μ/w) , 4 here

x y and μ = . s+ξ s+ξ From this expansion, and the fact that s ≤ s + ξ ≤ 2s, it follows that w=

τ |I | ≤ Cb 3 s +

/2

(A.243)

(A.244)

 √ 2 ∞   b 1 2 √ y 2 − 4 s − 2sx 1− xy √ e | y − x|γ × x y 4s 2 x

y − s

2 4 2 2 2 2 0  2 x y x x y − + d y. (A.245) + +O 1+ s s s y x

To estimate this integral, we let z =

√

y/x − 1, and λ = x/s, obtaining:

γ √ ∞ 1 λ 2 τx 2 λ |I | ≤ Cb (z + 1)b− 2 e− 2 z |z|γ × 2 s

+

2 λ −1

  2 4 2 λ z + λz + O 1 + z +

1 1+z

 dz. (A.246)

1

If λ → 0, then the integral behaves like e− 4λ . As λ → ∞, an application of Laplace’s 1+γ method shows that the z-integral behaves like λ− 2 , which, in turn, establishes (6.49).  A.4 OFF-DIAGONAL AND LARGE-T BEHAVIOR We close this section with estimates valid for t, with positive real part, which do not use an assumption about the H¨older continuity of the data.

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APPENDIX A

L EMMA A.4.1. [L EMMA 6.1.21] For 0 < b < B, 0 < φ < a constant C j,B,φ so that if t ∈ Sφ , then ∞

j

|∂x ktb (x, y)|d y ≤ 0

and

∞

j

j

and j ∈ ⺞ there is

C j,B,φ , |t| j

|x 2 ∂x ktb (x, y)|d y ≤ 0

π 2,

C j,B j

|t| 2

(A.247)

.

(A.248)

P ROOF. The proof of this lemma is easier than results proved above for small t behavior. We observe that for 0 < b < B we can write ktb (x, y) =

1 x y F , , y t t

(A.249)

where, for z and ζ in the right half plane, F(ζ, z) = z b e−(z+ζ ) ψb (zζ ).

(A.250)

This expression easily implies that j

∂x kbt (x, y) =

1 j x y , . ∂ F yt j ζ t t

(A.251)

From the form of F and the fact that ∂z ψb (z) = ψb+1 (z),we see that a simple induction establishes: j    j j (−1) j −l z l ψb+l (zζ ). ∂ζ F(ζ, z) = z b e−(z+ζ ) (A.252) l l=0

We let t = |t|eiθ , with |θ | < π2 − φ, and set w = y/|t|, λ = x/|t|. To complete the proof of (A.247) it suffices to show that there are constants Cl,B,φ so that ∞

wb+l e− cos θ(w+λ) |ψb+l (wλe2θ )|

0

dw ≤ Cl,B,φ . w

(A.253)

When l = 0 the integral is bounded in Lemma 6.1.3, so we can assume that l ≥ 1. We need to estimate I j,b+l

1 = j |t|

∞ 0

wb+l e− cos θ(w+λ) |ψb+l (wλe2θ )|

dw . w

(A.254)

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293

We split the integral into the part from [0, 1/λ] and the rest; applying the asymptotic formula we obtain that ⎡ I j,b+l ≤

1

Cb+l ⎢ ⎢e− cos θλ |t| j ⎣

λ 0

⎤ ∞   b+l 1 w 2 − 4 − cos θ(√w−√λ)2 dw ⎥ dw . + wb+l e− cos θw e √ ⎥ w λ w⎦ 1 λ

(A.255) The first term in the brackets is bounded by (b + l)e− cos θλ , and the second √ is √ term rapidly decaying as λ → 0. To study the second term as λ → ∞, we let z = w− λ, to see that the second integral is bounded by

Cb+l

1  ∞  z b+l− 2 − cos θ z 2 1+ √ e dz. λ √

√1 − λ

(A.256)

λ

As b + l − 12 > 0, it follows easily that this integral is bounded as λ → ∞, which completes the proof of (A.247). The estimate in (A.248) for x/|t| < 1 follows immediately from this formula. To prove (A.248) for x/|t| ≥ 1 requires more careful consideration. Using (A.252) and the asymptotic expansions for ψb+l we see that  b−1 z 2 4 −(√z−√ζ )2 e × ζ ⎡ ⎡ ⎤⎤ ) ( l [ 2j ]     j 1 k z 2 ⎢ (−1) (b + l + k − 2 ) 1 ⎢ j ⎥⎥ (−1)l +O ⎣ ⎣ ⎦⎦ . k j l 1 ζ k 2 (zζ ) 4 l=0 k=0 4 (zζ ) (b + l − k − 2 )

j |∂x kt (x, y)|

(−1) j = tj

(A.257) We observe that the ratios of -functions are polynomials in l, which can be expressed as (b + l + k − 12 ) (b + l − k − 12 )

= pk,0 (b) +

2k 

pk,m (b)l(l − 1) · · · (l − m + 1).

(A.258)

m=1

The coefficients { pk,m (b)} are polynomials in b. Putting this expression into the previous formula and using the fact that j

(1 − u) =

j    j l=0

l

(−1)l u l ,

(A.259)

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APPENDIX A

we see there are polynomials, P j,k (b, u) in (b, u), so that j ∂x kt (x, y)d y

 b−1 z 2 4 −(√z−√ζ )2 e × ζ    j −2k z [ 2j ] 1 −   2  ζ z + P j,k b, k ζ 2 (zζ ) k=0 ⎡ ⎤ ) (  l j    2 z 1 j l ⎣ ⎦ . (−1) ·O j l ζ (zζ ) 4

(−1) j = tj

(A.260)

l=0

Using this expression and the analysis from the previous case we easily show that ∞

j

j

|x 2 ∂x ktb (x, y)|d y ≤ 0

Cb, j,φ j

|t| 2

,

(A.261) 

which completes the proof of the lemma. L EMMA A.4.2. [L EMMA 8.2.15] For j ∈ ⺞ and 0 < φ < C j,φ so that if t ∈ Sφ , then ∞

j

|∂x kte (x, y)|d y ≤ −∞

C j,φ j

|t| 2

π 2

.

there is a constant

(A.262)

P ROOF. These estimates, which are classical, follow easily from homogeneity considerations, and the formula j ∂x kte (x, y)

=

j 1  j

t2

 c j,l

l=0

x−y √ 2 t

l

kte (x, y).

(A.263) 

We consider the off-diagonal behavior. L EMMA A.4.3. [L EMMA 9.3.2] Let b > 0, η > 0 and for x ∈ ⺢+ define the set √ √ (A.264) Jx,η = {y ∈ ⺢+ : | x − y| ≥ η}. For 0 ≤ b < B, 0 < φ < π2 , and j ∈ ⺞0 there is a constant Cη, j,B,φ so that if t = |t|eiθ , with |θ | ≤ π2 − φ, then  Jx,η

η2

j |∂x ktb (x, y)|d y

e− cos θ 2|t| ≤ Cη, j,B,φ . |t| j

(A.265)

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For the Euclidean models we have L EMMA A.4.4. [L EMMA 9.3.3] Let η > 0 and for x ∈ ⺢ define the set Jx,η = {y ∈ ⺢ : |x − y| ≥ η}. π 2,

For j ∈ ⺞0 , 0 < φ < π 2 − φ, then



(A.266)

there is a constant Cη, j,φ so that if t = |t|eiθ , with |θ | ≤ η2

j |∂x kte (x, y)|d y

≤ Cη, j,φ

e− cos θ 8t

Jx,η

j

|t| 2

.

(A.267)

P ROOF OF L EMMA 9.3.2. Recall that for 0 < b, the kernel is given by ktb (x, y) =

xy  1  y b − x+y e t ψb 2 , y t t

where ψb (z) =

∞  j =0

zj . j !( j + b)

(A.268)

(A.269)

Using a simple inductive argument, and the fact that ∂zl ψb (z) = ψb+l (z),

(A.270)

we can show that there are constants {c j,l } so that j

∂x ktb (x, y) =

j  y l xy  1  y b − x+y  t e c j,l ψb+l 2 . j yt t t t

(A.271)

l=0

To prove the assertion of the lemma, it therefore suffices to prove it for each function, xy  1  y b − x+y  y l e t ψb+l 2 j yt t t t

(A.272)

where 0 ≤ l ≤ j. Letting w = y/|t| and λ = x/|t|, we see that we must estimate the integrals  1 dw . (A.273) Il, j (x, t, η) = j wb+l e− cos θ(w+λ) |ψb+l (wλe2θ )| |t| w |t |−1 Jx,η

There are two cases: if x < η2 , then |t|

−1

Jx,η

) / √ ( x + η)2 ,∞ ; = |t|

(A.274)

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APPENDIX A

otherwise,

0 / √ ) √ ( x − η)2 7 ( x + η)2 ,∞ . = 0, |t| |t| /

|t|

−1

Jx,η

(A.275)

Without loss of generality, we can assume that |t| < η2 . We first consider the case where x < η2 . Here again there are two cases to examine: √ 2 if |t| < x( x + η)2 (“small |t| case”), then we only need to use the asymptotic expansion forψ√ b+l ; otherwise  (“large |t| case”) we also need to separately estimate the ( x+η)2 −1 . We begin with the small |t| case. The product wλ > 1 ,λ integral over |t | and we can use the asymptotic expansion 1

b+l

z 4 − 2 2√ z ψb+l (z) ∼ √ e . 4π

(A.276)

There is a constant Cb,l so that 1 Il, j (x, t, η) ≤Cb,l j |t|

∞   b+l 1 w 2 − 4 − cos θ(√w−√λ)2 dw e √ λ w √

( x+η)2 |t|

(A.277)

1  ∞  z b+l− 2 − cos θ z 2 1 1+ √ e bz ≤Cb,l j |t| λ η √

|t|

√ √ √ where we set z = w− λ in the second line. Since η/ |t|λ ≤ 2η2 /|t|, an elementary integration by parts argument shows that ( Il, j (x, t, η) ≤ Cb,l,θ

η2 |t|

)b+l

2

e

− cos θ η|t|

|t| j

.

(A.278)

For the large |t| case we need to consider 1

1 Il, j (x, t, η) = j |t|

λ

dw . w

(A.279)

dw . w

(A.280)

wb+l e− cos θ(w+λ) ψb+l (wλe2θ )

√ ( x+η)2 |t|

In this case we approximate ψb+l (z) by a constant to obtain 1

Cb,l e− cos θλ Il, j (x, t, η) ≤ |t| j

λ √

( x+η)2 |t|

wb+l e− cos θw

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If b + l ≥ 1, then this is estimated by ( )b+l−1 2 η2 Cb,l,θ η2 Cb,l,θ − cos θ 2|t| − cos θ η|t| e ≤ e . |t| j |t| |t| j

(A.281)

If 0 < b + l < 1, then because η2 /|t| > 1, we have that η2

Cb,l,θ e− cos θ |t| ≤ . |t| j The other part of Il, j (x, t, η) is bounded by Il, j (x, t, η)

Il, j (x, t, η)

1 ≤ Cb,l j |t|

∞  1 λ

(A.282)

1 √ √ w  b+l 2 − 4 − cos θ( w− λ)2 dw e √ . λ w

(A.283)

Estimating the integral shows that Il, j (x, t, η) ≤ Cb,l,θ Since 1/λ >

√ ( x+η)2 , |t |

λ−(b+l) e− |t| j

cos θ λ

.

(A.284)

this is again easily seen to satisfy η2

e− cos θ 2|t| . Il, j (x, t, η) ≤ Cb,l,θ |t| j

(A.285)

This establishes the estimates η2

e− cos θ 2|t| , when x ≤ η2 . Il, j (x, t, η) ≤ Cb,l, j,θ |t| j

(A.286)

The constants Cb,l, j,θ are uniformly bounded for 0 < b < B, and |θ | < π2 − φ. 2 2 √ We now consider x ≥ η ; as before we assume that |t| < η , so that 1/λ < ( x + η)2 /|t|. We first estimate the non-compact part of the integral:  Il,k, j (x, t, η)

1 ≤Cb,l j |t|

≤Cb,l,θ

e

1  ∞  z b+l− 2 − cos θ z 2 1+ √ e bz λ η

√ |t|

− cos θ

|t| j

η2 |t|

(A.287)

.

This leaves only

Il, j (x, t, η)

1 = j |t|

√ ( x−η)2 |t|

 0

wb+l e− cos θ(w+λ) |ψb+l (wλe2θ )|

dw . w

(A.288)

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APPENDIX A

√ If |t|2 < x( x − η)2 , then we need to split this integral into two parts: from 0 to 1/λ and the rest. We first assume that there is just one part. If b + l ≥ 1, then we have the estimate

Il, j (x, t, η) ≤

√ ( x−η)2 |t|



Cb,l |t| j

wb+l e− cos θ(w+λ)

0

Cb,l,θ e− cos θλ dw ≤ . w |t| j

(A.289)

In this case the fact that x ≥ η2 , shows that there is a constant Cb,l,θ so that η2

Il, j (x, t, η)

e− cos θ |t| ≤ Cb,l,θ . |t| j

(A.290)

If l = 0 and b < 1, then we need to use the approximation ψb (z) =

1 + O(z) (b)

(A.291)

to see that

 I0, j (x, t, η) ≤

Cb,0,θ e− cos θλ |t| j

√ ( x−η)2 |t|



wb−1 e− cos θw

0



1 + O(wλ) dw, (A.292) (b)

which again implies that Il, j (x, t, η)

2

η − cos θ 2|t|

λe− cos θλ e  ≤ Cb,0,θ ≤ Cb,0,θ j |t| |t| j

.

(A.293)

 is bounded as b → 0. Here Cb,0,θ √ The only case that remains is when |t|2 < x( x − η)2 , wherein 1

Il, j (x, t, η) ≤

Cb,l |t| j

 λ

wb+l e− cos θ(w+λ) |ψb+l (wλe2θ )|

0 √ ( x−η)2 |t|

 1 λ

 w  b+l − 1 2

λ

4

dw + w

e− cos θ(

√ √ w− λ)2

dw √ . w

(A.294)

If b + l ≥ 1, then we can estimate ψb+l by a constant to see that the first term is bounded by η2

Cb,l,θ e− cos θ |t| Cb,l,θ e− cos θλ ≤ . |t| j |t| j

(A.295)

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PROOFS OF ESTIMATES FOR THE DEGENERATE 1-D MODEL

If l = 0 and b < 1, then, as before, we need to use (A.291) to see that this term is bounded by Cb,0,θ

η2

λe− cos θλ

≤

|t| j

 e− cos θ 2|t| Cb,0,θ

|t| j

,

(A.296)

 is bounded for b < B. This leaves only where again Cb,0,θ √ ( x−η)2 |t|



Cb,l |t| j

 w  b+l − 1 2

4

λ

1 λ

e− cos θ(

√ √ w− λ)2

√ λ− √1

Cb,l |t| j If b + l −

dw √ = w

1 2



λ

√η |t|

1   z b+l− 2 − cos θ z 2 1− √ e dz. (A.297) λ

> 0, then this is bounded by √ λ− √1

Cb,l |t| j



2

λ

e

− cos θ z 2

√η |t|

− cos θ η|t|

Cb,l,θ e dz ≤ |t| j

.

(A.298)

This leaves only the case l = 0, b < 12 . To obtain a good estimate in this case, as λ → ∞, we split the integral into two parts: 

2λ

 1  3 z b− 2 − cos θ z 2 1− √ e dz + λ η

√ |t|

√ λ− √1





The first term is bounded by Cb,0,θ e− cos θ

λ

  1 z b− 2 − cos θ z 2 1− √ e dz. λ

(A.299)

2λ 3 η2 t

; and the second by

√ η2 2λ − cos θ 2|t| Cb,0,θ λe− cos θ 3 ≤ Cb,0,θ e . Altogether we have shown that 2

η − cos θ 2|t|

Cb,l,θ e Il, j (x, t, η) ≤ |t| j

.

(A.300) 

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APPENDIX A

The proof of Lemma 9.3.3 is similar, but easier. It follows from the formula, valid for t ∈ S0 :   j 1  (x − y) l e j e ∂x kt (x, y) = j c j,l kt (x, y). (A.301) √ 2 t t 2 l=0 The details of the proof are left to the reader.

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Index [|g|]W F,0,γ , 75 [|g|]W F,0,γ ,T , 75

0, 21 1, 21 1-variable-at-a-time method, 18, 108

K tb,m , 144 kt0 , 82 ktb , 54, 81 ktb (x, y), 7 ktb,m , 54

α,R , 106 κtb,m , 144 ρe ( y, y ), 20 ρs (x, x  ), 20 ψb (z), 7

L b,m , 6 L H , 31 L Kim , 1 L, 235  L , 237

B R+ , 109 b P T (L), 37 b P (L), 37 b Pmin , 11 b Pter (L), 38

ᏹ(P), 237 ᏹ (P), 237

Ꮿ , 52  Ꮿ, 2 , 52 ˙ m , 52, 65 Ꮿ

Sn,m , 20 sup1 , 65

ᏯW F , 74

k,γ

Tα,R , 10

ᏯW F (P), 182

0,γ

x j , 107 x j , 107

k,2+γ ᏯW F (P), 183 k,γ ᏯW F (P), 183 0,2+γ ᏯW F (⺢+ ), 68 0,γ ᏯW F (⺢+ ), 67 k,2+γ ᏯW F (⺢+ ), 71 k,γ ᏯW F (⺢+ ), 71 0,2+γ ᏯW F (⺢+ × [0, T ]), 70 0,γ ᏯW F (⺢+ × [0, T ]), 70

adapted local coordinates, 32 adversible Kimura diffusion, 247 anisotropic metric, 7 asymptotic expansion, ψb , 59 backward Kolmogorov equation, 5 Bessel process, 9 boundary condition 1-dimensional, 6, 7 1-dimensional, no-flux, 6 adjoint, 7

Ᏸ2W F , 36

d W F , 20, 72 305

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306

INDEX

no-flux, 238 boundary defining function, 27 boundary measure, 238 boundary stratification, 4

WF, heat, 69 WF semi-norm, 72, 182 homogeneous Cauchy problem, 52 Hopf boundary maximum principle, 38

clean, L to b P, 37 components b P, 26 boundary stratification, 26 conic neighborhood, 218 corner of codimension , 26

indicial root, 6 inhomogeneous problem, 52 interior, 191 interior measure, 238 irregular solution, 247

defining function, 27 doubling, 28 doubling theorem, 190 equilibrium measure, 242 faces, 4 forward Kolmogorov equation, 5 Fredholm property, 18, 219 generalized Kimura diffusion, 5, 29 graph closure of L, 235 heat kernel asymptotics, 87 boundary parametrix, 191 Euclidean, 137 higher dimensional model problem, 54 interior parametrix, 188 Laplace transform, 9, 62, 218 L b,m , 56 model problem, 7 parametrix, 9 H¨older space heat, 66 higher order WF, 75 Leibniz formula, 70, 184 Leibniz rule, 68 local WF norms, 183 manifolds with corners, 183 standard, 66 WF, 11 WF, 1-dimensional, 67

L is tangent to, 30, 37 L is transverse to, 37 L meets b P cleanly, 37 manifold with corners, 4 martingale problem, 13 minimal boundary, 11, 38 model operator, 6 model problem, 6 NCC, 182 near diagonal, 232 normal cubic coordinate, 182 normal form for L, 32 regular polyhedron, 28 resolvent operator, 9, 218 model problem, 60 parametrix, 9, 220 restriction of L to H , 31 right half plane, 20 sector, Sφ , 20 semi-group on Ꮿ0 , 235 holomorphic, 62, 229 simplex, 1 small time localization property, 189 stratum of b P, 26 tangent, L to b P, 11, 37 terminal boundary, 11, 38 transverse, L to b P, 11, 37 uniformly degenerate operators, 15 Wright-Fisher metric, 7