Fourier Restriction for Hypersurfaces in Three Dimensions and Newton Polyhedra (AM-194) (Annals of Mathematics Studies, 194) [1 ed.] 0691170541, 9780691170541

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Fourier Restriction for Hypersurfaces in Three Dimensions and Newton Polyhedra

Isroil A. Ibromov

Detlef Muller

ANNALS

OF

MATHEMATICS

STUDIES

Annals of Mathematics Studies Number 194

Fourier Restriction for Hypersurfaces in Three Dimensions and Newton Polyhedra

Isroil A. Ikromov Detlef Müller

PRINCETON UNIVERSITY PRESS PRINCETON AND OXFORD 2016

c 2016 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 Names: Ikromov, Isroil A., 1961– author. |Müller, Detlef, 1954– author. Title: Fourier restriction for hypersurfaces in three dimensions and Newton polyhedra / Isroil A. Ikromov and Detlef Müller. Description: Princeton : Princeton University Press, [2016] | Series: Annals of mathematics studies ; number 194 | Includes bibliographical references and index. Identifiers: LCCN 2015041649 | ISBN 9780691170541 (hardcover : alk. paper) | ISBN 9780691170558 (pbk. : alk. paper) Subjects: LCSH: Hypersurfaces. | Polyhedra. | Surfaces, Algebraic. | Fourier analysis. Classification: LCC QA571 .I37 2016 | DDC 516.3/52—dc23 LC record available at http://lccn.loc.gov/2015041649 British Library Cataloging-in-Publication Data is available This book has been composed in Times Printed on acid-free paper. ∞ Typeset by S R Nova Pvt Ltd, Bangalore, India Printed in the United States of America 10 9 8 7 6 5 4 3 2 1

To Eli Stein

Contents

Chapter 1 Introduction

1.1 1.2 1.3 1.4 1.5 1.6

Newton Polyhedra Associated with φ, Adapted Coordinates, and Uniform Estimates for Oscillatory Integrals with Phase φ Fourier Restriction in the Presence of a Linear Coordinate System That Is Adapted to φ Fourier Restriction When No Linear Coordinate System Is Adapted to φ—the Analytic Case Smooth Hypersurfaces of Finite Type, Condition (R), and the General Restriction Theorem An Invariant Description of the Notion of r-Height. Organization of the Monograph and Strategy of Proof

Chapter 2 Auxiliary Results

2.1 2.2 2.3 2.4 2.5 2.6

1

5 10 11 17 23 24 29

Van der Corput–Type Estimates Airy-Type Integrals Integral Estimates of van der Corput Type Fourier Restriction via Real Interpolation Uniform Estimates for Families of Oscillatory Sums Normal Forms of φ under Linear Coordinate Changes When hlin (φ) < 2

30 31 34 38 40

Chapter 3 Reduction to Restriction Estimates near the Principal Root Jet

50

Chapter 4 Restriction for Surfaces with Linear Height below 2

57

4.1 4.2

Preliminary Reductions by Means of Littlewood-Paley Decompositions Restriction Estimates for Normalized Rescaled Measures When 22j δ3
1

Chapter 5 Improved Estimates by Means of Airy-Type Analysis

5.1 5.2 5.3

Airy-Type Decompositions Required for Proposition 4.2(c) The Endpoint in Proposition 4.2(c): Complex Interpolation Proof of Proposition 4.2(a), (b): Complex Interpolation

46

58 64 75

76 85 96

viii

CONTENTS

Chapter 6 The Case When hlin (φ) ≥ 2: Preparatory Results

6.1 6.2 6.3 6.4 6.5

The First Domain Decomposition Restriction Estimates in the Transition Domains El When hlin (φ) ≥ 2 Restriction Estimates in the Domains Dl , l < lpr , When hlin (φ)≥2 Restriction Estimates in the Domain Dpr When hlin (φ)≥ 5 Refined Domain Decomposition of Dpr: The Stopping-Time Algorithm

Chapter 7 How to Go beyond the Case hlin (φ) ≥ 5

7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 7.9 7.10

The Case When hlin (φ)≥ 2: Reminder of the Open Cases Restriction Estimates for the Domains D  (1) : Reduction to Normalized Measures ν δ Removal of the Term y2B−1 bB−1 (y1 ) in (7.7) Lower Bounds for hr (φ) Spectral Localization to Frequency Boxes Where |ξi |∼λi : The Case Where Not All λi s Are Comparable Interpolation Arguments for the Open Cases Where m = 2 and B =2 or B =3 The case where λ1 ∼λ2 ∼λ3 The case where B = 5 Collecting the Remaining Cases Restriction Estimates for the Domains D  (l) , l≥2

Chapter 8 The Remaining Cases Where m = 2 and B = 3 or B = 4

8.1 8.2 8.3 8.4 8.5 8.6 8.7 8.8

Preliminaries Refined Airy-Type Analysis ˜
1 The Case Where λρ(δ) The Case Where m = 2, B = 4, and A = 1 The Case Where m = 2, B = 4, and A = 0 The Case Where m= 2, B= 3, and A= 0: What Still Needs to Be Done Proof of Proposition 8.12(a): Complex Interpolation Proof of Proposition 8.12(b): Complex Interpolation

Chapter 9 Proofs of Propositions 1.7 and 1.17

9.1 9.2

Appendix A: Proof of Proposition 1.7 on the Characterization of Linearly Adapted Coordinates Appendix B: A Direct Proof of Proposition 1.17 on an Invariant Description of the Notion of r-Height

105

107 109 115 123 125 131

132 135 138 141 143 151 169 178 178 179 181

183 185 189 197 204 212 217 233 244

244 245

Bibliography

251

Index

257

Fourier Restriction for Hypersurfaces in Three Dimensions and Newton Polyhedra

Chapter One Introduction Let S be a smooth hypersurface in R3 with Riemannian surface measure dσ. We shall assume that S is of finite type, that is, that every tangent plane has finite order of contact with S. Consider the compactly supported measure dµ := ρdσ on S, where 0 ≤ ρ ∈ C0∞ (S). The central problem that we shall investigate in this monograph is the determination of the range of exponents p for which a Fourier restriction estimate 1/2
 |f|2 dµ ≤ Cp f Lp (R3 ) , f ∈ S(R3 ), (1.1) S

holds true. This problem is a special case of the more general Fourier restriction problem, which asks for the exact range of exponents p and q for which an Lp -Lq Fourier restriction estimate 1/q
 |f|q dµ ≤ Cp f Lp (Rn ) , f ∈ S(Rn ), (1.2) S

holds true and which can be formulated for much wider classes of subvarieties S in arbitrary dimension n and suitable measures dµ supported on S. In fact, as observed by G. Mockenhaupt [M00] (see also the more recent work by I. Łaba and M. Pramanik [LB09]), it makes sense in much wider settings, even for measures dµ supported on “thin” subsets S of Rn , such as Salem subsets of the real line. The Fourier restriction problem presents one important instance of a wide circle of related problems, such as the boundedness properties of Bochner Riesz means, dimensional properties of Kakeya type sets, smoothing effects of averaging over time intervals for solutions to the wave equation (or more general dispersive equations), or the study of maximal averages along hypersurfaces. The common question underlying all these problems asks for the understanding of the interplay between the Fourier transform and properties of thin sets in Euclidean space, for instance geometric properties of subvarieties. Some of these aspects have been outlined in the survey article [M14], from which parts of this introduction have been taken. The idea of Fourier restriction goes back to E. M. Stein, and a first instance of this concept is the determination of the sharp range of Lp -Lq Fourier restriction estimates for the circle in the plane through work by C. Fefferman and E. M. Stein [F70] and A. Zygmund [Z74], who obtained the endpoint estimates (see also L. Hörmander [H73] and L. Carleson and P. Sjölin [CS72] for estimates on more general related oscillatory integral operators). For subvarieties of higher dimension, the first fundamental result was obtained (in various steps) for Euclidean spheres

2

CHAPTER 1

S n−1 by E. M. Stein and P. A. Tomas [To75], who proved that an Lp -L2 Fourier restriction estimate holds true for S n−1 , n ≥ 3, if and only if p ≥ 2(2/(n − 1) + 1), where p denotes the exponent conjugate to p, that is, 1/p + 1/p = 1 (cf. [S93] for the history of this result). A crucial property of Euclidean spheres which is essential for this result is the non-vanishing of the Gaussian curvature on these spheres, and indeed an analogous result holds true for every smooth hypersurface S with nonvanishing Gaussian curvature (see [Gl81]). Fourier restriction estimates have turned out to have numerous applications to other fields. For instance, their great importance to the study of dispersive partial differential equations became evident through R. Strichartz’ article [Str77], and in the PDE-literature dual versions which invoke also Plancherel’s theorem are often called Strichartz estimates. The question as to which Lp -Lq Fourier restriction estimates hold true for Euclidean spheres is still widely open. It is conjectured that estimate (1.2) holds true for S = S n−1 if and only if p  > 2n/(n − 1) and p ≥ q(2/(n − 1) + 1), and there has been a lot of deep work on this and related problems by numerous mathematicians, including J. Bourgain, T. Wolff, A. Moyua, A. Vargas, L. Vega, and T. Tao (see, e.g., [Bou91], [Bou95], [W95], [MVV96], [TVV98], [W00], [TV00], [T03], and [T04] for a few of the relevant articles, but this list is far from being complete). There has been a lot of work also on conic hypersurfaces and some on even more general classes of hypersurfaces with vanishing Gaussian curvature, for instance in Barcelo [Ba85], [Ba86], in Tao, Vargas, and Vega [TVV98], in Wolff [W01], and in Tao and Vargas [TV00], and more recently by A. Vargas and S. Lee [LV10] and S. Buschenhenke [Bu12]. Again, these citations give only a sample of what has been published on this subject. Recent work by J. Bourgain and L. Guth [BG11], making use also of multilinear estimates from work by J. Bennett, A. Carbery, and T. Tao [BCT06], has led to further important progress. Nevertheless, this and related problems continue to represent one of the major challenges in Euclidean harmonic analysis, bearing various deep connections with other important open problems, such as the Bochner-Riesz conjecture, the Kakeya conjecture and C. Sogge’s local smoothing conjecture for solutions to the wave equation. We refer to Stein’s book [S93] for more information on and additional references to these topics and their history until 1993, and to more recent related essays by Tao, for instance in [T04]. As explained before, we shall restrict ourselves to the study of the Stein-Tomastype estimates (1.1). For convex hypersurfaces of finite line type, a good understanding of this type of restriction estimates is available, even in arbitrary dimension (we refer to the article [I99] by A. Iosevich, which is based on work by J. Bruna, A. Nagel and S. Wainger [BNW88], providing sharp estimates for the Fourier transform of the surface measure on convex hypersurfaces). However, our emphasis will be to allow for very general classes of hypersurfaces S ⊂ R3 , not necessarily convex, whose Gaussian curvature may vanish on small, or even large subsets. Given such a hypersurface S, one may ask in terms of which quantities one should describe the range of ps for which (1.1) holds true. It turns out that an extremely useful concept to answer this question is the notion of

3

INTRODUCTION

Newton polyhedron. The importance of this concept to various problems in analysis and related fields has been revealed by V.I. Arnol’d and his school, in particular through groundbreaking work by A. N. Varchenko [V76] and subsequent work by V. N. Karpushkin [K84] on estimates for oscillatory integrals, and came up again in the seminal article [PS97] by D. H. Phong and E. M. Stein on oscillatory integral operators. Indeed, there is a close connection between estimates for oscillatory integrals and Lp -L2 Fourier restriction estimates, which had become evident already through the aforementioned work by Stein and Tomas. The underlying principles had been formalized in a subsequent article by A. Greenleaf [Gl81]. For the case of hypersurfaces, Greenleaf’s classical restriction estimate reads as follows:  )|
|ξ |−1/ h . Then the restriction Theorem 1.1 (Greenleaf). Assume that |dµ(ξ estimate (1.1) holds true for every p ≥ 1 such that p ≥ 2(h + 1). Observe next that in order to establish the restriction estimate (1.1), we may localize the estimate to a sufficiently small neighborhood of a given point x 0 on S. Notice also that if estimate (1.1) holds for the hypersurface S, then it is valid also for every affine-linear image of S, possibly with a different constant if the Jacobian of this map is not one. By applying a suitable Euclidean motion of R3 we may and shall therefore assume in the sequel that x 0 = (0, 0, 0) and that S is the graph S = Sφ = {(x1 , x2 , φ(x1 , x2 )) : (x1 , x2 ) ∈ } of a smooth function φ defined on a sufficiently small neighborhood  of the origin, such that φ(0, 0) = 0 and ∇φ(0, 0) = 0.  ) as an oscillatory integral, Then we may write dµ(ξ   ) = J (ξ ) := e−i(ξ3 φ(x1 ,x2 )+ξ1 x1 +ξ2 x2 ) η(x) dx1 dx2 , ξ ∈ R3 , dµ(ξ 

C0∞ ().

Since ∇φ(0, 0) = 0, the complete phase in this oscillatory inwhere η ∈ tegral will have no critical point on the support of η unless |ξ1 | + |ξ2 | |ξ3 |, provided  is chosen sufficiently small. Integrations by parts then show that  µ(ξ ) = O(|ξ |−N ) as |ξ | → ∞, for every N ∈ N, unless |ξ1 | + |ξ2 | |ξ3 |. We may thus focus on the latter case. In this case, by writing λ = −ξ3 and ξj = −sj λ, j = 1, 2, we are reduced to estimating two-dimensional oscillatory integrals of the form  I (λ; s) := eiλ(φ(x1 ,x2 )+s1 x1 +s2 x2 ) η(x1 , x2 ) dx1 dx2 , where we may assume without loss of generality that λ  1 and that s = (s1 , s2 ) ∈ R2 is sufficiently small, provided that η is supported in a sufficiently small neighborhood of the origin. The complete phase function is thus a small, linear perturbation of the function φ. s = 0, then the function I (λ; 0) is given by an oscillatory integral of the form  Ifiλφ(x) η(x) dx, and it is well known ([BG69], [At70]) that for any analytic phase e function φ defined on a neighborhood of the origin in Rn satisfying φ(0) = 0, such

4

CHAPTER 1

an integral admits an asymptotic expansion as λ → ∞ of the form ∞  n−1 

aj,k (η)λ−rk log(λ)j ,

(1.3)

k=0 j =0

provided the support of η is sufficiently small. Here, the rk form an increasing sequence of rational numbers consisting of a finite number of arithmetic progressions, which depends only on the zero set of φ, and the aj,k (η) are distributions with respect to the cutoff function η. The proof is based on Hironaka’s theorem on the resolution of singularities. We are interested in the case n = 2. If we denote the leading exponent r0 in (1.3) by r0 = 1/ h, then we find that the following estimate holds true: |I (λ; 0)| ≤ Cλ−1/ h log(λ)ν ,

λ  1,

(1.4)

where ν may be 0, or 1. Assuming that this estimate is stable under sufficiently small analytic perturbations of φ, then we find in particular that I (λ; s) satisfies the same estimate (1.4) for |s| sufficiently small, so that we obtain the following  uniform estimate for dµ,  )| ≤ C(1 + |ξ |)−1/ h log(2 + |ξ |)ν , |dµ(ξ

ξ ∈ R3 ,

(1.5)

provided the support of ρ is sufficiently small. Greenleaf’s theorem then shows that the Fourier restriction estimate (1.1) holds true for p  ≥ 2(h + 1), if ν = 0, and at least for p > 2(h + 1), if ν = 1, where 1/ h denotes the decay rate of  Moreover, for instance for the oscillatory integral I (λ; 0), hence ultimately of dµ. hypersurfaces with nonvanishing Gaussian curvature, this yields the sharp restriction result mentioned before. However, as we shall see, for large classes of hypersurfaces, the relation between the decay rate of the Fourier transform of dµ and the range of p  s for which (1.1) holds true will not be so close anymore. Nevertheless, uniform decay estimates of the form (1.5) will still play an important role. The first major question that arises is thus the following one: given a smooth phase function φ, how can one determine the sharp decay rate 1/ h and the exponent ν in the estimate (1.4) for the oscillatory integral I (λ; 0) = eiλφ(x) η(x) dx? This question has been answered by Varchenko for analytic φ in [V76], where he identified h as the so-called height of the Newton polyhedron associated to φ in “adapted” coordinates and also gave a corresponding interpretation of the exponent ν. Subsequently, Karpushkin [K84] showed that the estimates given by Varchenko are stable under small analytic perturbations of the phase function φ, which in particular leads to uniform estimates of the form (1.5). More recently, in [IM11b], we proved, by a quite different method, that Karpushkin’s result remains valid even for smooth, finite-type functions φ, at least for linear perturbations, which is sufficient in order to establish uniform estimates of the form (1.5). In order to present these results in more detail, let us review some basic notations and results concerning Newton polyhedra (see [V76], [IM11a]).

5

INTRODUCTION

1.1 NEWTON POLYHEDRA ASSOCIATED WITH φ, ADAPTED COORDINATES, AND UNIFORM ESTIMATES FOR OSCILLATORY INTEGRALS WITH PHASE φ We shall build on the results and technics developed in [IM11a] and [IKM10], which will be our main sources, also for references to earlier and related work. Let us first recall some basic notions from [IM11a], which essentially go back to Arnol’d (cf. [Arn73], [AGV88]) and his school, most notably Varchenko [V76]. If φ is given as before, consider the associated Taylor series ∞  cα1 ,α2 x1α1 x2α2 φ(x1 , x2 ) ∼ α1 ,α2 =0

of φ centered at the origin. The set  T (φ) := (α1 , α2 ) ∈ N2 : cα1 ,α2 =

1 α1 α2 ∂ ∂ φ(0, 0) = 0 α1 !α2 ! 1 2 will be called the Taylor support of φ at (0, 0). We shall always assume that the function φ is of finite type at every point, that is, that the associated graph S of φ is of finite type. Since we are also assuming that φ(0, 0) = 0 and ∇φ(0, 0) = 0, the finite-type assumption at the origin just means that T (φ) = ∅.

The Newton polyhedron N (φ) of φ at the origin is defined to be the convex hull of the union of all the quadrants (α1 , α2 ) + R2+ in R2 , with (α1 , α2 ) ∈ T (φ). The associated Newton diagram Nd (φ) of φ in the sense of Varchenko [V76] is the union of all compact faces of the Newton polyhedron; here, by a face, we shall mean an edge or a vertex. We shall use coordinates (t1 , t2 ) for points in the plane containing the Newton polyhedron, in order to distinguish this plane from the (x1 , x2 )-plane. The Newton distance in the sense of Varchenko, or shorter distance, d = d(φ) between the Newton polyhedron and the origin is given by the coordinate d of the point (d, d) at which the bisectrix t1 = t2 intersects the boundary of the Newton polyhedron. (See Figure 1.1.) The principal face π(φ) of the Newton polyhedron of φ is the face of minimal dimension containing the point (d, d). Deviating from the notation in [V76], we shall call the series  cα1 ,α2 x1α1 x2α2 φpr (x1 , x2 ) := (α1 ,α2 )∈π(φ)

the principal part of φ. In case that π(φ) is compact, φpr is a mixed homogeneous polynomial; otherwise, we shall consider φpr as a formal power series. Note that the distance between the Newton polyhedron and the origin depends on the chosen local coordinate system in which φ is expressed. By a local coordinate system (at the origin) we shall mean a smooth coordinate system defined near the origin which preserves 0. The height of the smooth function φ is defined by h(φ) := sup{dy },

6

CHAPTER 1

N (φ)

1/κ2 Nd (φ) π(φ) d(φ)

d(φ)

1/κ1

Figure 1.1 Newton polyhedron

where the supremum is taken over all local coordinate systems y = (y1 , y2 ) at the origin and where dy is the distance between the Newton polyhedron and the origin in the coordinates y. A given coordinate system x is said to be adapted to φ if h(φ) = dx . In [IM11a] we proved that one can always find an adapted local coordinate system in two dimensions, thus generalizing the fundamental work by Varchenko [V76] who worked in the setting of real-analytic functions φ (see also [PSS99]). Notice that if the principal face of the Newton polyhedron N (φ) is a compact edge, then it lies on a unique principal line L := {(t1 , t2 ) ∈ R2 : κ1 t1 + κ2 t2 = 1}, with κ1 , κ2 > 0. By permuting the coordinates x1 and x2 , if necessary, we shall always assume that κ1 ≤ κ2 . The weight κ = (κ1 , κ2 ) will be called the principal weight associated with φ. It induces dilations δr (x1 , x2 ) := (r κ1 x1 , r κ2 x2 ), r > 0, on R2 , so that the principal part φpr of φ is κ-homogeneous of degree one with respect to these dilations, that is, φpr (δr (x1 , x2 )) = rφpr (x1 , x2 ) for every r > 0, and we find that d=

1 1 = . κ 1 + κ2 |κ|

7

INTRODUCTION

It can then easily be shown (cf. Proposition 2.2 in [IM11a]) that φpr can be factored as φpr (x1 , x2 ) = cx1ν1 x2ν2

M

q

p

(x2 − λl x1 )nl ,

(1.6)

l=1

with M ≥ 1, distinct nontrivial “roots” λl ∈ C \ {0} of multiplicities nl ∈ N \ {0}, and trivial roots of multiplicities ν1 , ν2 ∈ N at the coordinate axes. Here, p and q are positive integers without common divisor, and κ2 /κ1 = p/q. More generally, assume that κ = (κ1 , κ2 ) is any weight with 0 < κ1 ≤ κ2 such that the line Lκ := {(t1 , t2 ) ∈ R2 : κ1 t1 + κ2 t2 = 1} is a supporting line to the Newton polyhedron N (φ) of φ (recall that a supporting line to a convex set K in the plane is a line such that K is contained in one of the two closed half planes into which the line divides the plane and such that this line intersects the boundary of K). Then Lκ ∩ N (φ) is a face of N (φ), i.e., either a compact edge or a vertex, and the κ-principal part of φ  cα1 ,α2 x1α1 x2α2 φκ (x1 , x2 ) := (α1 ,α2 )∈Lκ

is a nontrivial polynomial which is κ-homogeneous of degree 1 with respect to the dilations associated to this weight as before, which can be factored in a similar way as in (1.6). By definition, we then have φ(x1 , x2 ) = φκ (x1 , x2 ) + terms of higher κ-degree. Adaptedness of a given coordinate system can be verified by means of the following proposition (see [IM11a]): If P is any given polynomial that is κ-homogeneous of degree one (such as P = φpr ), then we denote by n(P ) := ord S 1 P

(1.7)

the maximal order of vanishing of P along the unit circle S 1 . Observe that by homogeneity, the Taylor support T (P ) of P is contained in the face Lκ ∩ N (P ) of N (P ). We therefore define the homogeneous distance of P by dh (P ) := 1/(κ1 +κ2 ) = 1/|κ|. Notice that (dh (P ), dh (P )) is just the point of intersection of the line Lκ with the bisectrix t1 = t2 , and that dh (P ) = d(P ) if and only if Lκ ∩ N (P ) intersects the bisectrix. We remark that the height of P can then easily be computed by means of the formula h(P ) = max{n(P ), dh (P )}

(1.8)

(see Corollary 3.4 in [IM11a]). Moreover, in [IM11a] (Corollary 4.3 and Corollary 5.3), we also proved the following characterization of adaptedness of a given coordinate system. Proposition 1.2. The coordinates x are adapted to φ if and only if one of the following conditions is satisfied: (a) The principal face π(φ) of the Newton polyhedron is a compact edge, and n(φpr ) ≤ d(φ).

8

CHAPTER 1

(b) π(φ) is a vertex. (c) π(φ) is an unbounded edge. These conditions had already been introduced by Varchenko, who has shown that they are sufficient for adaptedness when φ is analytic. We also note that in case (a) we have h(φ) = h(φpr ) = dh (φpr ). Moreover, it can / N; be shown that we are in case (a) whenever π(φ) is a compact edge and κ2 /κ1 ∈ in this case we even have n(φpr ) < d(φ) (cf. [IM11a], Corollary 2.3). 1.1.1 Construction of adapted coordinates In the case where the coordinates (x1 , x2 ) are not adapted to φ, the previous results show that the principal face π(φ) must be a compact edge, that m := κ2 /κ1 ∈ N, and that n(φpr ) > d(φ). One easily verifies that this implies that p = m and q = 1 in (1.6), and that there is at least one nontrivial, real root x2 = λl x1m of φpr of multiplicity nl = n(φpr ) > d(φ). Indeed, one can show that this root is unique. Putting b1 := λl , we shall denote the corresponding root x2 = b1 x1m of φpr as its principal root. Changing coordinates y1 := x1 , y2 := x2 − b1 x1m , we arrive at a “better” coordinate system y = (y1 , y2 ). Indeed, this change of co ordinates will transform φpr into a function φ pr , where the principal face of φ pr  will be a horizontal half line at level t2 = n(φpr ), so that d(φpr ) > d(φ), and cor˜ > d(φ) if φ˜ expresses φ in the coordinates y (cf. respondingly one finds that d(φ) [IM11a]). In particular, if the new coordinates y are still not adapted, then the principal ˜ will again be a compact edge, associated to a weight κ˜ = (κ˜ 1 , κ˜ 2 ) face of N (φ) ˜ > m ≥ 1. such that m ˜ := κ˜ 2 /κ˜ 1 is again an integer and m Somewhat oversimplifying, by iterating this procedure, we essentially arrive at Varchenko’s algorithm for the construction of an adapted coordinate system (cf. [IM11a] for details). In conclusion, one can show (compare Theorem 5.1 in [IM11a]) that there exists a smooth real-valued function ψ (which we may choose as the principal root jet of φ) of the form ψ(x1 ) = b1 x1m + O(x1m+1 ),

(1.9)

with b1 ∈ R \ {0}, defined on a neighborhood of the origin such that an adapted coordinate system (y1 , y2 ) for φ is given locally near the origin by means of the (in general nonlinear) shear y1 := x1 , y2 := x2 − ψ(x1 ).

(1.10)

In these adapted coordinates, φ is given by φ a (y) := φ(y1 , y2 + ψ(y1 )).

(1.11)

9

INTRODUCTION

n

N (φa ) h(φ)

d(φ)

N (φ)

mn Figure 1.2 φ(x1 , x2 ) := (x2 − x1m )n + x1

l ( > mn)

Example 1.3. φ(x1 , x2 ) := (x2 − x1m )n + x1 . Assume that  > mn. Then the coordinates are not adapted. Indeed, φpr (x1 , x2 ) =(x2 − x1m )n , d(φ) = 1/(1/n + 1/(mn)) = mn/(m + 1) and n(φpr ) = n > d(φ). Adapted coordinates are given by y1 := x1 , y2 := x2 − x1m , in which φ is expressed by φ a (y) = y2n + y1 . (See Figure 1.2.) Remark 1.4. An alternative proof of Varchenko’s theorem on the existence of adapted coordinates for analytic functions φ of two variables has been given by Phong, Sturm, and Stein in [PSS99], by means of Puiseux series expansions of the roots of φ. We are now in the position to identify the exponents h and ν in (1.4) and (1.5) in terms of Newton polyhedra associated to φ: If there exists an adapted local coordinate system y near the origin such that the principal face π(φ a ) of φ, when expressed by the function φ a in the new coordinates, is a vertex, and if h(φ) ≥ 2, then we put ν(φ) := 1; otherwise, we put ν(φ) := 0. We remark [IM11b] that the first condition is equivalent to the following one: If y is any adapted local coordinate system at the origin, then either π(φ a ) is a ) = d(φ a ). a vertex or a compact edge, and n(φpr

10

CHAPTER 1

Varchenko [V76] has shown for analytic φ that the leading exponent in (1.3) is given by r0 = 1/ h(φ), and ν(φ) is the maximal j for which aj,0 (η) = 0. Correspondingly, in [IM11b] we prove, by means of a quite different method, that estimate (1.5) holds true with h = h(φ) and ν = ν(φ), that is, that the following estimate holds true for φ smooth and of finite type:  )| ≤ C(1 + |ξ |)−1/ h(φ) log(2 + |ξ |)ν(φ) , |dµ(ξ

ξ ∈ R3 .

(1.12)

The special case where ξ = (0, 0, ξ3 ) is normal to S at the origin is due to Greenblatt [Gb09]. One can also show that this estimate is sharp in the exponents even when φ is not analytic, except for the case where the principal face π(φ a ) is an unbounded edge (see [IM11b], [M14]).

1.2 FOURIER RESTRICTION IN THE PRESENCE OF A LINEAR COORDINATE SYSTEM THAT IS ADAPTED TO φ Coming back to the restriction estimate (1.1) for our hypersurface S in R3 , we begin with the case where there exists a linear coordinate system that is adapted to the function φ. For this case, the following complete answer was given in [IM11b]. Theorem 1.5. Let S ⊂ R3 be a smooth hypersurface of finite type, and fix a point x 0 ∈ S. After applying a suitable Euclidean motion of R3 , let us assume that x 0 = 0 and that near x 0 we may view S as the graph Sφ of a smooth function φ of finite type satisfying φ(0, 0) = 0 and ∇φ(0, 0) = 0. Assume that, after applying a suitable linear change of coordinates, the coordinates (x1 , x2 ) are adapted to φ. We then define the critical exponent pc by pc := 2h(φ) + 2, where p  denotes the exponent conjugate to p, that is, 1/p + 1/p = 1. Then there exists a neighborhood U ⊂ S of the point x 0 such that for every non-negative density ρ ∈ C0∞ (U ), the Fourier restriction estimate (1.1) holds true for every p such that 1 ≤ p ≤ pc .

(1.13)

Moreover, if ρ(x ) = 0, then the condition (1.13) on p is also necessary for the validity of (1.1). 0

Earlier results for particular classes of hypersurfaces in R3 (which can be seen to satisfy the assumptions of this theorem) are, for instance, in the work by E. Ferreyra and M. Urciuolo [FU04], [FU08] and [FU09], who studied certain classes of quasi-homogeneous hypersurfaces, for which they were able to prove Lp -Lq -restriction estimates when p < 43 . For further progress in the study of these classes of hypersurfaces, we refer to the work by S. Buschenhenke, A. Vargas and the second named author [BMV15]. We also like to mention work by A. Magyar [M09] on Lp -L2 Fourier restriction estimates for some classes of analytic hypersurfaces, which preceeded [IM11b] .

11

INTRODUCTION

As shown in [IM11b], the necessity of condition (1.13) follows easily by means of Knapp type examples (a related discussion of Knapp type arguments is given in Section 1.4). It is here where we need to assume that there is a linear coordinate system which is adapted to φ. The sufficiency of condition (1.13) is immediate from Greenleaf’s Theorem 1.1 in combination with (1.12), in the case where ν(φ) = 0. Notice that this is true, no matter whether or not there exists a linear coordinate system that is adapted to φ. If ν(φ) = 1, then we just miss the endpoint p = pc , which ultimately can be dealt with by means of Littlewood-Paley theory (for more details, we refer the reader to [IM11b] and also the survey article [M14]). An analogous argument based on Littlewood-Paley theory will appear in Chapter 3. In view of Theorem 1.5, from now on we shall always make the following assumption, unless stated explicitly otherwise. Assumption 1.6 (NLA). There is no linear coordinate system that is adapted to φ. Our main goal will be to understand which Fourier restriction estimates of the form (1.1) will hold under this assumption.

1.3 FOURIER RESTRICTION WHEN NO LINEAR COORDINATE SYSTEM IS ADAPTED TO φ—THE ANALYTIC CASE Under the preceding assumption, but not yet assuming that φ is analytic, let us have another look at the first step of Varchenko’s algorithm. If here m = κ2 /κ1 = 1, then this leads to a linear change of coordinates of the form y1 = x1 , y2 = x2 − b1 x1 , which will transform φ into a function φ˜ for which, by our assumption, the coor˜ it is also immediate that dinates (y1 , y2 ) are still not adapted. Replacing φ by φ, estimate (1.1) will hold for the graph of φ if and only if it holds for the graph of ˜ Replacing φ by φ, ˜ we may and shall therefore always assume that our original φ. coordinate system (x1 , x2 ) is chosen so that κ2 ∈N and m ≥ 2. (1.14) m= κ1 The next proposition will show that such a linear coordinate system is linearly adapted to φ in the following sense. In analogy with Varchenko’s notion of height, we can introduce the notion of linear height of φ, which measures the upper limit of all Newton distances of φ in linear coordinate systems: h lin (φ) := sup{d(φ ◦ T ) : T ∈ GL(2, R)}. Note that d(φ) ≤ h lin (φ) ≤ h(φ). We also say that a linear coordinate system y = (y1 , y2 ) is linearly adapted to φ if dy = h lin (φ). Clearly, if there is a linear coordinate system that is adapted to φ, it

12

CHAPTER 1

is in particular linearly adapted to φ. The following proposition, whose proof will be given in Chapter 9 (Appendix A), gives a characterization of linearly adapted coordinates under Assumption 1.6 (NLA). Proposition 1.7. If φ satisfies Assumption (NLA) and if φ = φ(x), then the following are equivalent: (a) The coordinates x are linearly adapted to φ. (b) If the principal face π(φ) is contained in the line L = {(t1 , t2 ) ∈ R2 : κ1 t1 + κ2 t2 = 1}, then either κ2 /κ1 ≥ 2 or κ1 /κ2 ≥ 2. Moreover, in all linearly adapted coordinates x for which κ2 /κ1 > 1, the principal face of the Newton polyhedron is the same, so that in particular the numbers m := κ2 /κ1 and dx do not depend on the choice of the linearly adapted coordinate system x = (x1 , x2 ). This result shows in particular that linearly adapted coordinates always do exist under Assumption (NLA), since either the original coordinates for φ are already linearly adapted or we arrive at such coordinates after applying the first step in Varchenko’s algorithm (when κ2 /κ1 = 1 in the original coordinates). Let us then look at the Newton polyhedron N (φ a ) of φ a , which expresses φ in the adapted coordinates (y1 , y2 ) of (1.11), and denote the vertices of the Newton polyhedron N (φ a ) by (Al , Bl ), l = 0, . . . , n, where we assume that they are ordered so that Al−1 < Al , l = 1, . . . , n, with associated compact edges given by the intervals γl := [(Al−1 , Bl−1 ), (Al , Bl )], l = 1, . . . , n. The unbounded horizontal edge with left endpoint (An , Bn ) will be denoted by γn+1 . To each of these edges γl , we associate the weight κ l = (κ1l , κ2l ) so that γl is contained in the line Ll := {(t1 , t2 ) ∈ R2 : κ1l t1 + κ2l t2 = 1}. For l = n + 1, we have κ1n+1 := 0, κ2n+1 = 1/Bn . We denote by al :=

κ2l , κ1l

l = 1, . . . , n,

the reciprocal of the (modulus of the) slope of the line Ll . For l = n + 1, we formally set an+1 := ∞. (See Figure 1.3.) If l ≤ n, then the κ l -principal part φκ l of φ corresponding to the supporting line Ll can easily be shown to be of the form  Nα


A (1.15) φκ l (y) = cl y1 l−1 y2Bl y2 − clα y1al α

(cf. Proposition 2.2 in [IM11a]; also [IKM10]). Remark 1.8. When φ is analytic, then this expression is linked to the Puiseux series expansion of roots of φ a as follows [PS97] (compare also [IM11a]): We may then factor

(y2 − r(y1 )), φ a (y1 , y2 ) = U (y1 , y2 )y1ν1 y2ν2 r

13

INTRODUCTION

1/κ22 N (φa )

(A0 , B0 ) γ1

(A1 , B1 ) γ2

(A2 , B2 ) (An , Bn )

γn

γn+1

1/κ21 Figure 1.3 Edges and weights

where the product is indexed by the set of all nontrivial roots r = r(y1 ) of φ a (which may also be empty) and where U is analytic, with U (0, 0) = 0. Moreover, these roots can be expressed in a small neighborhood of 0 as Puiseux series α

al 1l

al

α ···α

α1 ···αp−1 ···lp

al

r(y1 ) = clα11 y1 1 + clα11l2α2 y1 1 2 + · · · + cl11···lp p y1 1

+ ··· ,

where α ···α

β

α ···α

cl11···lp p−1 = cl11···lp p−1 α ···α

γ

for

β = γ ,

α ···α

p−2 , al11···lp p−1 > al11···lp−1

α ···α

with strictly positive exponents al11···lp p−1 > 0 (which are all multiples of a fixed α ···α positive rational number) and nonzero complex coefficients cl11···lp p = 0 and where we have kept enough terms to distinguish between all the nonidentical roots of φ a . The leading exponents in these series are the numbers a1 < a2 < · · · < an . One can therefore group the roots into the clusters of roots [l], l = 1, . . . , n, where the lth cluster [l] consists of all roots with leading exponent al .

14

CHAPTER 1

Correspondingly, we can decompose a

φ (y1 , y2 ) =

U (y1 , y2 )y1ν1 y2ν2

n

[l] (y1 , y2 ),

l=1

where [l] (y1 , y2 ) :=



(y2 − r(y1 )).

r∈[l]



α ··· α More generally, by the cluster l 1 . . . l p , we shall designate all the roots 1 p r(y1 ), counted with their multiplicities, that satisfy α1 ···αp−1  α
al 1l al α ···α al ···l = O(y1b ) r(y1 ) − clα11 y1 1 + clα11l2α2 y1 1 2 + · · · + cl11···lp p y1 1 p α ···α

for some exponent b > al11···lp p−1 . Then the cluster [l] = [l1 ] will split into the clus

α α α ters l 1 , these cluster into the finer “subclusters” l 1 l 2 , and so on. 1 1 2 l l Observe also the following: If δsl (y1 , y2 ) = (s κ1 y1 , s κ2 y2 ), s > 0, denote the dilations associated to the weight κ l , and if r ∈ [l1 ] is a root in the cluster [l1 ], then l one easily checks that for y = (y1 , y2 ) in a bounded set we have δsl y2 = s κ2 y2 and l l al r(δsl y1 ) = s al1 κ1 clα11 y1 1 + O(s al1 κ1 +ε ) as s → 0, for some ε > 0. Consequently,  l l al −s al1 κ1 clα11 y1 1 + O(s al1 κ1 +ε ), if l1 < l,    l l δsl y2 − r(δsl y1 ) = s κ2 (y2 − clαl y1al ) + O(s κ2 +ε ), if l1 = l,    κ2l l s y2 + O(s κ2 +ε ), if l1 > l. This shows that the κ l -principal part of φ a is given by   ν1 + l l |[l1 ]|

1 1 y2 (y2 − clα1 y1al )Nl,α1 , φκal = Cl y1

(1.16)

α1

where Nl,α1 denotes the number of roots in the cluster [l] with leading term clα1 y1al , and where by |M| we denote the cardinality of a set M. A look at the Newton polyhedron reveals that the exponents of y1 and y2 in (1.16) can be expressed in terms of the vertices (Aj , Bj ) of the Newton polyhedron:   |[l1 ]|al1 = Al−1 , ν2 + |[l1 ]| = Bl . ν1 + l1 l

(y2 − clα1 y1al )Nl,α1 = ([l] )κ l . α1

Comparing this with (1.15), the close relation between the Newton polyhedron of φ a and the Puiseux series expansion of roots becomes evident, and accordingly we say that the edge γl := [(Al−1 , Bl−1 ), (Al , Bl )] is associated to the cluster of roots [l].

15

INTRODUCTION

Consider next the line parallel to the bisectrix (m) := {(t, t + m + 1) : t ∈ R}. For any edge γl ⊂ Ll := {(t1 , t2 ) ∈ R2 : κ1l t1 + κ2l t2 = 1}, define hl by (m) ∩ Ll = {(hl − m, hl + 1)}, that is, hl =

1 + mκ1l − κ2l , κ1l + κ2l

(1.17)

and define the restriction height, or for short, r-height, of φ by hr (φ) := max(d,

max

{l=1,...,n+1: al >m}

hl ).

Here, d = dx denotes again the Newton distance d(φ) of φ with respect to our original, linearly adapted coordinates x = (x1 , x2 ). Recall also that by Proposition 1.7 the numbers m and dx are well defined, that is, they do not depend on the chosen linearly adapted coordinate system x. For a more invariant description of hr (φ), we refer the reader to Proposition 1.17. Remarks 1.9. (a) For L in place of Ll and κ in place of κ l , one has m = κ2 /κ1 and d = 1/(κ1 + κ2 ), so that one gets d in place of hl in (1.17). (b) Since m < al , we have hl < 1/(κ1l + κ2l ), hence hr (φ) < h(φ). On the other hand, since the line (m) lies above the bisectrix, it is obvious that hr (φ) + 1 ≥ h(φ), so that h(φ) − 1 ≤ hr (φ) < h(φ).

(1.18)

(See Figure 1.4.) It is easy to see from Remark 1.9(a) that the r-height admits the following geometric interpretation. By following Varchenko’s algorithm (cf. Subsection 8.2 of [IKM10]), one realizes that the Newton polyhedron of φ a intersects the line L of the Newton polyhedron of φ in a compact face, either in a single vertex or a compact edge. That is, the intersection contains at least one and at most two vertices of φ a , and we choose (Al0 −1 , Bl0 −1 ) as the one with smallest second coordinate. Then l0 is the smallest index l such that γl has a slope smaller than the slope of L, that is, al0 −1 ≤ m < al0 . We may thus consider the augmented Newton polyhedron N r (φ a ) of φ a , which is the convex hull of the union of N (φ a ) with the half line L+ ⊂ L with right endpoint (Al0 −1 , Bl0 −1 ). Then hr (φ) + 1 is the second coordinate of the point at which the line (m) intersects the boundary of N r (φ a ). We remind the reader that all notions that we have introduced so far (with the exception of those discussed in Remark 1.8) make perfect sense for arbitrary smooth functions φ of finite type, in particular, for analytic φ. For real analytic hypersurfaces, it turns out that we now have all necessary notions at hand in order to formulate the central result of this monograph. The extension to more general classes of smooth, finite-type hypersurfaces will require further notions and will be discussed in the next section (compare Theorem 1.14).

16

CHAPTER 1

1/κ2

∆(m)

N (φa )

h (φ) + 1 r

π(φa )

d+1 m+1 L

1/κ1 Figure 1.4 r-height

Theorem 1.10. Let S ⊂ R3 be a real analytic hypersurface of finite type, and fix a point x 0 ∈ S. After applying a suitable Euclidean motion of R3 , let us assume that x 0 = 0 and that near x 0 we may view S as the graph Sφ of a real analytic function φ satisfying φ(0, 0) = 0 and ∇φ(0, 0) = 0. Assume that there is no linear coordinate system adapted to φ. Then there exists a neighborhood U ⊂ S of x 0 such that for every nonnegative density ρ ∈ C0∞ (U ), the Fourier restriction estimate (1.1) holds true for every p ≥ 1 such that p ≥ pc := 2hr (φ) + 2. Remarks 1.11. (a) An application of Greenleaf’s result would imply, at best, that the condition p ≥ 2h(φ) + 2 is sufficient for (1.1) to hold, which is a strictly stronger condition than p  ≥ pc . (b) In a preprint, which regretfully has remained unpublished and which has been brought to our attention by A. Seeger after we had found our results, H. Schulz [Sc90] had already observed this kind of phenomenon for particular examples of surfaces of revolution. Example 1.12. φ(x1 , x2 ) := (x2 − x1m )n ,

n, m ≥ 2.

The coordinates (x1 , x2 ) are not adapted. Adapted coordinates are y1 := x1 , y2 := x2 − x1m , in which φ is given by φ a (y1 , y2 ) = y2n .

17

INTRODUCTION

Here κ1 =

1 , mn

d : = d(φ) = and

 pc

=

κ2 =

1 , n

nm 1 = , κ 1 + κ2 m+1

2d + 2,

if n ≤ m + 1,

2n,

if n > m + 1.

On the other hand, h := h(φ) = n, so that 2h + 2 = 2n + 2 > pc . 1.4 SMOOTH HYPERSURFACES OF FINITE TYPE, CONDITION (R), AND THE GENERAL RESTRICTION THEOREM Theorem 1.10 can be extended to smooth, finite-type functions φ under an additional Condition (R), which, however, is always satisfied when φ is real analytic. To state this more general result and in order to prepare a more invariant description of the notion of r-height, we need to introduce more notation. Again, we shall assume that the coordinates (x1 , x2 ) are linearly adapted to φ. 1.4.1 Fractional shears and condition (R) We need a few more definitions. Consider the half lines R± := {x1 ∈ R : ±x1 > 0}, and denote by H ± := R± × R the corresponding right (respectively, left) half plane. We say that a function f = f (x1 ) defined in U ∩ R+ (respectively, U ∩ R− ), where U is an open neighborhood of the origin, is fractionally smooth if there exist a smooth function g on U and a positive integer q such that f (x1 ) = g(|x1 |1/q ) for x1 ∈ U ∩ R+ (respectively, x1 ∈ U ∩ R− ). Moreover, we shall say that a fractionally smooth function f is flat if f (x1 ) = O(|x1 |N ) as x1 → 0, for every N ∈ N. Notice that this notion of flatness describes only the behavior at the origin. Observe also that a fractionally smooth function that is flat is even smooth. Two fractionally smooth functions f and g defined on a neighborhood of the origin will be called equivalent, and we shall write f ∼ g, if f − g is flat. Finally, a fractional shear in H ± will be a change of coordinates of the form y1 := x1 , y2 := x2 − f (x1 ), where f is real valued and fractionally smooth but not flat. If we express the smooth function φ on, say, the half plane H + as a function of y = (y1 , y2 ), the resulting function φ f (y) := φ(y1 , y2 + f (y1 )) will in general no longer be smooth at the origin, but fractionally smooth in the sense described next.

18

CHAPTER 1

For such functions, there are straightforward generalizations of the notions of Newton polyhedron, and so forth. Namely, following [IKM10] and assuming without loss of generality that we are in H + where x1 > 0, let  be any smooth function 1/q of the variables x1 and x2 near the origin; that is, there exists a smooth function 1/q [q] near the origin such that (x) = [q] (x1 , x2 ) (more generally, one could 1/q 1/p assume that  is a smooth function of the variables x1 and x2 , where p and q are positive integers, but we won’t need this generality here and shall, therefore, always assume that p = 1). Such functions  will also be called fractionally smooth. If the formal Taylor series of [q] is given by [q] (x1 , x2 ) ∼

∞ 

cα1 ,α2 x1α1 x2α2 ,

α1 ,α2 =0

then  has the formal Puiseux series expansion (x1 , x2 ) ∼

∞ 

α /q

cα1 ,α2 x1 1 x2α2 .

α1 ,α2 =0

We therefore define the Taylor-Puiseux support, or Taylor support, of  by T () := {( αq1 , α2 ) ∈ N2q : cα1 ,α2 = 0}, where N2q := ( q1 N) × N. The Newton-Puiseux polyhedron (or Newton polyhedron) N () of  at the origin is then defined to be the convex hull of the union of all the quadrants (α1 /q, α2 ) + R2+ in R2 , with (α1 /q, α2 ) ∈ T (), and other notions, such as the notion of principal face, Newton distance, or homogenous distance, are defined in analogy with our previous definitions for smooth functions φ. Coming back to our fractional shear f, assume that f (x1 ) has the formal Puiseux series expansion (say for x1 > 0)  m cj x1 j , (1.19) f (x1 ) ∼ j ≥0

with nonzero coefficients cj and exponents mj , which are growing with j and are all multiples of 1/q. We then isolate the leading exponent m0 and choose the weight f f κ f so that κ2 /κ1 = m0 and such that the line f

f

Lf := {(t1 , t2 ) ∈ R2 : κ1 t1 + κ2 t2 = 1} is a supporting line to N (φ f ). In analogy with hr (φ), by replacing the exponent m by m0 and the line L by Lf , we can then define the r-height hf (φ) associated with f by putting f

hf (φ) = max(d f , max hl ), {l: al >m0 }

(1.20)

where (d f , d f ) is the point of intersection of the line Lf with the bisectrix and f where hl is associated to the edge γl of N (φ f ) by the analogue of formula (1.17),

19

INTRODUCTION

that is, f

hl =

1 + m0 κ1l − κ2l , κ1l + κ2l

(1.21)

if γl is again contained in the line Ll defined by the weight κ l . In a similar way as for the notion of r-height, we can reinterpret hf (φ) geometrically as follows. We define the augmented Newton polyhedron N f (φ f ) as the convex hull of the union of N (φ f ) with the half line (Lf )+ ⊂ Lf , whose right endpoint is the vertex of N (φ f ) ∩ Lf with the smallest second coordinate. Then hf (φ) + 1 is the second coordinate of the point at which the line (m0 ) intersects the boundary of N f (φ f ). Finally, let us say that a fractionally smooth function f (x1 ) agrees with the principal root jet ψ(x1 ) up to terms of higher order if the following holds: if ψ is not a polynomial, then we require that f ∼ ψ, and if ψ is a polynomial of degree D, then we require that the leading exponent in the formal Puiseux series expansion of f − ψ is strictly bigger than D. Such functions f will indeed arise in Section 6.5 in the course of an algorithm that will allow us to analyze the fine splitting of certain roots near the principal root jet. It is this fine splitting that will lead to terms of higher order that have to be added to ψ. We can now formulate the extra “root” condition that we need when φ is nonanalytic. Condition (R). For every fractionally smooth, real function f (x1 ) that agrees with the principal root jet ψ(x1 ) up to terms of higher order, the following holds true: If B ∈ N is maximal such that N (φ f ) ⊂ {(t1 , t2 ) : t2 ≥ B}, and if B ≥ 1, then φ ˜ 1 , x2 ), where f˜ ∼ f and where φ˜ is fracfactors as φ(x1 , x2 ) = (x2 − f˜(x1 ))B φ(x tionally smooth. Examples 1.13. (a) If ϕ(x1 ) is flat and nontrivial and if m ≥ 2 and B ≥ 2, then the function φ1 (x1 , x2 ) := (x2 − x1m − ϕ(x1 ))B does satisfy Condition (R), whereas (R) fails for φ2 (x1 , x2 ) := (x2 − x1m )B + ϕ(x1 ). In these examples, we have ψ(x1 ) = x1m .   (b) φ3 (x1 , x2 ) := (x2 − x12 − ϕ1 (x1 ))5 (x2 − 3x12 )2 + ϕ2 (x1 ) does satisfy Condition (R) for arbitrary flat functions ϕ1 (x1 ) and ϕ2 (x1 ). Indeed, in this example the principal face π(φ3 ) is the interval [(0, 7), (14, 0)], and so one 1 ) = 14 and ψ(x1 ) = x12 . Notice that ψ(x1 ) is easily finds that d = 1/( 17 + 14 3 of (φ3 )pr , and hence the principal root. a root of multiplicity 5 > 14 3 Real analytic functions φ are easily seen to satisfy Condition (R). Indeed, the definition of B implies that φ f (y1 , y2 ) = y2B h(y1 , y2 ) + ϕ(y1 , y2 ), where ϕ(y1 , y2 ) is flat in y1 and where the mapping y1  → h(y1 , 0) is of finite type. In particular, g(x1 ) := φ(x1 , f (x1 )) = φ f (y1 , 0) = ϕ(y1 , 0) is flat. On the other

20

CHAPTER 1

hand, if the function φ is analytic, then we may factor it as φ(x1 , x2 ) =

U (x1 , x2 )x1ν1 x2ν2

J

(x2 − rj (x1 ))nj ,

j =1

where the rj = rj (x1 ) denote the distinct nontrivial roots of φ and nj their multiplicities, and where U is analytic near the origin such that U (0, 0) = 0. Moreover, the roots rj admit Puiseux series expansions near the origin (compare Remark 1.8). But then J

(f (x1 ) − rj (x1 ))nj , g(x1 ) = U (x1 , f (x1 ))x1ν1 f (x1 )ν2 j =1

and since g is flat, this shows that necessarily f − rk is flat for some k ∈ {1, . . . , J }, that is, f ∼ rk . Then rk must be a real root of φ, and the identity J   ν

n φ f (y1 , y2 ) = U (y1 , y2 + f (y1 ))y1ν1 y2 + f (y1 ) 2 y2 + f (y1 ) − rj (y1 ) j j =1

shows that B = nk . By choosing f˜ := rk , we thus find that  indeed φ(x1 , x2 ) = ˜ 1 , x2 ) := U (x1 , x2 )x1ν1 x2ν2 j =k (x2 − rj (x1 ))nj . ˜ 1 , x2 ), with φ(x (x2 − f˜(x1 ))B φ(x 1.4.2 The general restriction theorem and sharpness of its conditions We can now state our main result. Theorem 1.14. Let S ⊂ R3 be a smooth hypersurface of finite type, and fix a point x 0 ∈ S. After applying a suitable Euclidean motion of R3 , let us assume that x 0 = 0 and that near x 0 we may view S as the graph Sφ of a smooth function φ of finite type satisfying φ(0, 0) = 0 and ∇φ(0, 0) = 0. Assume that the coordinates (x1 , x2 ) are linearly adapted to φ but not adapted and that Condition (R) is satisfied. Then there exists a neighborhood U ⊂ S of x 0 such that for every nonnegative density ρ ∈ C0∞ (U ), the Fourier restriction estimate (1.1) holds true for every p ≥ 1 such that p ≥ pc := 2hr (φ) + 2. The main body of this monograph will be devoted to the proof of this theorem. A question that remains open at this stage is whether Condition (R) is really needed in this theorem, or whether it can be removed completely, respectively replaced by a weaker condition. Before returning to the proof, we shall show by means of Knapp type examples that the conditions in this theorem are sharp in the following sense: Theorem 1.15. Let φ be smooth of finite type, and assume that the Fourier restriction estimate (1.1) holds true in a neighborhood of x 0 . Then, if ρ(x 0 ) = 0, necessarily p ≥ pc . Since the proof will also help to illuminate the notion of r-height, we shall give it right away. In fact, we shall prove the following more general result (notice that we are making no assumption on adaptedness of φ here).

21

INTRODUCTION

Proposition 1.16. Assume that the coordinates x = (x1 , x2 ) are linearly adapted to φ and that the restriction estimate (1.1) holds true in a neighborhood of x 0 = 0, where ρ(x 0 ) = 0. Consider any fractional shear, say on H + , given by y1 := x1 , y2 := x2 − f (x1 ), where f is real valued and fractionally smooth but not flat. Let φ f (y) = φ(y1 , y2 + f (y1 )) be the function expressing φ in the coordinates y = (y1 , y2 ). Then, necessarily, p  ≥ 2hf (φ) + 2. Theorem 1.15 will follow by choosing for f the principal root jet ψ. Proof. The proof will be based on suitable Knapp-type arguments. Let us use the same notation for the Newton polyhedron of φ f as we did for φ a in Section 1.3, that is, the vertices of the Newton polyhedron N (φ f ) will be denoted by (Al , Bl ), l = 0, . . . , n, where we assume that they are ordered so that Al−1 < Al , l = 1, . . . , n, with associated compact edges γl := [(Al−1 , Bl−1 ), (Al , Bl )], l = 1, . . . , n, contained in the supporting lines Ll to N (φ f ) and associated with the weights κ l . The unbounded horizontal edge with left endpoint (An , Bn ) will be denoted by γn+1 . For l = n + 1, we have κ1n+1 := 0, κ2n+1 = 1/Bn . Again, we put al := κ2l /κ1l , and an+1 := ∞. Because of (1.20), we have to prove the following estimates: p ≥ 2d f + 2;

(1.22)

f 2hl

(1.23)



p ≥

+ 2 for every l such that al > m0 ,

where, according to (1.21), f

hl =

1 + m0 κ1l − κ2l . κ1l + κ2l

Consider first any nonhorizontal edge γl of N (φ f ) with al > m0 , and denote by the region

f Dε

l

l

Dεf := {y ∈ R2 : |y1 | ≤ εκ1 , |y2 | ≤ εκ2 },

ε > 0,

in the coordinates y. In the original coordinates x, it corresponds to l

l

Dε := {x ∈ R2 : |x1 | ≤ εκ1 , |x2 − f (x1 )| ≤ εκ2 }. Assume that ε is sufficiently small. Since l

l

f

φ f (εκ1 y1 , εκ2 y2 ) = ε(φκ l (y1 , y2 ) + O(εδ )), f

for some δ > 0, where φκ l denotes the κ l -principal part of φ f , we have that |φ f (y)| f ≤ Cε for every y ∈ Dε , that is, |φ(x)| ≤ Cε

for every x ∈ Dε .

(1.24)

Moreover, for x ∈ Dε , because |f (x1 )|
|x1 |m0 and m0 ≤ al = κ2l /κ1l , we have l

l

l

l

|x2 | ≤ εκ2 + |f (x1 )|
εκ2 + εm0 κ1
εm0 κ1 .

22

CHAPTER 1 l

We may thus assume that Dε is contained in the box where |x1 | ≤ 2εκ1 , |x2 | ≤ m0 κ1l . Choose a Schwartz function ϕε such that 2ε
x 
x 
x  1 2 3 , χ0 χ0 ϕε (x1 , x2 , x3 ) = χ0 l l κ m κ 0 Cε 1 1 ε ε where χ0 is a smooth cutoff function supported in [−2, 2] and identically 1 on [−1, 1]. Then by (1.24) we see that ϕε (x1 , x2 , φ(x1 , x2 )) ≥ 1 on Dε ; hence, if ρ(0) = 0, then 1/2
 l l |ϕε |2 ρ dσ ≥ C1 |Dε |1/2 = C1 ε(κ1 +κ2 )/2 , S

l



where C1 > 0 is a positive constant. Since ϕε p  ε((1+m0 )κ1 +1)/p , we find that the restriction estimate (1.1) can hold only if p ≥ 2

(1 + m0 )κ1l + 1 f = 2hl + 2. κ1l + κ2l f

The case l = n + 1, where γl is the horizontal edge for which hl = Bn − 1 (with Bn = 1/κl2 ), requires a minor modification of this argument. Observe that, by Taylor expansion, in this case φ f can be written as φ f (y1 , y2 ) = y2Bn h(y1 , y2 ) +

B n −1

j

y2 gj (y1 ),

(1.25)

j =0

where the functions gj are flat and h is fractionally smooth and continuous at the origin. Choose a small δ > 0, and define l

Dεf := {y ∈ R2 : |y1 | ≤ εδ , |y2 | ≤ εκ2 },

ε > 0. f

Then (1.25) shows that again |φ f (y)| ≤ Cε for every y ∈ Dε , so that (1.24) holds true again. Moreover, for x ∈ Dε , we now find that l

l

|x2 | ≤ εκ2 + |f (x1 )|
εκ2 + εm0 δ
εm0 δ for δ sufficiently small. Choosing ϕε (x1 , x2 , x3 ) = χ0


x 
x 
x  1 2 3 , χ0 m δ χ0 δ 0 ε ε Cε

and arguing as before, we find that here (1.1) implies that p ≥ 2 f

(1 + m0 )δ + 1 δ + κ2l

for every δ > 0;

hence, p ≥ 2Bn = 2hln+1 + 2. This finishes the proof of (1.23). Notice finally that the argument for the nonhorizontal edges still works if we replace the line Ll by the line Lf and the weight κ l by the weight κ f associated f f Q.E.D. with that line. Since here m0 κ1 = κ2 , this leads to condition (1.22).

23

INTRODUCTION

1.5 AN INVARIANT DESCRIPTION OF THE NOTION OF r-HEIGHT Finally, we can also give a more invariant description of the notion of r-height, which conceptually resembles more closely Varchenko’s definition of the notion of height, only that we restrict the admissible changes of coordinates to the class of fractional shears in the half planes H + and H − . In this section, we do not work under Assumption (NLA), but we may and shall again assume that our initial coordinates (x1 , x2 ) are linearly adapted to φ. Then we set h˜ r (φ) := sup hf (φ), f

where the supremum is taken over all nonflat fractionally smooth, real functions f (x1 ) of x1 > 0 (corresponding to a fractional shear in H + ) or of x1 < 0 (corresponding to a fractional shear in H − ). Then, obviously, hr (φ) ≤ h˜ r (φ), but in fact there is equality. Proposition 1.17. Assume that the coordinates (x1 , x2 ) are linearly adapted to φ, where φ is smooth and of finite type and satisfies φ(0, 0) = 0, ∇φ(0, 0) = 0. (a) If the coordinates (x1 , x2 ) are not adapted to φ, then for every nonflat fractionally smooth, real function f (x1 ) and the corresponding fractional shear in H + (respectively, H − ), we have hf (φ) ≤ hr (φ). Consequently, hr (φ) = h˜ r (φ). (b) If the coordinates (x1 , x2 ) are adapted to φ, then h˜ r (φ) = d(φ) = h(φ). In particular, the critical exponent for the restriction estimate (1.1) is in all cases given by pc := 2h˜ r (φ) + 2. Let us content ourselves at this stage with a short, but admittedly indirect, proof of part (b) and of part (a) under the assumption that φ is analytic. Our arguments will again be based on Proposition 1.16. Since these arguments will rely on the validity of Theorems 1.5 and 1.10, which is somewhat unsatisfactory, we shall give a direct, but lengthier, proof in Chapter 9, which will in addition not require analyticity of φ. On the proof of Proposition 1.17. Recall that we assume that the original coordinates (x1 , x2 ) are linearly adapted to φ. In order to prove (a) for analytic φ, assume furthermore that the coordinates (x1 , x2 ) are not adapted to φ, and let f (x1 ) be any nonflat fractionally smooth, real-valued function of x1 , with corresponding fractional shear, say in H + . We have to show that hf (φ) ≤ hr (φ).

(1.26)

According to Theorem 1.10 the restriction estimate (1.1) holds true for p = pc , where pc = 2hr (φ) + 2. Moreover, choosing ρ so that ρ(x 0 ) = 0, then

24

CHAPTER 1

Proposition 1.16 implies that p  ≥ 2hf (φ) + 2. Combining these estimates, we obtain (1.26). In order to prove (b), we assume that the coordinates (x1 , x2 ) are adapted to φ, so that d(φ) = h(φ). We have to prove that h˜ r (φ) = d(φ).

(1.27)

Let us first observe that Theorem 1.5 and Proposition 1.16 imply, in a similar way as in the proof of (a), that 2h(φ) + 2 ≥ 2hf (φ) + 2; hence d(φ) ≥ hf (φ). We thus see that h˜ r (φ) ≤ d(φ). On the other hand, when the principal face π(φ) is compact, then we can choose a supporting line L = {(t1 , t2 ) ∈ R2 : κ1 t1 + κ2 t2 = 1} to the Newton polyhedron of φ containing π(φ) and such that 0 < κ1 ≤ κ2 . We then put f (x1 ) := x1m0 , where m0 := κ2 /κ1 . Then d(φ) = 1/(κ1 + κ2 ) = d f ≤ hf (φ) ≤ h˜ r (φ), and we obtain (1.27). Assume finally that π(φ) is an unbounded horizontal half line, with left endpoint (A, B), where A < B. We then choose fn (x1 ) := x1n , n ∈ N. Then it is easy to see that for n sufficiently large, the line Lfn will pass through the point (A, B), and thus limn→∞ hfn (φ) = B = d(φ). Therefore, h˜ r (φ) ≥ d(φ), which shows that (1.27) is also valid in this case. Q.E.D.

1.6 ORGANIZATION OF THE MONOGRAPH AND STRATEGY OF PROOF In Chapter 2 we shall begin to prepare the proof of Theorem 1.14 by compiling various auxiliary results. This will include variants of van der Corput type estimates for one-dimensional oscillatory integrals and related sublevel estimates through “integrals of sublevel type,” which will be used all over the place. In some situations, we shall also need more specific information, in particular on Airy-type oscillatory integrals, as well as on some special classes of integrals of sublevel type, which will also be provided. We shall also derive a straightforward variant of a beautiful real interpolation method that has been devised by Bak and Seeger in [BS11] and that will allow us in some cases to replace the more classical complex interpolation methods in the proof of Stein-Tomas-type Fourier restriction estimates by substantially shorter arguments. Nevertheless, complex interpolation methods will play a major role in many other situations, and a crucial tool in our application of Stein’s interpolation theorem for analytic families of operators will be provided by certain uniform estimates for oscillatory sums, respectively, double sums (cf. Lemmas 2.7 and 2.9). Last, we shall derive normal forms for phase functions φ of linear height < 2 for which no linear coordinate system adapted to φ does exist. These normal forms will provide the basis for our discussion of the case h lin (φ) < 2 in Chapter 4.

INTRODUCTION

25

As a first step in our proof of Theorem 1.14, in which we shall always assume that the coordinates x are linearly adapted, but not adapted to φ, we shall show in Chapter 3 that one may reduce the desired Fourier restriction estimate to a piece Sψ of the surface S lying above a small, “horn-shaped” neighborhood Dψ of the principal root jet ψ, on which |x2 − ψ(x1 )| ≤ εx1m . Here, ε > 0 can be chosen as small as we wish. This step works in all cases, no matter which value h lin (φ) takes. The proof will give us the opportunity to introduce some of the basic tools which will be applied frequently, such as dyadic domain decompositions, rescaling arguments based on the dilations associated to a given edge of the Newton polyhedron (in this chapter given by the principal face π(φ) of the Newton polyhedron of φ), in combination with Greenleaf’s restriction Theorem 1.1 and Littlewood-Paley theory, which will allow us to sum the estimates that we obtain for the dyadic pieces. From here on, following our approach in [IKM10] and [IM11b], it will be natural to distinguish between the cases where h lin (φ) < 2 and where h lin (φ) ≥ 2, since, in contrast to the first case, in the latter case a reduction to estimates for one-dimensional oscillatory integrals will be possible in many situations, which in return can be performed by means of the van der Corput–type Lemma 2.1. Chapters 4 and 5 will be devoted to the case where h lin (φ) < 2. The starting point for our discussion will be the normal forms for φ provided by Proposition 2.11. Some of the main tools will again consist of various kinds of dyadic domain decompositions in combination with Littlewood-Paley theory and rescaling arguments. In addition, we shall have to localize frequencies to dyadic intervals in each component, which then also leads us to distinguish a variety of different cases, depending on the relative sizes of these dyadic intervals as well as of another parameter related to the Littlewood-Paley decomposition. It turns out that the particular case where m = 2 in (1.9) and (1.14) will require, in some situations (these are listed in Proposition 4.2), a substantially more refined analysis than the case m ≥ 3. Indeed, in some cases, namely, those described in Proposition 4.2(a) and (b), our arguments from Chapter 4 will almost give the complete answer, except that we miss the endpoint p = pc . In order to capture also the corresponding endpoint estimates, we shall devise rather intricate complex interpolation arguments in Section 5.3. These will be prototypical for many more arguments of this type that we shall devise in later chapters. Even more of a problem will be presented by the cases described in Proposition 4.2(c). In these situations, we not only miss the endpoint estimate in our discussion in Chapter 4, but it turns out that we even have to close a large gap in the Lp range that we need to cover. In order to overcome this problem, we shall perform a further dyadic decomposition in frequency space with respect to the distance to a certain “Airy cone.” This refined Airy-type analysis will be developed in Sections 5.1 and 5.2. Again, in order to capture also the endpoint p = pc , we need to apply a complex interpolation argument. Useful tools in these complex interpolation arguments will be Lemmas 2.7 and 2.9 on oscillatory sums and double sums. In Chapter 6 we shall turn to the case where h lin (φ) ≥ 2. In a first step, following some ideas from the article [PS97] by Phong and Stein (compare also [IKM10] and [IM11b]), we shall perform a decomposition of the remaining piece Sψ of

26

CHAPTER 1

the surface S, which will be adapted in some sense to the “root structure” of the function φ within the domain Dψ . When speaking of roots, we will, in fact, always have the case of analytic φ in mind; for nonanalytic φ, these statements may no longer make strict sense, but the ideas from the analytic case may still serve as a very useful guideline. In order to understand the root structure within our narrow horn-shaped neighborhood Dψ of the principal root jet ψ, it is natural to look at the Newton polyhedron N (φ a ) of φ when expressed in the adapted coordinates (y1 , y2 ) given by (1.10), (1.11). More precisely, we shall associate to every edge γl of N (φ a ) lying above the bisectrix a domain Dl , which will be homogeneous in the adapted l l coordinates (y1 , y2 ) under the natural dilations (y1 , y2 )  → (r κ1 y1 , r κ2 y2 ), r > 0, defined by the weight κ l that we had associated to the edge γl . We shall then partition the domain Dψ into these domains Dl , intermediate domains El , and a residue domain Dpr and consider the corresponding decomposition of the surface S. The remaining domain Dpr , which contains the principal root jet x2 = ψ(x1 ), will in some sense be associated with the principal face π(φ a ) of the Newton polyhedron of φ a and, hence, homogeneous in the coordinates (y1 , y2 ). Each domain El can be viewed as a “transition” domain between two different types of homogeneity (in adapted coordinates). In the domains Dl we can again apply our dyadic decomposition techniques in combination with rescaling arguments, making use of the dilations associated with the weight κ l , but serious new problems do arise, caused by the nonlinear change from the coordinates (x1 , x2 ) to the adapted coordinates (y1 , y2 ). Following again [PS97], the discussion of the transition domains El will be based on bidyadic domain decompositions in the coordinates (y1 , y2 ). In our discussion of the domains Dl , we shall have to distinguish three cases, Cases 1–3, depending on the behavior of the κ l -principal part φκal of φ a near a given point v = 0. The first case will be easy to handle, and the same is true even for the residue domain Dpr . However, in the other two cases, a large difference between the domains Dl and the domain Dpr will appear. The reason for this is that for the edges γl lying above the bisectrix, we shall be able to prove a favorable control on the multiplicities of the roots of φκal (more precisely of ∂2 φκal ), but this breaks down on Dpr . We shall, therefore, start to have a closer look at the domain Dpr in Section 6.4. In order to handle Case 3, which is the case where φκalpr has a critical point at v, in Section 6.5 we shall devise a further decomposition of the domain Dpr into various subdomains of “type” D(l) and E(l) , where each domain D(l) will be homogeneous in suitable “modified adapted” coordinates, and the domains E(l) can again be viewed as “transition domains.” This domain decomposition algorithm, roughly speaking, reflects the “fine splitting” of roots of ∂2 φ a . The new transition domains E(l) can be treated in a similar way as the domains El before, and in the end we shall be left with domains of type D(l) . Now, under the assumption that h lin (φ) ≥ 5, it turns out that these remaining domains can be handled by means of a fibration of the given piece of surface into a family of curves, in combination with Drury’s Fourier restriction theorem for curves with nonvanishing torsion [Dru85]. However, that method breaks down when h lin (φ) < 5, so that in the subsequent two

INTRODUCTION

27

chapters we shall devise an alternative approach for dealing with these remaining domains D(l) . That approach will work equally well whenever h lin (φ) ≥ 2. In Chapter 7 we shall mostly consider the domains of type D(1) , which are in some sense “closest” to the principal root jet, since it will turn out that the other domains D(l) with l ≥ 2 are easier to handle (compare Section 7.10). Within the domains of type D(1) , we shall have to deal with functions φ which, in suitable “modified” adapted coordinates, look like φ˜ a (y1 , y2 ) = y2B bB (y1 , y2 ) + y1n α(y1 ) + B−1 j j =1 y2 bj (y1 ), where B ≥ 2 and bB is nonvanishing. In a first step, by means of some lower bounds on the r-height, we shall be able to establish favorable restriction estimates in most situations, with the exception of certain cases where m = 2 and B = 3 or B = 4. Along the way, in some cases we shall have to apply interpolation arguments in order to capture the endpoint estimates for p = pc . Sometimes this can be achieved by means of the aforementioned variant of the Fourier restriction theorem by Bak and Seeger, whose assumptions, when satisfied, are easily checked. However, in most of these cases we shall have to apply complex interpolation, in a similar way as we did before in Section 5.3. Eventually we can thus reduce considerations to certain cases where m = 2 and B = 3 or B = 4. The most difficult situations will occur when frequencies ξ = (ξ1 , ξ2 , ξ3 ) are localized to domains on which all components ξi of ξ are comparable in size. These cases, which will be discussed in Chapter 8, turn out to be the most challenging ones among all, the case B = 3 being the worst. Again, we shall have to apply a refined Airy-type analysis in combination with complex interpolation arguments as in Chapter 5, but further methods are needed— for instance, rescaling arguments going back to Duistermaat [Dui74]— in order to control the dependence of certain classes of oscillatory integrals on some small parameters, and the technical complexity will become quite demanding. The monograph will conclude with the proof of Proposition 1.7 on the characterization of linearly adapted coordinates in Appendix A and a direct proof of Proposition 1.17 on an invariant description of the notion of r-height in Appendix B of Chapter 9. Conventions: Throughout this monograph, we shall use the “variable constant” notation, that is, many constants appearing in the course of our arguments, often denoted by C, will typically have different values on different lines. Moreover, we shall use symbols such as ∼,
, or in order to avoid writing down constants. By A ∼ B we mean that there are constants 0 < C1 ≤ C2 such that C1 A ≤ B ≤ C2 A, and these constants will not depend on the relevant parameters arising in the context in which the quantities A and B appear. Similarly, by A
B we mean that there is a (possibly large) constant C1 > 0 such that A ≤ C1 B, and by A B we mean that there is a sufficiently small constant c1 > 0 such that A ≤ c1 B, and again these constants do not depend on the relevant parameters. By χ0 and χ1 we shall always denote smooth cutoff functions with compact support on Rn , where χ0 will be supported in a neighborhood of the origin and usually be identically 1 near the origin, whereas χ1 = χ1 (x) will be support away from the origin, sometimes in each of its coordinates xj , that is, |xj | ∼ 1 for j = 1, . . . , n, for every x in the support of χ1 . These cutoff functions may also vary from line

28

CHAPTER 1

to line and may, in some instances, where several of such functions of different variables appear within the same formula, even designate different functions. Also, if we speak of the slope of a line such as a supporting line to a Newton polyhedron, then we shall actually mean the modulus of the slope. Finally, when speaking of domain decompositions, we shall not always keep to the mathematical convention of a domain being on open connected set but shall occasionally use the word domain in a more colloquial way. Finally, by N× , Q× , R× , and so on, we shall denote the set of nonzero elements in N, Q, R, and so on.

ACKNOWLEDGMENTS We thank the referees for numerous suggestions which greatly helped to improve the exposition. Moreover, we are indepted to Eugen Zimmermann for providing some of the graphics for this monograph. Finally, we gratefully acknowledge the support for this work by the Deutsche Forschungsgemeinschaft (DFG). We also wish to acknowledge permission to reprint excerpts from the following previously published materials: “Uniform Estimates for the Fourier Transform of Surface Carried Measures in R3 and an Application to Fourier Restriction” by Isroil A. Ikromov and Detlef Müller from Journal of Fourier Analysis and Applications, December 2011, c Springer Science+ Volume 17, Issue 6, pp. 1292–1332 (16 July 2011)  Business Media, LLC 2011. Published with permission of Springer Science+ Business Media. “On Adapted Coordinate Systems” by Isroil A. Ikromov and Detlef Müller. First published in Transactions of the American Mathematical Society in 363 (2011), 2821–2848, published by the American Mathematical Society. Reprinted by permission of the American Mathematical Society. “Estimates for maximal functions associated to hypersurfaces in R3 and related problems of harmonic analysis”; I.A. Ikromov, M. Kempe and D. Müller, Acta Mathematica 204 (2010), 151–271. Reprinted by permission of Acta Mathematica. Advances in Analysis: The Legacy of Elias M. Stein edited by Charles Fefferman, Alexandru Ionescu, D.H. Phong, and Stephen Wainger, Princeton University Press, 2014. Reprinted by permission of Princeton University Press and Stephen Wainger, Charles Fefferman, D.H. Phong, and Alexandru Ionescu.

Chapter Two Auxiliary Results In the course of the proof of the main Theorem 1.14, various auxiliary results will be useful. These will be compiled in this chapter. First, we shall recall some classical
van der Corput–type estimates for oneiλf (s) g(s) ds and related integrals dimensional oscillatory integrals I e
G(λf (s)) ds, which will be frequently needed. Integrals of the latter type (and I related integrals) will be called integrals of sublevel type, because if we choose for G the characteristic function of the interval [−1, 1], then the integral computes the measure of the sublevel set where |f (s)| ≤ 1/λ in I. In some situations, we shall need more precise information on the asymptotic behavior of oscillatory integrals. In particular, we shall frequently make use of the method of stationary phase (in higher dimension), which covers the case of phase functions with nondegenerate critical points (see, e.g., [H90],[So93] or [S93] for references). We shall also encounter phase functions with degenerate critical points, in particular Airy-type integrals, in the presence of perturbation terms, for which precise asymptotic information will be crucial. Therefore, we shall analyze a model class of such Airy-type integrals (and integrals with even higher order degeneracies) in Section 2.2. A rather specific class of integrals of sublevel type will be studied in Section 2.3. These will become quite important to various complex interpolation arguments of Chapters 7 and 8 in order to prove certain endpoint results of Theorem 1.14 when 2 ≤ h lin (φ) < 5. Next, as mentioned before, in some situations we may work with a variant of the real interpolation method devised by J. G. Bak and A. Seeger [BS11] in place of a complex interpolation argument in order to establish certain endpoint estimates. This variant of the results by Bak and Seeger will be stated in Section 2.4. Whenever it is possible to apply this method, it will allow for substantially simplified proofs, as for instance in Chapters 5 and 7. Yet another important tool for our complex interpolation arguments in Chapters 7 and 8 will be provided by the uniform estimates for oscillatory sums and doublesums of Section 2.5. Such sums will arise from dyadic and bidyadic frequency domain decompositions (also with respect to the distance to certain “Airy cones”) in Chapters 5, 7, and 8. Finally, for our treatment of the case where h lin (φ) < 2, we shall make use of normal forms for phase functions φ of linear height < 2 for which no linear coordinate system adapted to φ does exist. These will be derived in the last Section 2.6. When φ has only isolated critical points, they will actually correspond to singularities of type An and Dn in Arnol’ds classification of singularities.

30

CHAPTER 2

2.1 VAN DER CORPUT–TYPE ESTIMATES We shall often make use of van der Corput–type estimates. These include the classical van der Corput lemma [vdC21] (see also [S93]) as well as variants of it, going back to J. E. Björk (see [D77]) and G. I. Arhipov [AKC79], and also related classical sublevel estimates, which originated in work by van der Corput too [vdC21] (see also [AKC79], [CaCW99], and [G09]). Lemma 2.1. Let M ≥ 2 (M ∈ N), and let f be a real-valued function of class C M defined on an interval I ⊂ R. Assume that either (i) |f (M) (s)| ≥ 1 on I , or that (ii) f is of polynomial type M ≥ 2, that is, there are positive constants c1 , c2 > 0 such that c1 ≤

M 

|f (j ) (s)| ≤ c2

for every s ∈ I,

j =1

and I is compact. Then the following hold true: For every λ ∈ R, (a)

     eiλf (s) g(s) ds  ≤ C(gL∞ (I ) + g  L1 (I ) ) (1 + |λ|)−1/M , I

where the constant C depends only on M in case (i), and on M, c1 , c2 and I in case (ii). (b) If G ∈ L1 (I ) is a nonnegative function which is majorized by a function H ∈ L1 (I ) such that Hˆ ∈ L1 (R), then  G(λf (s)) ds ≤ C|λ|−1/M , I

where the constant C depends only on M and H 1 + Hˆ 1 in case (i), and on M, c1 , c2 , I and H 1 + Hˆ 1 in case (ii). Proof. For (a), we refer to [vdC21], [S93], [D77], and [AKC79]. Moreover, it is well known (see [vdC21]) that (b) is an immediate consequence of (a). Indeed, by means of the Fourier inversion formula and Fubini’s theorem, we may estimate      1   G(λf (s)) ds ≤  Hˆ (ξ ) eiξ λf (s) ds dξ  ≤ C|λ|−1/M |Hˆ (ξ )||ξ |−1/M dξ. 2π I I R Q.E.D. We remark that the conditions on the function G in (b) are satisfied in particular if G = |ϕ|, where ϕ is of Schwartz class.

31

AUXILIARY RESULTS

2.2 AIRY-TYPE INTEGRALS In various situations, we shall need not only estimates but more precise information on the asymptotic behavior of certain oscillatory integrals of Airy type. The next lemma and its proof will be prototypical for the situations that will arise. We shall actually only make use of the case B = 3 of Airy-type integrals, but since the method of proof works in the same way for arbitrary B ≥ 3, we shall state it more generally. We shall sketch a proof, since the method of proof will later also be applied to similar situations that are not completely covered by the lemma and also since we shall need somewhat more specific results than those we found in the literature (compare, for instance, Lemma 1 in [R69] or [Dui74]). Lemma 2.2. Let B ≥ 3 be an integer, and let  B−1 B j eiλ(b(t,s)t −ut− j =2 bj (u)t ) a(t, s) dt, J (λ, u, s) := R

λ ≥ 1, u ∈ R, |u|
1,

where a, b are smooth, real-valued functions of (t, s) on an open neighborhood of I × K, where I is a compact neighborhood of the origin in R and K is a compact subset of Rm . The functions bj are assumed to be real valued and also smooth. Assume also that b(t, s)  = 0 on I × K, that |t| ≤ ε on the support of a, and that |bj (u)| ≤ C|u|,

j = 2, . . . , B − 1.

If ε > 0 is chosen sufficiently small and λ sufficiently large, then the following hold true: (a) If λ(B−1)/B |u|
1, then J (λ, u, s) = λ−1/B g(λ(B−1)/B u, λ, s), where g(v, λ, s) is a smooth function of (v, λ, s) whose derivates of any order are uniformly bounded on its natural domain. (b) If λ(B−1)/B |u| 1, let us assume first that u and b have the same sign, and that B is odd. Then u J (λ, u, s) = λ−1/2 |u|−(B−2)/(2B−2) χ0 ε  1/(B−1) iλ|u|B/(B−1) q+ (|u|1/(B−1) ,s) × a+ (|u| , s) e B/(B−1) 1/(B−1) q− (|u|1/(B−1) ,s) +a− (|u| , s) eiλ|u| + (λ|u|)−1 E(λ|u|B/(B−1) , |u|1/(B−1) , s), where a± , q± are smooth functions and where E is smooth and satisfies estimates |∂µα ∂vβ ∂sγ E(µ, v, s)| ≤ CN,α,β,γ |v|−β |µ|−N ,

∀N, α, β, γ ∈ N.

32

CHAPTER 2

Moreover, when |u| is sufficiently small, then q± (v, s) = ∓sgn b(0, s)|b(0, s)|1/(B−1) ρ(v, s), where ρ is smooth and ρ(0, s) = (B − 1) · B −B/(B−1) . Finally, if u and b have opposite signs, then the same formula remains valid, even with a+ ≡ 0, a− ≡ 0. And, if B is even, we do have a similar result, but without the presence of the term containing a− . Proof. In Case (a), scaling in t by the factor λ−1/B allows us to rewrite    −1/B B (B−1)/B (B−j )/B ut− B−1 bj (u)t j j =2 λ J (λ, u, s) = λ−1/B ei b(λ t,s)t −λ a(λ−1/B t, s) dt. Choose a smooth cutoff function χ0 on R that is identically 1 on [−1, 1], and M 1, and decompose λ1/B J (λ, u, s) = G0 (λ(B−1)/B u, λ, s) + G∞ (λ(B−1)/B u, λ, s), where, for |v|
1,

   t i(b(λ−1/B t,s)t B −vt− B−1 λ(B−j )/B bj (λ(1−B)/B v)t j ) j =2 G0 (v, λ, s) := e χ0 a(λ−1/B t, s) dt, M  B−1 (B−j )/B −1/B B bj (λ(1−B)/B v)t j ) G∞ (v, λ, s) := ei(b(λ t,s)t −vt− j =2 λ

 t × 1 − χ0 a(λ−1/B t, s) dt. M Notice that for j ≥ 2, |λ(B−j )/B bj (λ(1−B)/B v)| ≤ Cλ(B−j )/B λ(1−B)/B |v|
λ−1/B . It is then easy to see that G0 is a smooth function of (v, λ, s) whose derivates of any order are uniformly bounded on its natural domain, and the same can easily be verified for G∞ by means of iterated integrations by parts. This proves (a). In order to prove (b), consider first the case where |u| ≥ ε. If = (t) denotes the complete phase in the oscillatory integral defining J (λ, u, s), recalling that |t| ≤ ε, we easily see that |  (t)| ≥ Cλ|u|, provided we choose ε sufficiently small. Integrations by parts then show that we can represent J (λ, u, s) by the third term (λ|u|)−1 E(λ|u|B/(B−1) , |u|1/(B−1) , λ, s). Let us therefore assume that |u| < ε. We shall also assume that u > 0; the case u < 0 can be treated in a similar way. Here, we scale t by the factor u1/(B−1) , and rewrite   B/(B−1) −(B−j )/(B−1) (b(u1/(B−1) t,s)t B −t− B−1 bj (u)t j ) j =2 u J (λ, u, s) = u1/(B−1) eiλu × a(u1/(B−1) t, s) dt. Again, we decompose this as J (λ, u, s) = J0 (λ, u1/(B−1) , s) + J∞ (λ, u1/(B−1) , s),

33

AUXILIARY RESULTS

where, with v := u1/(B−1) ,

   t iλv B (b(vt,s)t B −t− B−1 v −(B−j ) bj (v B−1 )t j ) j =2 χ0 a(vt, s) dt, J0 (λ, v, s) := v e M

  B−1 −(B−j ) t B B bj (v B−1 )t j ) a(vt, s) dt. 1 − χ0 J∞ (λ, v, s) := v eiλv (b(vt,s)t −t− j =2 v M Observe that |v −(B−j ) bj (v B−1 )| ≤ Cv j −1
ε1/(B−1) ,

j = 2, . . . , B − 1.

Assume that ε is sufficiently small. If B is odd, then, in the first integral J0 , the phase has exactly two nondegenerate critical points t± (v, s) ∼ ±1 if b > 0, and thus the method of stationary phase shows that J0 (λ, v, s) = v(λv B )−1/2 a+ (v, s)eiλv

B

q+ (v,s)

+ v(λv B )−1/2 a− (v, s)eiλv

B

q− (v,s)

+ vE1 (λv B , v, s),

where a± are smooth functions and where E1 is smooth and rapidly decaying with respect to the first variable. If b < 0, then there are no critical points, and we get the term E1 only. Moreover, q± (v, s) = b(vt± (v, s), s)t± (v, s)B − t± (v, s) + O(v). Note that if v = 0, then t± (0, s) = ±(Bb(0, s))−1/(B−1) , so that q± (0, s) = ∓

B −1 B B/(B−1) b(0, s)1/(B−1)

 = 0,

which proves the statement about q± . A similar discussion applies when B is even. In this case, there is only one critical point, namely, t+ (v, s). In the second integral J∞ , we may apply integrations by parts in order to rewrite it as  B−1 −(B−j ) B B bj (v B−1 )t j ) J∞ (λ, v, s) := v(λv B )−N eiλv (b(vt,s)t −t− j =2 v × aN (t, v, s) dt,

N ∈ N,

where aN is supported where |t| ≥ M and |aN (t, v, s)| ≤ CN |t|−2N . Similarly, if we take derivatives with respect to s, we produce additional powers of t in the integrand, which, however, can be compensated by integrations by parts. Analogous considerations apply to derivatives with respect to v (where we produce negative powers of v) and with respect to λv B . Altogether, we find that J∞ (λ, v, s) =

1 E2 (λv B , v, s), λv B−1

where E2 is smooth and |∂µα ∂vβ ∂sγ E2 (µ, v, s)| ≤ CN,α,β,γ |v|−β |µ|−N ,

∀N, α, β, γ ∈ N.

Summing up all terms, and putting E := E1 + E2 , we obtain the statements in (b). Q.E.D.

34

CHAPTER 2

The following remark, which will become relevant, for instance, in Chapter 5, can be verified easily by well-known versions of the method of stationary phase for oscillatory integrals whose amplitude depends also on the parameter λ as symbols of order 0 (see, e.g., [So93]). Remark 2.3. We may even allow in Lemma 2.2 that the function a(t, s) depends on λ too, that is, a = a(t, s; λ), in such a way that it is a symbol of order 0 with respect to λ, uniformly in the other parameters; that is,        ∂ α ∂ β1 ∂ β2   a(t, s; λ) ≤ Cα,β (1 + λ)−α    ∂λ ∂t ∂s for all α, β1 , β2 ∈ N. Then the same conclusions hold true, only with a± and E depending additionally on λ as symbols of order 0 in a uniform way. 2.3 INTEGRAL ESTIMATES OF VAN DER CORPUT TYPE Lemma 2.4. Let b = b(y1 , y2 ) be a C 2 -function on R × [−1, 1] such that b(0, 0)  = 0, b(y1 , ·)C 2 ([−1,1]) ≤ c1 for every y1 ∈ R and      |b(y1 , y2 ) − b(0, 0)| ≤ ε, and c2 |y2 |3−j ≤ ∂yj2 b(y1 , y2 )y23 , j = 1, 2, (2.1) for every (y1 , y2 ) ∈ R × [−1, 1], where 0 < c1 ≤ c2 . Furthermore, let Q = Q(y2 ) be a smooth function on [−1, 1] such that QC 2 ([−1,1]) ≤ c1 , and let A, B, T be real numbers so that max{|A|, |B|} ≥ L, T ≥ L and |A| ≤ T 3 ,

|B| ≤ T 2 .

(2.2)

Moreover, let ri = ri (y1 ), i = 1, 2, be measurable functions on R such that |ri (y1 )| ≤ c(1 + |y1 |). For  ≥ 0 and N ≥ 2, we put I (A, B, T ) :=

T ∞ 

−T ∞

    y  y2  3 2  y2 − b y1 , y2 1 + A − B + Q T T

−N y2  (1 + |y1 |)−N |y2 | dy1 dy2 . r2 (y1 ) T Then, for N sufficiently large, there are constants C > 0 and ε0 > 0, which depend only on the constants c, c1 , and c2 , such that for all functions b and q and all A, B, T with the assumed properties and all ε ≤ ε0 and L ≥ C, we have + r1 (y1 ) +

|I (A, B, T )| ≤ C max{|A|1/3 , |B|1/2 }−1/2 . Proof. It will be convenient for the proof to call a constant C admissible if it depends only on the constants ε, , b(0, 0) and c, c1 , c2 from the statement of the lemma. All constants C appearing within the proof will be admissible but may change from line to line. We begin by the observation that the second assumption in (2.1) implies also that      (2.3) c2 |y2 |3−j ≤ ∂yj2 b(y1 , τy2 )y23 , j = 1, 2, |y2 | < τ −1 ,

35

AUXILIARY RESULTS

for every τ > 0. Indeed, if we fix y1 and put ψτ (y2 ) := b(y1 , τy2 )y23 , then (j ) (j ) ψτ (y2 ) = ψ1 (τy2 )τ −3 , so that ψτ (y2 ) = ψ1 (τy2 )τ j −3 . This immediately leads to (2.3). Let us first assume that |A| ≥ L ≥ 1. Without loss of generality, we may assume that A, B ≥ 0 (if necessary, we may change the signs of r or b, q as well as of y2 ). We may then choose α, β ≥ 0 so that A = α 3 , B = β 2 . Next, by convolving (1 + | · |)−N with a suitable smooth bump function, we may choose a smooth, nonnegative function ρ on R that is integrable and such that its Fourier transform is also integrable and so that (1 + |x|)−N ≤ ρ(x) ≤ 2(1 + |x|)−N and put T ∞    y   y2  3 2 ρ α3 − β 2 + Q J (α, β, T ) := y2 − b y1 , y2 T T −T ∞

+ r1 (y1 ) +

 y2 r2 (y1 ) (1 + |y1 |)−N |y2 | dy1 dy2 . T

It then suffices to prove that |J (α, β, T )| ≤ C max{α, β}−1/2

(2.4)

whenever L1/3 ≤ α ≤ T , 0 ≤ β ≤ T (recall (2.2)). To this end, performing the change of variables y2 = αs, we rewrite ∞ T /α J (α, β, T ) = α





ρ α

1+ ∞ −T /α

+ r1 (y1 ) + s

3





1 α  s 1 − γ + 2Q α T

 α  3 s − b y1 , s s T

 −N α |s| ds dy1 , r2 (y1 ) 1 + |y1 | T

where γ :=

 β 2 α

(2.5)

.

Case 1. γ ≥ 1, that is, β ≥ α. Changing variables s = γ 1/2 t, we rewrite ∞ T /β J (α, β, T ) = β





ρ α −β

1+

3

∞ −T /β

+ r1 (y1 ) + t

3



  β β 1 t t − b y1 , t t 3 1+ 2Q β T T

 β r2 (y1 ) (1 + |y1 |)−N |t| dt dy1 . T

Now, there are admissible constants C1 , C2 ≥ 1 so that if |t| ≥ C1 , then 



 

 3  β β 1 α − β 3 t t − b y1 , t t 3  ≥ C2 β 3 |t|3 . 1 + 2Q  β T T Notice also that |r1 (y1 ) + t Tβ r2 (y1 )| ≤ 2c(1 + |y1 |). Integrating separately in y1 over the sets where 1 + |y1 | ≤ C2 β 3 |t|3 /(4c) and where 1 + |y1 | > C2 β 3 |t|3 /(4c),

36

CHAPTER 2

one easily finds that the contribution JI (α, β, T ) by the region where |t| ≥ C1 to the integral J (α, β, T ) can be estimated by  1−N  ≤ 2Cβ 4+−3N . |JI (α, β, T )| ≤ Cβ 1+ β 3 )−N + (β 3 Assume next that |t| < C1 . We then choose χ ∈ C0∞ (R) such that χ (t) = 1 when |t| ≤ C1 and χ (t) = 0 when |t| ≥ 2C1 , with corresponding control of the derivatives of χ . The contribution by the region where |t| < C1 to the integral J (α, β, T ) can then be estimated by   ρ α 3 − β 3 φy1 (t) + r1 (y1 ) (1 + |y1 |)−N χ (t)|t| dt dy1 , JI I (α, β, T ) := β 1+ where we have set

 

 1 β β r2 (y1 ) t − 2 t − b y1 , t t 3 , φy1 (t) := 1 + 2 Q β T β T T

|t| ≤ C1 .

Recall here that T /β ≥ 1 (for 1 ≤ |t| ≤ 2C1 , we may extend the function b in a suitable way, if necessary). By Fourier inversion, this can be estimated by    3 3 1+  e−iξβ φy1 (t) χ (t)|t| dt eiξ(α +r1 (y1 )) |JI I (α, β, T )| ≤ Cβ    −N  × 1 + |y1 | ρ(ξ ˆ ) dξ dy1      −N 3   1+ |ρ(ξ ˆ )| dξ dy1 . ≤ Cβ  e−iξβ φy1 (t) χ (t)|t| dt  1 + |y1 | Now, if |r2 (y1 )/(β 2 T )| ≥ 12 , then c(1 + |y1 |) ≥ β 2 T /2 ≥ 1, so trivially the integration in y1 yields that |JI I (α, β, T )| ≤ Cβ 1+ (β 2 T )1−N ≤ Cβ 3+−2N for every N ∈ N. So, assume that |r2 (y1 )/(β 2 T )| < 12 . Then the phase φy1 (t) has no degenerate critical point on the support of χ (t) if we assume ε to be sufficiently small, since then b is a small perturbation of the constant function b(0, 0) in the sense of (2.1). It is then easily verified that our assumptions (in particular, (2.3)) imply that φy1 (t) satisfies the hypotheses of the van der Corput–type Lemma 2.1(ii), with M = 2 and constants c1 , c2 > 0, which are admissible provided we choose L sufficiently large. Therefore, the lemma shows that the inner integral with respect to t is bounded by C(1 + |ξ |β 3 )−1/2 , which implies that |JI I (α, β, T )| ≤ Cβ 1+ (β 3 )−1/2 = Cβ −1/2 . Case 2. γ < 1, that is, β < α. Then there are admissible constants C3 , C4 ≥ 1 so that if |s| ≥ C3 , then α 3 |1 − (γ + α12 Q( Tα s)) s − b(y1 , Tα s) s 3 | ≥ C4 α 3 |s|3 in (2.5). Arguing in a similar way as in the first case, this implies that the contribution JI I I (α, β, T ) by the region where |s| ≥ C3 to the integral J (α, β, T ) in (2.5) can be estimated by |JI I I (α, β, T )| ≤ Cα 4+−3N .

37

AUXILIARY RESULTS

Similarly, there are admissible constants C5 , C6 > 0 so that if |s| ≤ C5 , then |1 − (γ + α12 Q( Tα s)) s − b(y1 , Tα s) s 3 | ≥ C6 in (2.5), and this implies that the contribution JI V (α, β, T ) by the region where |s| ≤ C5 to the integral J (α, β, T ) can be estimated by |JI V (α, β, T )| ≤ Cα 4+−3N . 1) Finally, on the set where C5 < |s| < C3 , the phase φy1 (s) := ( rα2 (y 2 T − (γ + 1 α α 3 Q( T s)))s − b(y1 , T s)s has again no degenerate critical point, and we conclude α2 in a similar way as in the first case that the contribution JV (α, β, T ) by this region can be estimated by

|JV (α, β, T )| ≤ Cα 1+ (α 3 )−1/2 = Cα −1/2 . Combining all these estimates, we arrive at (2.4), which concludes the proof of the lemma when |A| ≥ L. Finally, when |A| ≤ L and |B| ≥ L, then we may indeed assume without loss of generality that α = 0. Arguments very similar to those that we have applied before then show that |J (α, β, T )| ≤ Cβ −1/2 , which concludes the proof of the lemma. Q.E.D. Lemma 2.5. Let b = b(y) be a C 2 -function on [−1, 1] such that b(0)  = 0 and bC 2 ([−1,1]) ≤ c1 . Furthermore, let A and B be real numbers, and let δ0 ∈]0, 1[. For T ≥ L and δ > 0 such that δ < 1 and δT ≤ δ0 , we put       ρ A + By + b(δy)y 3  χ0 y dy , I (A, B) := T R where ρ ∈ S(R) denotes a fixed Schwartz function and χ0 ∈ C0∞ (R), a nonnegative bump function supported in the interval [−1, 1]. Then, for δ0 sufficiently small and L sufficiently large, we have −1/2  , (2.6) |I (A, B)| ≤ C 1 + max{|A|1/3 , |B|1/2 } where the constant C depends only on c1 , δ0 , ρ, and χ0 but not on A, B, T , and δ. Proof. We may dominate the function |ρ| by a nonnegative Schwartz function. Let us therefore assume without loss of generality that ρ ≥ 0. Let us first assume that max{|A|, |B|} ≥ L. If |A| ≤ T 3 /δ03 and |B| ≤ T 2 /δ02 , estimate (2.6) follows easily from the previous Lemma 2.4 (replace χ0 (·/T ) by the characteristic function of the interval [−1/δ, 1/δ], choose r1 ≡ 0 and r2 ≡ 0, and put T := 1/δ in the previous lemma). Next, if |B| > T 2 /δ02 , using Fourier inversion, we may estimate    y    |I (A, B)| ≤ C  eisφ(y) χ0 dy  |ρ(s)| ˆ ds, (2.7) T where φ(y) := By + b(δy)y 3 . And, by means of integration by parts, we obtain that      T       φ (y)   1    +   eisφ(y) χ0 y dy  ≤ C|s|−1   φ  (y)2   T φ  (y)  dy.  T −T

38

CHAPTER 2

Since for |y| ≤ T we have  

  |B| δy   2  and |φ  (y)| ≤ CT , |φ (y)| = B + y 3b(δy) + b (δy)  ≥ 3 2 
  for δ0 if we choose δ0 sufficiently small, we see that  eisφ(y)χ
0 Ty dy  ≤ C/|sB|   sufficiently small. On the other hand, trivially we have  eisφ(y) χ0 Ty dy  ≤ CT , and taking the geometric mean of these estimates and using T < δ0 |B|1/2 , we find that       eisφ(y) χ0 y dy  ≤ C|s|−1/2 |B|−1/4 .   T  In combination with (2.7) this leads to the estimate |I (A, B)| ≤ C 1 + |B|)−1/4 . Next, if |A| > T 3 /δ03 and |B| ≤ |A|2/3 , then we may estimate |A + By + b(δy)y 3 | ≥ |A| − |B|T − b∞ T 3 ≥ |A| − |A|2/3 δ0 |A|1/3 − b∞ δ0 |A| ≥

|A| , 2

provided δ0 is sufficiently small. This implies that |I (A, B)| ≤ C|A|−N T ≤ C|A|−N+1/3 for every N ∈ N, which is stronger than what we need for (2.6). We have thus confirmed estimate (2.6) when max{|A|, |B|} ≥ L. Finally, if max{|A|, |B|} < L, then the rapid decay of ρ easily implies that |I (A, B)| ≤ C. Q.E.D.

2.4 FOURIER RESTRICTION VIA REAL INTERPOLATION In order to establish the restriction estimate (1.1) at the endpoint p = pc , we shall have to apply interpolation arguments in several situations. Indeed, our approach will be based on various kinds of dyadic frequency domain decompositions, and it turns out that the estimates for the operators corresponding to the dyadic frequency domains arising in this way do not sum at the endpoint. Classically, one uses complex interpolation methods, based on Stein’s interpolation theorem for analytic families of operators (Theorem 4.1 in [SW71]), as in the seminal work by Stein and Tomas on the Euclidean sphere (see [To75]). An alternative “classical” approach, which has turned out to be quite useful in order to obtain even mixed Lp -estimates in time and space for solutions to dispersive partial differential equations, is based on the Hardy-Littlewood-Sobolev inequality. This approach has been used in work by J. Ginibre and G. Velo [GV92] and further developed, for instance, by M. Keel and T. Tao [KT98]. However, it does not seem clear at all how to adapt the underlying method to our problem, so that we shall not pursue this question here, even though it might be interesting to investigate which mixed Lp -norm estimates do hold in our context. Yet another, alternative approach, based on real interpolation, has been devised more recently by J. G. Bak and A. Seeger [BS11], which yields an even stronger endpoint estimate in terms of Lorentz spaces (see Theorem 1.1 in [BS11]) and whose assumptions often can be verified quite easily.

39

AUXILIARY RESULTS

Regretfully, in the majority of situations that we shall encounter in the course of the proof of Theorem 1.14, it does not seem possible to make use of this result (respectively, the method of proof), so the last chapters of this monograph will be devoted to the study of various rather intricate analytic families of operators constructed by summing various pieces of dyadic decompositions that have been performed earlier, to which we shall apply Stein’s interpolation theorem. Nevertheless, in a few cases, we shall be able to make use of the following variant of Theorem 1.1 in [BS11], whose proof follows directly by means of the methods developed in [BS11]. What prevents us from applying Theorem 1.1 in [BS11] directly is that we shall encounter complex measures, whereas in the work by Bak and Seeger it is important to start with a positive measure. In the sequel, if µ is any bounded, complex Borel measure on Rd , then we shall often denote by Tµ the convolution operator Tµ : ϕ  → ϕ ∗ µ. ˆ Proposition 2.6. Let µ be a positive Borel measure on Rd of total mass µ1 ≤ 1, and let p0 ∈ [1, 2[. Assume that µ can be decomposed into a finite sum  µ = µb + µi i∈I b

of bounded complex Borel measures µ and µi , i ∈ I, such that the following hold true. There is a constant A ≥ 0 such that: 

(a) The operator Tµb is bounded from Lp0 (Rd ) to Lp0 (Rd ), with Tµb p0 →p0 ≤ A.

(2.8)

(b) Each of the measures µi decomposes as µi =

Ki 

µij =

j =1

Ki 

µ ∗ φji ,

j =1

where Ki ∈ N ∪ {∞}, and where the φji are integrable functions such that φji 1 ≤ 1.

(2.9)

Assume also that there are constants ai > 0, bi > 0 such that for all i and j, µij ∞ ≤ A2j ai ;

(2.10)

i  ≤ A2−j bi ; µ j ∞

(2.11)

p0 = 2 that is, if θ ai − (1 − θ )bi = 0, then

1 p0

ai + bi ; 2ai + bi =

θ 2

(2.12)

+ (1 − θ ).

Then there is a constant C that depends only on d and any given compact interval in ]0, ∞[ containing the ai and bi such that for every i, Tµi f Lp0 ≤ CAf Lp0 ,

(2.13)

40 and consequently

CHAPTER 2



|fˆ|2 dµ ≤ CAf 2Lp0 (Rd ) .

(2.14)

Proof. We shall denote by Lp,s the Lorentz space of type (p, s). By essentially following the proof of Proposition 2.1 in [BS11], we define the interpolation parameter θ := 2/p0 . Observe that by (2.12) we have θ = bi /(ai + bi ); hence (1 − θ )(−bi ) + θ ai = 0 for every i. Thus, the two inequalities (2.10) and (2.11) allow us to apply an interpolation trick due to Bourgain [Bou85] and to conclude that each of the operators Tµi is of restricted weak-type (p0 , p0 ), with operator norm ≤ CA. Moreover, if J is any compact subinterval of ]0, ∞[, then for ai , bi ∈ J we may chose the constant C so that it depends only on J. In combination with (2.8), this implies that also Tµ is of restricted weak-type (p0 , p0 ), with operator norm ≤ CA, where C may be different from the previous constant but with similar properties. By applying Tomas’ R ∗ R-argument for the restriction operator R, we get  |fˆ|2 dµ ≤ CAf 2Lp0 ,1 . In combination with Plancherel’s theorem and (2.10) and (2.9) we can next use this estimate as in [BS11] to control i 2 = cfˇµi 2 ≤ A2j ai φ i  CAf 2 p ,1 ≤ CA2 2j ai f 2 p ,1 . Tµij f 22 = f ∗ µ j 2 j 1 j 2 L 0 L 0 Here, we have denoted by fˇ the inverse Fourier transform of f. It is in this estimate that we make use of the positivity of the measure µ in an essential way. The remaining part of the argument in [BS11] does not require positivity of the underlying measure, so that it applies to each of the complex measures µi as well, and we may conclude that for any s ∈ [0, ∞], Tµi f Lp0 ,s ≤ CAf Lp0 ,s (compare Proposition 2.1, (2.2), in [BS11]). Choosing s = 2, so that p0 ≤ 2 ≤ p0 , by the nesting properties of the scale of Lorentz spaces this implies in particular that Tµi f Lp0 ,p0 ≤ CAf Lp0 ,p0 and, hence, (2.13). The same type of estimate then holds also for the operator Tµ , and Tomas’ argument then implies (2.14). Q.E.D.

2.5 UNIFORM ESTIMATES FOR FAMILIES OF OSCILLATORY SUMS The following simple lemma, which may also be of independent interest, will become crucial in several of our complex interpolation arguments, in order to control certain sums corresponding to the dyadic decompositions mention before in the previous section.

41

AUXILIARY RESULTS

n

Lemma 2.7. Let Q = j =1 [−Rk , Rk ] ⊂ Rn be a compact cuboid, with Rk > 0, k = 1, . . . , n, and let H be a C 1 -function on an open neighborhood of Q. Moreover, let α, β 1 , . . . , β n ∈ R× be given. For any given real numbers a1 , . . . , an ∈ R× and M ∈ N, we put F (t) :=

M 

 1 n 2iαlt (H χQ ) 2β l a1 , . . . , 2β l an .

(2.15)

l=0

Then there is a constant C depending on Q and the numbers α and β k but not on H, the ak , M, and t, such that H C 1 (Q) , for all t ∈ R, a1 , . . . , a2 ∈ R× and M ∈ N. (2.16) |F (t)| ≤ C iαt |2 − 1| Proof. For y = (y1 , . . . , yn ) in an open neighborhood of Q, Taylor’s integral for mula allows us to write H (y) = H (0) + nk=1 yk Hk (y), with continuous functions 1 Hk whose C 0 -norms on Q are controlled  by the C (Q)-norm of H. Accordingly, we shall decompose F (t) = F0 (t) + k Fk (t), where F0 (t) := H (0)

M 

 1 n 2iαlt χQ 2β l a1 , . . . , 2β l an ,

l=0

Fk (t) :=

M 

 1 n 2iαlt (yk Hk χQ ) 2β l a1 , . . . , 2β l an ,

k = 1, . . . , n.

l=0

It will thus suffice to establish estimates of the form (2.16) for each of these functions F0 and Fk , k = 1, . . . , n. We begin with F0 . Observe that in the sum defining F0 (t), we are effectively summing a geometric series over the integers contained in an interval, l ∈ {M1 , M1 + 1, . . . , M2 − 1, M2 }, where M1 , M2 ∈ N depend on M, the ak s, and the β k s, and therefore F0 (t) = H (0)

2iα(M2 +1)t − 2iαM1 t . 2iαt − 1

This implies an estimate of the form (2.16) for F0 (t). Next, if k ≥ 1, then trivially  k 2β l |ak | ≤ CRk , |Fk (t)| ≤ C  {l:2β k l |ak |≤Rk }

by summing a geometric series. Again this implies an estimate of the form (2.16). Q.E.D. Remark 2.8. The estimate in (2.16) can be sharpened as follows: Assume that there are constants  ∈ ]0, 1] and Ck , k = 1, . . . , n, such that  1   ∂H  (sy) ds ≤ Ck |yk |−1 , for all y ∈ Q. (2.17)  ∂yk 0 Then, under the hypotheses of Lemma 2.7, there is a constant C depending on Q, the numbers α and β k and , but not on H, the ak , M, and t, such that  |H (0)| + k Ck |F (t)| ≤ C , for all t ∈ R, a1 , . . . , a2 ∈ R× and M ∈ N. |2iαt − 1|

42

CHAPTER 2

Indeed, Taylor’s integral formula and (2.17) imply that |yk Hk (y)| ≤ Ck |yk | , which is sufficient for us to reach a conclusion in a similar way as before. We shall also need the following analogue of Lemma 2.7 for oscillatory double sums. Its proof follows similar ideas but is technically more involved.  Lemma 2.9. Let Q = nj=1 [−Rk , Rk ] ⊂ Rn be a compact cuboid, with Rk > 0, k = 1, . . . , n, and let H be a C 2 -function on an open neighborhood of Q. Moreover, let α1 , α2 ∈ Q× and β1k , β2k ∈ Q such that the vectors (α1 , α2 ) and (β1k , β2k ) are linearly independent, for every k = 1, . . . , n, that is, α1 β2k − α2 β1k  = 0,

k = 1, . . . , n.

(2.18)

For any given real numbers a1 , . . . , an ∈ R× and M1 , M2 ∈ N, we then put F (t) :=

M2 M1  

 1 n n 1 2i(α1 m1 +α2 m2 )t (H χQ ) 2(β1 m1 +β2 m2 ) a1 , . . . , 2(β1 m1 +β2 m2 ) an .

m1 =0 m2 =0

(2.19) Then there is a constant C depending on Q and the numbers αi and βik , but not on H, the ak , M1 , M2 , and t, and a number N ∈ N× depending on the βik such that H C 2 (Q) , for all t ∈ R, a1 , . . . , a2 ∈ R× and M1 , M2 ∈ N, |ρ(t)| (2.20)  ˜ ρ(−νt), ˜ with where ρ(t) := N ν=1 ρ(νt) |F (t)| ≤ C

ρ(t) ˜ := (2iα1 t − 1)(2iα2 t − 1)

n 

k

k

k

k

(2i(α1 β2 −α2 β1 )t − 1).

k=1

Remark 2.10. For ζ ∈ C and 0 < θ < 1, let us put γ˜ (ζ ) := (2α1 ζ − 1)(2α2 ζ − 1)

n 

(2(α1 β2 −α2 β1 )ζ − 1)

k=1

and γθ (ζ ) :=

 γ˜ (ν(ζ − 1))γ˜ (−ν(ζ − 1)) γ˜ (ν(θ − 1))γ˜ (−ν(θ − 1)) 1≤ν≤N

(notice that for ν = 1, . . . , N, we have γ˜ (±ν(θ − 1))  = 0). Then γθ is a welldefined entire analytic function such that γθ (θ ) = 1. Moreover, for ζ in the complex strip  := {ζ ∈ C : 0 ≤ Re ζ ≤ 1}, this function is uniformly bounded and γθ (1 + it) = cθ ρ(t), so that     for all t ∈ R, a1 , . . . , a2 ∈ R× and M1 , M2 ∈ N γ (1 + it)F (t) ≤ C if F (t) is defined as in (2.19). Proof. The basic idea of the proof becomes most transparent under the additional assumption that also the vectors (β1k , β2k ), k = 1, . . . , n, are pairwise linearly independent, that is, β1l β2k − β1k β2l  = 0,

for all l  = k.

(2.21)

43

AUXILIARY RESULTS

We shall, therefore, begin with this case and later indicate the modifications needed for the general case. of Q, Taylor’s integral formula For y = (y1 , . . . , yn ) in an open neighborhood  allows us to write H (y) = H (0) + nk=1 yk Hk (y), with C 1 -functions Hk whose C 1 -norms on Q are controlled by the C 2 (Q)-norm of H. Similarly, by putting  hk (yk ) := Hk (0, . . . , 0, yk , 0, . . . , 0), we may decompose Hk (y) = hk (yk ) + {l:l=k} yl Hkl (y), with continuous functions Hkl whose C(Q)norms are controlled by the C 2 (Q)-norm of H. This allows to write   H (y) = H (0) + yk hk (yk ) + yk yl Hkl (y). k

l=k

Accordingly, we shall decompose F (t) = F0 (t) + F0 (t) := H (0)

M2 M1  

 k

Fk (t) +

 l=k

Fkl (t), where

 1 n n 1 2i(α1 m1 +α2 m2 )t χQ 2(β1 m1 +β2 m2 ) a1 , . . . , 2(β1 m1 +β2 m2 ) an ,

m1 =0 m2 =0

Fk (t) :=

M1 

M2 

k

k

2i(α1 m1 +α2 m2 )t (yk hk )(2(β1 m1 +β2 m2 ) ak )

m1 =0 m2 =0

 1 n n 1 × χQ 2(β1 m1 +β2 m2 ) a1 , . . . , 2(β1 m1 +β2 m2 ) an ,

Fkl (t) :=

M2 M1  

 1 n n 1 2i(α1 m1 +α2 m2 )t (yk yl Hkl χQ ) 2(β1 m1 +β2 m2 ) a1 , . . . , 2(β1 m1 +β2 m2 ) an .

m1 =0 m2 =0

It will therefore suffice to establish estimates of the form (2.20) for each of these functions F0 , Fk and Fkl . We begin with F0 . We may choose r ∈ N× so that every βik can be written as βik = pik /r, with pik ∈ Z. Let us assume that there is a least one β2k  = 0 (otherwise, we find some β1k  = 0 and may proceedwith the roles of the indices i = 1 and i = 2 interchanged). We then let p2 := | k:p2k =0 p2k | and qk := (p1k p2 )/p2k whenever p2k  = 0, so that qk ∈ Z. Observe next that we may write every m1 ∈ N uniquely in the form m1 = α + j1 p2 , with α ∈ {0, . . . , p2 − 1} and j1 ∈ Z. This allows us to decompose F0 (t) = p2 −1 α α α=0 F0 (t), where F0 (t) is defined like F0 (t), but the summation in m1 is restricted to those m1 that are congruent to α modulo p2 . Next, an easy computation shows that if β2k  = 0, then β1k (α + j1 p2 ) + β2k m2 = β1k α + β2k (m2 + qk j1 ). Therefore, if we write Rk /ak = (sgn ak )2bk , then the restriction imposed by χQ on the coordinate of k leads to the condition β1k α + β2k (m2 + qk j1 ) ≤ bk . This means that m2 lies in an “interval” of the form {0, . . . , dk − qk j1 } or {ek − qk j1 , . . . , M2 } (depending on the sign of β2k ) for every k such that β2k  = 0 (by an interval we mean here the set of integer points within a real interval). We may, therefore, decompose the set of j1 s over which we are summing into a finite number of (at most (n!)2 ) pairwise disjoint intervals Js such that for each

44

CHAPTER 2

given s there are indices ks , ks such that for j1 ∈ Is , m2 will run through an interval of the form {es − us j1 , . . . , ds − vs j1 }, {0, . . . , ds − vs j1 } or {es − us j1 , . . . , M2 }, where es := eks , us := qks and ds := dks , vs := qks . We may thus reduce ourselves to considering, for each fixed s, the corresponding part Fsα of F α given by summation over the interval Is , that is,   Fsα (t) := H (0) 2i(αα1 +p2 α1 j1 +α2 m2 )t {j1 ∈Is :0≤α+j1 p2 ≤M1 } m2 ∈Is



×

k

χ[−Rk ,Rk ] (2β1 (α+j1 p2 ) ak ).

{k:β2k =0}

We may split the sum over m2 ∈ Is into the difference of at most two sums, in which we either sum over m2 ∈ {0, . . . , M2 }, or over m2 ∈ {0, . . . , fs − ws j1 }, with fs = ds and ws = vs or fs = es − 1 and ws = us . Let us look at the latter case, assuming that fs = es − 1 and ws = us . Evaluating the corresponding geometric sum in m2 , we see that the corresponding contribution is given by 

α (t) = H (0) Fs,1



2iαα1 t

{j1 ∈Is :0≤α+j1 p2 ≤M1 }

×



2i(α2 es +j1 (α1 p2 −α2 us ))t − 2ip2 α1 j1 t 2iα2 t − 1

k

χ[−Rk ,Rk ] (2β1 (α+j1 p2 ) ak ).

{k:β2k =0}

An analogous term arises when fs = ds and ws = vs . But, by assumption (2.18), α1 p2 − α2 us = (α1 β2ks − α2 β1ks )nks  = 0, where nks := p2 /β2ks ∈ Z× , and the characteristic functions of the intervals [−Rk , Rk ] again localize the summation over the j1 s to the summation over some interval, which shows that we may estimate α (t)| ≤ |Fs,1

C |2iα2 t

−

1||2iα1 p2 t

−

ks ks 1||2i(α1 β2 −α2 β1 )nks t

− 1|

≤

C . |ρ(t)|

The case where we sum over m2 ∈ {0, . . . , M2 } is even easier to treat, and we can α (t) to Fsα (t) by the right-hand again estimate the corresponding contribution Fs,2 side of the preceding inequality. This establishes the desired estimate for F0 (t). We next turn to Fk (t). Given k, let us assume again without loss of generality that β2k  = 0. Then we may write m1 , m2 ∈ Z in a unique way as m1 = α + j1 p2k , m2 = j2 − j1 p1k ,

with

α ∈ {0, . . . , |p2k |},

(2.22)

with integers j1 , j2 ∈ Z. Observe that then β1l m1 + β2l m2 = β1l α + j2 β2l + j1 (β1l β2k − β2l β1k )r, α1 m1 + α2 m2 = α1 α + j2 α2 + j1 (α1 β2k − α2 β1k )r. In particular, β1k m1 + β2k m2 = β1k α + j2 β2k does not depend on j1 . Moreover, for l l given α and j2 , the localizations given by the conditions |2(β1 m1 +β2 m2 ) al | ≤ Rl , l  = k, reduce the summation over j1 to the summation over an interval I (α, j2 ),

45

AUXILIARY RESULTS

and summing a geometric sum with respect to j1 , we thus see that |Fk (t)| ≤

k

C i(α1 β2k −α2 β1k )rt

|2

− 1|

|p2 | 



α=0

{j2 :|2β1 α+j2 β2 ak |≤Rk }

k

k

k

|2β1 α+j2 β2 ak | ≤

k

CRk . |ρ(t)|

Consider finally Fkl (t), for k  = l. We may simply estimate  k k l l |2(β1 m1 +β2 m2 ) ak | |2(β1 m1 +β2 m2 ) al |, |Fkl (t)| ≤ C (m1 ,m2 )∈Jk,l k

k

where Jkl is the set of all (m1 , m2 ) ∈ N2 satisfying |2(β1 m1 +β2 m2 ) ak | ≤ Rk and l l in |2(β1 m1 +β2 m2 ) al | ≤ Rl . By comparing with an integral and changing

k variables  β1 β2k the integral (recall that by our assumption (2.21) the matrix is nonβ1l β2l degenerate) this leads to the estimate  k k l l  |Fkl (t)| ≤ C |2(β1 s1 +β2 s2 ) ak | |2(β1 s1 +β2 s2 ) al | ds1 ds2 ≤C





Ik,l

log2 (Rl /|al |)

−∞



log2 (Rk /|ak |)

−∞

|2x1 ak ||2x2 al | dx1 dx2 ≤ CRk Rl , k

k

where Ikl denotes the set of all (s1 , s2 ) ∈ R2+ satisfying |2(β1 s1 +β2 s2 ) ak | ≤ Rk and l l |2(β1 s1 +β2 s2 ) al | ≤ Rl . This concludes the proof of the lemma under our additional hypotheses (2.21). Let us finally indicate how to remove assumptions (2.21). To this end, let us write j j β j := (β1 , β2 ). In the general case, we may decompose the index set {1, . . . , n} into pairwise disjoint subsets I1 , . . . , Ih such that the following hold true. There are nontrivial vectors γ k = (γ1k , γ2k ), k = 1, . . . , h, in Q2 and rational numbers rj  = 0, j = 1, . . . , n, such that (a) if j ∈ Ik , then β j = rj γ k ; (b) for k  = l, the vectors γ k and γ l are linearly independent. Let us accordingly define the vectors Yk := (yj )j ∈Ik ∈ RIk , k = 1, . . . , h. We may assume (possibly after a permutation of coordinates) that y = (Y1 , . . . , Yh ). Following h t the first step of the previous proof, weIk then decompose H (y) = H (0) + k=1 Yk · Hk (y), where now Hk maps into R . Next, we set hk (Yk ) := Hk (0, . . . , 0, Yk , 0, . . . , 0) ∈ RIk and apply Taylor’s formula in order to write H (y) = H (0) +

h  k=1

t

Yk · hk (Yk ) +



t

Yk · Hkl (y) · Yl ,

k=l

where here Hkl is a matrix-valued function. Correspondingly, we define the functions F0 (t), Fk (t) and Fkl (t) as before, only with yk hk (yk ) replaced by t Yk · hk (Yk ) and yk yl Hkl (y) by t Yk · Hkl (y) · Yl , respectively. Notice that terms t Yk · hk (Yk ) arise only when h ≥ 2.

46

CHAPTER 2

The discussion of F0 (t) remains unchanged, and the same applies essentially also to the discussion of Fkl (t), because of property (b). Finally, for the estimation of Fk (t), notice that for a given, fixed k, if j ∈ Ik , then by (a) we see that the arguk k ments at which t Yk · hk is evaluated are all of the form 2rj (γ1 m1 +γ2 m2 ) aj . Therefore, in the coordinates given by α, j1 , j2 from (2.22), they all will not depend on j1 . We may therefore proceed in the estimation of Fk (t) essentially as before, which concludes the proof of Lemma 2.9, also in the general case. Q.E.D. 2.6 NORMAL FORMS OF φ UNDER LINEAR COORDINATE CHANGES WHEN hlin (φ) < 2 Our discussion of the case where h lin (φ) < 2 will be based on normal forms of φ under linear coordinate changes. In the analytic setting, such normal forms are due to Siersma [Si74], but we shall need them also for smooth, finite-type φ. The designation of the type of singularity that we list next corresponds to Arnol’d’s classification of singularities in the case of analytic functions (cf. [AGV88] and [Dui74]), that is, in the analytic case, nonlinear analytic changes of coordinates would allow to further reduce φ to Arnol’d’s normal forms. Proposition 2.11. Assume that h lin (φ) < 2, where φ satisfies Assumption (NLA). Then, after applying a suitable linear change of coordinates, φ can be written in the following form on a sufficiently small neighborhood of the origin: φ(x1 , x2 ) = b(x1 , x2 )(x2 − ψ(x1 ))2 + b0 (x1 ),

(2.23)

where b, b0 and ψ are smooth functions and where ψ(x1 ) = cx1m + O(x1m+1 ), with c  = 0 and m ≥ 2. Moreover, we can distinguish two cases: Case a. b(0, 0)  = 0. Then either (i) b0 is flat (singularity of type A∞ ), or (ii) b0 (x1 ) = x1n β(x1 ), where β(0)  = 0 and n ≥ 2m + 1 (singularity of type An−1 ). In these cases we say that φ is of type A. Case b. b(0, 0) = 0. Then we may assume that b(x1 , x2 ) = x1 b1 (x1 , x2 ) + x22 b2 (x2 ), where b1 and b2 are smooth functions, with b1 (0, 0)  = 0. Moreover, either (singularity of type D∞ ), (i) b0 is flat or (ii) b0 (x1 ) = x1n β(x1 ), where β(0)  = 0 and n ≥ 2m + 2. (singularity of type Dn+1 ). In these cases we say that φ is of type D.

47

AUXILIARY RESULTS

Remark 2.12. (a) It is easy to see that the principal weight κ and the Newton distance d = d(φ) for these normal forms are given by

κ=

κ=

1 1 , 2m 2

 and

1 m , 2m + 1 2m + 1

d=

2m , m+1

if φ is of type A,

d=

2m + 1 , m+1

if φ is of type D,

 and

and by Proposition 1.7 that h lin (φ) = d, that is, that the coordinates x are linearly adapted. (b) Similarly, the coordinates y1 := x1 , y2 := x2 − ψ(x1 ) are adapted to φ, and we can choose ψ as the principal root jet. Comparing this with (1.9), we see that here the leading coefficient b1 of the principal root jet ψ is given by the constant c. (c) When φ has a singularity of type A∞ or D∞ and satisfies Condition (R), then necessarily b0 ≡ 0. Proof. If D 2 φ(0, 0) had full rank 2, then the principal part φpr of φ would be a non-degenerate quadratic form, and by Proposition 1.2 one would easily see that the coordinates x would already be adapted to φ. This would contradict our assumppart tions. Therefore, rank D 2 φ(0, 0) ≤ 1. Let us denote by Pn the homogeneous  j of degree n of the Taylor polynomial of φ, that is, Pn (x1 , x2 ) = j +k=n cj k x1 x2k . Case 1. rankD 2 φ(0, 0) = 1. In this case, by passing to a suitable linear coordinate system, we may assume that P2 (x1 , x2 ) = ax22 , where a  = 0. Consider the equation ∂2 φ(x1 , x2 ) = 0. By the implicit function theorem, locally it has a unique, smooth solution x2 = ψ(x1 ), that is, ∂2 φ(x1 , ψ(x1 )) = 0. A Taylor series expansion of the function φ(x1 , x2 ) with respect to the variable x2 around ψ(x1 ) then shows that φ(x1 , x2 ) = b(x1 , x2 )(x2 − ψ(x1 ))2 + b0 (x1 ), where b and b0 are smooth functions and b(0, 0) = 12 ∂22 φ(0, 0) = a  = 0, whereas b0 (x1 ) = O(x12 ), since φ(0, 0) = 0, ∇φ(0, 0) = 0 (this is a special instance of what would follow from a classical division theorem; see, for example, [H90]). Now, either b0 is flat, which leads to type A∞ , or otherwise we may write b0 (x1 ) = x1n β(x1 ), where β(0)  = 0 and n ≥ 2, which leads to type An−1 . Observe also that the function ψ cannot be flat, for otherwise the Newton polyhedron of φ would be the set (0, 2) + R2+ , in case that b0 is flat, or its principal edge would be the compact line segment with vertices (0, 2) and (n, 0). In the latter case, the principal part of φ is given by φpr (x1 , x2 ) = ax22 + β(0)x1n , so that the maximal multiplicity n(φpr ) of any real root of φpr along the unit circle is at most 1, whereas the Newton distance is given by d = 1/( 12 + n1 ) ≥ 1. Therefore, in both cases, the coordinates x would already be adapted to φ, according to Proposition 1.2. Notice also that the same argument shows that the coordinates y

48

CHAPTER 2

introduced in (1.10) are adapted to φ, so that, in particular, h = 2 (in case that b0 is flat) or h = 1/( 12 + n1 ) < 2 (if b0 (x1 ) = x1n β(x1 )). In particular, since ψ(0) = 0, we can write ψ(x1 ) = cx1m + O(x1m+1 ) for some m ∈ N× , where c  = 0. Note that indeed m ≥ 2, since P2 (x1 , x2 ) = ax22 . Finally, when b0 (x1 ) = x1n β(x1 ), a similar reasoning as before shows that the coordinates x are already adapted if 2m ≥ n, so that under Assumption 1.6 we must have n ≥ 2m + 1. Case 2. D 2 φ(0, 0) = 0. Then P2 = 0, and P3  = 0, for otherwise we had h lin ≥ d ≥ 1/( 14 + 14 ) = 2, which would contradict our assumption that h lin < 2. Notice also that P3  = 0 is homogeneous of odd degree 3, so that necessarily the multiplicity of roots (cf. (1.7)) satisfies n(P3 ) ≥ 1. Assume first that n(P3 ) = 1. Then, passing to a suitable linear coordinate system, we may assume that P3 (x1 , x2 ) = x1 (x2 − αx1 )(x2 − βx1 ), where either α  = β are both real or α = β are nonreal. Then one checks easily that the Newton diagram of P3 is a compact edge intersecting the bisectrix in its interior and contained in the line given by 13 t1 + 13 t2 = 1. Consequently, it agrees with the principal face π(φ), so that P3 = φpr . We thus find that the Newton distance d in this linear coordinate system satisfies d = 3/2 > n(φpr ), so that these coordinates would already be adapted, contradicting our assumptions. Assume next that n(P3 ) = 3. Then, in a suitable linear coordinate system, P3 (x1 , x2 ) = x23 . These coordinates are then adapted to P3 , so that h(P3 ) = d(P3 ) = 3 > 2. However, as has been shown in [IKM10], page 217, under Assumption 1.6 this implies that the Taylor support of φ is contained in the region where 16 t1 + 13 t2 ≥ 1. This, in turn, implies that h lin ≥ d ≥ 1/( 16 + 13 ) = 2, in contrast to what we assumed. We have thus seen that, necessarily, n(P3 ) = 2. Then, after applying a suitable linear change of coordinates, we may assume that P3 (x1 , x2 ) = x1 x22 , that is, φ(x1 , x2 ) = x1 x22 + R(x1 , x2 ), where R is a smooth function such that ∂ α R(0, 0) = 0 for |α| ≤ 3. Consider here the equation ∂2 φ(x1 , x2 ) = 0 with respect to x2 . We claim that it has a smooth solution x2 = ψ(x1 ), with ψ(0) = ψ  (0) = 0, near the origin. Indeed, we have ∂2 φ(x1 , x2 ) = 2x1 x2 + R1 (x1 , x2 ), where R1 is a smooth function such that ∂ α R1 (0, 0) = 0 for |α| ≤ 2, so that we have to solve the equation 2x1 x2 + R1 (x1 , x2 ) = 0.

(2.24)

To this end, let us write, for x1  = 0, x2 = x1 z. Then (2.24) is equivalent to 2x12 z + R1 (x1 , x1 z) = 0.

(2.25)

Clearly, by the properties of R1 , we may factor R1 (x1 , x1 z) = x13 g(x1 , z), with a smooth function g(x1 , z) defined near the origin, and thus for x1  = 0,

49

AUXILIARY RESULTS

equation (2.25) is equivalent to 2z + x1 g(x1 , z) = 0. Regard this as an equation near the origin in (x1 , z). We can now apply the implicit function theorem to conclude that locally near the origin this equation has a unique, smooth solution z = ψ1 (x1 ). In particular, we find that 2x12 ψ1 (x1 ) + R1 (x1 , x1 ψ1 (x1 )) = 0 near the origin in x1 . Setting ψ(x1 ) := x1 ψ1 (x1 ), we then find that indeed ψ(0) = ψ  (0) = 0 and ∂2 φ(x1 , ψ(x1 )) ≡ 0.

(2.26)

By means of a Taylor expansion of the function φ(x1 , x2 ) with respect to the variable x2 around x2 = ψ(x1 ), this implies that φ(x1 , x2 ) = b(x1 , x2 )(x2 − ψ(x1 ))2 + b2 (x1 )x2 + b0 (x1 ), where b, b0 , and b2 are smooth functions. Again, we have that ψ(x1 ) = cx1m + O(x1m+1 ), with m ≥ 2. Observe that (2.26) implies that b2 = 0; hence, φ(x1 , x2 ) = b(x1 , x2 )(x2 − ψ(x1 ))2 + b0 (x1 ). Moreover, since ∂22 φ(0, 0) = 0, ∂1 ∂22 φ(0, 0)  = 0, ∂23 φ(0, 0) = 0, we have that b(0, 0) = 0,

∂1 b(0, 0)  = 0

and ∂2 b(0, 0) = 0.

By Taylor’s formula, this implies that b(x1 , x2 ) = x1 b1 (x1 , x2 ) + x22 b2 (x2 ), where b1 and b2 are smooth functions, with b1 (0, 0)  = 0. In a similar way as in Case 1, one can see that the coordinates from (1.10) are adapted to φ. Moreover, if b0 is flat, which leads to case D∞ , then h = 2, and 2n < 2. Finally, one if b0 (x1 ) = x1n β(x1 ), which leads to case Dn+1 , then h = n+1 also checks easily that the coordinates x in (1.10) are already adapted to φ, if 2m + 1 ≥ n, so that under our assumption we must have n ≥ 2m + 2. This concludes the proof of Proposition 2.11. Q.E.D. Corollary 2.13. Assume that φ satisfies Assumption (NLA). By passing to a suitable linear coordinate system, let us also assume that the coordinates x are linearly adapted to φ. Then, if d = d(φ) < 2, the critical exponent in Theorem 1.14 is given by pc = 2d + 2. Proof. Proposition 2.11 shows that the principal face π(φ) of the Newton polyhedron of φ is a compact edge whose “upper” vertex v is one the following points (0, 2) or (1, 2), which both lie below the line H := {(t1 , t2 ) : t2 = 3} within the positive quadrant. On the other hand, m + 1 ≥ 3. It is then clear from the geometry of the lines H , the line L that contains π(φ), and the line (m) that (m) will intersect L above the vertex v. Since, by Varchenko’s algorithm, the point v will also be a vertex of the Newton polyhedron of φ a , this easily implies that hr (φ) = d (compare Figure 1.4). This proves the claim. Q.E.D.

Chapter Three Reduction to Restriction Estimates near the Principal Root Jet We now turn to the proof of Theorem 1.14 (which includes Theorem 1.10). As a first step, we shall reduce considerations to a small, “horn-shaped” neighborhood of the principal root jet ψ (cf. Subsection 1.1.1). This reduction will be achieved by means of a dyadic decomposition of the complementary region, making use of the dilations associated to the principal face π(φ), in combination with Greenleaf’s restriction Theorem 1.1 and Littlewood-Paley theory, which will allow us to sum the estimates for the dyadic pieces that we obtain. It turns out that for the contribution by this complementary region, the restriction estimate (1.1) will hold true even for the possibly wider range of ps given by p
≥ 2d + 2. For the case where d = h lin (φ) > 2, this will be an easy consequence of the fact that every root of φpr that does not agree with the principal root x2 = b1 x1m1 does have multiplicity strictly less than d. The discussion of the case where d = h lin (φ) ≤ 2 will require more refined estimates for oscillatory integrals, based on the normal forms provided by Proposition 2.6. Recall that our coordinates x are assumed to be linearly adapted but not adapted to φ. By decomposing the plane R2 into two half-planes, we may and shall in the sequel always restrict ourselves to the closed right half plane where x1 ≥ 0, that is, we shall assume that the surface carried measure dµ = ρdσ is of the form
f (x, φ(x)) η(x) dx, f ∈ C0 (R3 ), µ, f  = x1 ≥0

 where η(x) := ρ(x, φ(x)) 1 + |∇φ(x)|2 is smooth and supported in a neighborhood  of the origin, which we may assume to be sufficiently small. The contribution by the other half plane where x1 ≤ 0 can be treated in an analogous way. If F is any integrable function defined on , we put
F F f (x, φ(x)) η(x)F (x) dx. µ := (F ⊗ 1)µ, i.e., µ , f  = x1 ≥0

Recall from (1.9) that ψ(x1 ) = b1 x1m + O(x1m+1 ). We choose a nonnegative  bump  function χ0 ∈ C0∞ (R) that is supported in [−1, 1] and identically 1 on − 12 , 12 , and put ρ1 (x1 , x2 ) := χ0

 x − b xm  2 1 1 , εx1m

51

REGIONS AWAY FROM THE PRINCIPAL ROOT JET

where ε > 0 is a small parameter to be determined later. Notice that ρ1 is supported in the κ-homogeneous subset of  ∩ H + where |x2 − b1 x1m | ≤ εx1m ,

(3.1)

which contains the curve x2 = ψ(x1 ) when  is sufficiently small. In this chapter, we shall prove the following result, which will allow us to reduce our considerations to a domain of the form (3.1). Proposition 3.1. Let ε > 0. If we choose the support of µ sufficiently small, then 1/2 
|f|2 dµ1−ρ1 ≤ Cp,ε f Lp (R3 ) , f ∈ S(R3 ), S

whenever p
≥ 2d + 2. In particular, this estimate is valid whenever p
≥ pc
. The strategy of the proof will, by and large, follow the one of the proof of Theorem 1.7 in [IM11b]. By {δr }r>0 we shall again denote the dilations associated to the principal weight κ. We fix a suitable smooth cutoff function χ ≥ 0 on R2 supported in a closed annulus A ⊂ R2 on which |x| ∼ 1 in such a way that the functions χk := χ ◦ δ2k form a partition of unity, and decompose the measure µ1−ρ1 dyadically as  µk , (3.2) µ1−ρ1 = k≥k0

where µk := µχk (1−ρ1 ) . Let us extend the dilations δr to R3 by putting δre (x1 , x2 , x3 ) := (r κ1 x1 , r κ2 x2 , rx3 ). We rescale the measure µk by defining νk := 2−k µk ◦ δ2e−k , that is,
|κ|k e f (x, φ k (x)) η(δ2−k x)χ (x)(1 − ρ1 (x)) dx, νk , f  = 2 µk , f ◦ δ2k  = x1 ≥0

with φ k (x) := 2k φ(δ2−k x) = φκ (x) + error terms of order O(2−δk ),

(3.3)

where δ > 0. Recall here that the principal part φpr of φ agrees with φκ . Let us denote by S k the smooth hypersurface which is defined as the graph of φ k . Then (3.3) shows that the measures νk are supported on S k , that their total variations are uniformly bounded, i.e., supk νk 1 < ∞, and that they are approaching the surface carried measure ν∞ on S defined by
f (x, φκ (x)) η(0)χ (x)(1 − ρ1 (x)) dx ν∞ , f  := x1 ≥0

as k → ∞. Following [IM11b], the key point in proving Proposition 3.1 will be to establish the following uniform estimates for the Fourier transforms of the measures νk . Lemma 3.2. If k0 ∈ N is sufficiently large, then there exists a constant C > 0 such that | νk (ξ )| ≤ C(1 + |ξ |)−1/d

for every ξ ∈ R3 , k ≥ k0 .

52

CHAPTER 3

Proof. Assume first that h lin = h lin (φ) ≥ 2. Then h(φ) > 2 by Assumption (NLA). Thus, in this case, the proof of Lemma 2.3 in [IM11b] shows that indeed the estimate in Lemma 3.2 holds true. The main argument used here is that every root of φpr that does not agree with the principal root x2 = b1 x1m1 must have multiplicity strictly less than d = d(φ), as can be seen from Corollary 2.3 in [IM11a]. We may, therefore, assume from now on that h lin < 2. Then we may even assume that φ is given by one of the normal forms appearing in Proposition 2.11 and that h lin = d is the Newton distance. Moreover the leading coefficient b1 of the principal root jet ψ is given by the constant c of Proposition 2.11. We shall work from now on with c in place of b1 in order to avoid possible confusion with the coefficient function b1 (x1 , x2 ). Let us rewrite
k e−i(ξ1 x1 +ξ2 x2 +ξ3 φ (x1 ,x2 )) η(δ2−k x)χ (x)(1 − ρ1 (x)) dx νk (ξ ) = x1 ≥0

and observe that, by a partition of unity argument, it will suffice to prove the following. Given any point v = (v1 , v2 ) in the compact set A ∩ H + such that v2 − cv1m  = 0,

(3.4)

there is neighborhood V of v such that for every bump function χv ∈ C0∞ (R2 ) supported in V we have |J χv (ξ )| ≤ C(1 + |ξ |)−1/d where χv




J (ξ ) :=

x1 ≥0

for every ξ ∈ R3 , k ≥ k0 ,

e−i(ξ1 x1 +ξ2 x2 +ξ3 φ

k

(x1 ,x2 ))

(3.5)

η(δ2−k x)χv (x) dx.

To prove this, we shall distinguish Cases a and b from Proposition 2.11. 1 1 Case a (φ of type A). In this case, we see that κ = ( 2m , 2 ) and

φκ (x1 , x2 ) = φpr (x1 , x2 ) = b(0, 0)(x2 − cx1m )2 , so that 1/d = 1/2 + 1/2m. After applying a suitable linear change of coordinates (and possibly complex conjugation to the integral J χv (ξ )), we may assume that b(0, 0) = 1. Then the Hessian determinant of φκ is given by Hess(φκ )(x1 , x2 ) = −4m(m − 1)cx1m−2 (x2 − cx1m ). Therefore, by (3.4), if m = 2, or v1  = 0, then Hess(φκ )(v)  = 0. In this case, in view of (3.3) we can apply the method of stationary phase for phase functions depending on small parameters and easily obtain |J χv (ξ )| ≤ C(1 + |ξ |)−1

for every ξ ∈ R3 , k ≥ k0 ,

provided V is sufficiently small and k0 sufficiently large. Since d ≥ 1, this yields (3.5).

53

REGIONS AWAY FROM THE PRINCIPAL ROOT JET

We are left with the case where m > 2 and v1 = 0. Since v = (v1 , v2 ) ∈ A, this implies that v2  = 0. Putting φ˜ k (y1 , y2 ) := φ k (y1 , v2 + y2 ), we may rewrite J χv (ξ ) as


˜k J χv (ξ ) = e−iv2 ξ2 e−i(ξ1 y1 +ξ2 y2 +ξ3 φ (y1 ,y2 )) η δ2−k (y1 , v2 + y2 ) χ˜ 0 (y) dy, y1 ≥0

where χ˜ 0 is now supported in a sufficiently small neighborhood of the origin. But, φ˜ k (y1 , y2 ) = (v2 + y2 − cy1m )2 + O(2−δk )

= v22 + 2v2 y2 + y22 − 2cv2 y1m + c2 y12m − 2cy2 y1m + O(2−δk ). The main nonlinear term here is (y22 − 2cv2 y1m ), which shows that the phase has a singularity of type Am−1 in the sense of Arnol’d. By means of a linear change of variables in ξ -space, which replaces ξ2 + 2v2 ξ3 by ξ2 , we may thus reduce this to assuming that the complete phase in the oscillatory integral J χv (ξ ) is given by

ξ1 y1 + ξ2 y2 + ξ3 y22 − 2cv2 y1m + c2 y12m − 2cy2 y1m + O(2−δk ) . We claim that |J χv (ξ )| ≤ C(1 + |ξ |)−(1/2+1/m)

for every ξ ∈ R3 , k ≥ k0 ,

which is even stronger than (3.5). Indeed, if |ξ3 |  max{|ξ1 |, |ξ2 |}, then this follows easily by integration by parts, so let us assume that |ξ3 | ≥ M max{|ξ1 |, |ξ2 |} for some constant M > 0. Then |ξ3 | ∼ |ξ |. Consequently, by first applying the method of stationary phase to the integration in y2 and then van der Corput’s estimate to the y1 integration, we obtain the preceding estimate. Observe here that these types of estimates are stable under small, smooth perturbations. 1

m Case b (φ of type D). In this case, we see that κ = 2m+1 and , 2m+1 φκ (x1 , x2 ) = φpr (x1 , x2 ) = b1 (0, 0)x1 (x2 − cx1m )2 , so that 1/d = (m + 1)/(2m + 1). Again, we may assume without loss of generality that the coefficient function b1 (x1 , x2 ) satisfies b1 (0, 0) = 1, so that φκ (x1 , x2 ) = x1 x22 − 2cx1m+1 x2 + c2 x12m+1 . Straightforward computations show that ∂12 φκ (x) = −2cm(m + 1)x1m−1 x2 + c2 2m(2m + 1)x12m−1 , ∂1 ∂2 φκ (x) = 2x2 − 2c(m + 1)x1m , hence,

∂22 φκ (x) = 2x1 ;

Hess(φκ )(v) := −4(x2 − cx1m ) x2 + c(m2 − m − 1)x1m .

In view of (3.4), we see that Hess(φκ )(v)  = 0, if v2 + c(m2 − m − 1)v1m  = 0, so that we can again estimate J χv (ξ ) by means of the method of stationary phase.

54

CHAPTER 3

Let us therefore assume that Hess(φκ )(v) = 0, that is, v2 = −c(m2 − m − 1)v1m .

(3.6)

Observe that then v1  = 0, v2  = 0. Denote by  1 ∂ α φκ (v)y α Pj (y) := α! |α|=j the homogeneous Taylor polynomial of φκ of degree j, centered at v. Then clearly 

2 (v2 − c(m + 1)v1m )y1 2 P2 (y) = v1 y2 + = v1 y2 − cm2 v1m−1 y1 . v1 Moreover, by (3.6)

P3 (y) = −y1 13 c2 m2 (m3 − m2 + 2m + 1)v12m−2 y12 − cm(m + 1)v1m−1 y1 y2 + y22 = −y1 Q(y). Passing to the linear coordinates z 1 := y1 , z 2 := y2 − cm2 v1m−1 y1 , one finds that P2 = v1 z 22 ,

˜ P3 = −z 1 Q(z),

˜ = z 2 + 2β1 z 1 z 2 + β2 z 2 is again a quadratic form. Moreover, straightwhere Q 2 1 forward computations show that c2 2 m (m − 1)(m2 − 1)v12m−2  = 0. 3 Applying Taylor’s formula, we thus find that, in the coordinates z, β2 =

˜ φ(z) : = φκ (v1 + y1 , v2 + y2 ) = c0 + c1 z 1 + c2 z 2 + (v1 z 22 − β2 z 13 ) −(z 1 z 22 + 2β1 z 12 z 2 ) + O(|z|4 ). Let us put φ v (z) := φ(z) − (c0 + c1 z 1 + c2 z 2 ), so that φ v (0, 0) = 0, ∇φ v (0, 0) = 0. Then one finds that the principal part of φ v is given by v (z) = v1 z 22 − β2 z 13 , φpr

where β2  = 0.

We can now argue in a very similar way as in the previous case. Indeed, by passing from the variables (x1 , x2 ) in the integral defining J χv (ξ ) to the variables (z 1 , z 2 ), and then applying first the method of stationary phase to the integration in z 2 , and subsequently van der Corput’s estimate to the z 1 integration (in the case where |ξ3 | ≥ M max{|ξ1 |, |ξ2 |}), we obtain the estimate |J χv (ξ )| ≤ C(1 + |ξ |)−(1/2+1/3)

for every ξ ∈ R3 , k ≥ k0 .

Again, this is a stronger estimate than (3.5), since here 1 1 1 1 1 = + ≤ + . d 2 4m + 2 2 3

Q.E.D.

We can now conclude the proof of Proposition 3.1. According to Greenleaf’s Theorem 1.1, the estimates in Lemma 3.2 imply the restriction estimates 
1/2 |fˆ(x)|2 dνk (x) ≤ Cf p , f ∈ S(R3 ), (3.7)

55

REGIONS AWAY FROM THE PRINCIPAL ROOT JET

if p
≥ 2d + 2, and the proof of Theorem 1 in [Gl81] reveals that the constant C can be chosen independently of k. Let us rescale these estimates by putting f(r) (x) := r |κ|/2 f (δre x),

r > 0,

◦ δre−1 , and (3.7) implies for any function f on R3 . Then f(r) = r −|κ|/2−1 f


|fˆ(x)|2 dµk (x) = |f(2−k ) (x)|2 dνk (x) ≤ C 2 2(|κ|/2+1)k f ◦ δ2ek 2p ; hence,




|fˆ(x)|2 dµk (x) ≤ C 2 f 2p ,

(3.8)

with a constant C that does not depend on k. Fix a cutoff function χ˜ ∈ C0∞ (R2 ) supported in an annulus centered at the origin such that χ˜ = 1 on the support of χ , and define dyadic decomposition operators 
k by


˜ (δ2k x
) fˆ(x
, x3 ).  k f (x) := χ 


f (x)|2 dµk (x), so that (3.8) yields Then |fˆ(x)|2 dµk (x) = | k
2


|fˆ(x)|2 dµk (x) ≤ C 2  k f p ,

for any k ≥ k0 . In combination with Minkowski’s inequality, this implies  1/2  1/2 1/2 

 |fˆ(x)|2 dµk (x) ≤ C  = 
k f 2p  |fˆ(x)|2 dµ1−ρ1 (x) k≥k0

k≥k0

  1/p 1/2    2/p p/2 
          ≤ C  =C |
k f (x)|2   |
k f (x)|p dx   k≥k0  k≥k0  

,

Lp (R3 )

since p < 2. Thus, by Littlewood-Paley theory [S93], we obtain the estimate in Proposition 3.1. Proposition 3.1 shows that we are left with proving a Fourier restriction estimate for the measure µρ1 , which is supported in a small neighborhood of the form (3.1) of the principal root jet, that is, with the verification of the following. Proposition 3.3. Assume that φ satisfies the assumptions of Theorem 1.14. If ε > 0 is sufficiently small, then we have 1/2 
|f|2 dµρ1 ≤ Cp,ε f Lp (R3 ) , f ∈ S(R3 ), S

whenever p
≥ pc
.

56

CHAPTER 3

Notice that by interpolation with the trivial L1 -L2 -restriction estimate, it will suffice to prove this estimate for the endpoint p = pc . We shall distinguish between the cases where h lin < 2 and h lin ≥ 2 since their treatments will require somewhat different approaches. For h lin ≥ 2, our ultimate approach will require many different steps and rather intricate interpolation arguments. Luckily enough, a substantially simpler approach is available for h lin ≥ 5, which is based on restriction estimates for curves with nonvanishing torsion originating from the seminal work by S. W. Drury [Dru85]. Since the discussion of this approach will allow us to explain some of the basic ideas that will also be needed when h lin < 5 but in a less complicated setting, we shall first explain this simpler approach. The proof for the general case h lin ≥ 2 will finally occupy the last three chapters.

Chapter Four Restriction for Surfaces with Linear Height below 2 In this chapter and the following one we shall study the case where h lin (φ) < 2. We may—and shall—then assume that φ is given by one of the normal forms listed in Proposition 2.11. Recall also from Corollary 2.13 that for h lin (φ) < 2, we have pc
= 2d + 2. Moreover, since we are assuming Condition (R) holds, the term b0 in (2.23) vanishes identically if φ is of type A∞ or D∞ (cf. Remark 2.12(c)). In order to prove Proposition 3.3 in this case, in a first step we shall follow the approach of the previous chapter and perform a dyadic decomposition of the domain (3.1) and the corresponding measure dµρ1 by means of the dilations associated to the principal weight κ. Littlewood-Paley theory then again shows that it suffices to prove uniform Fourier restriction estimates for the corresponding dyadic constituents dµk of the measure dµρ1 , which, in return, are equivalent to similar estimates for a family of rescaled and normalized measures dνk (compare (4.1)). The latter measures will be supported where x1 ∼ 1 and |x2 | ∼ 1. Next, given a measure dνk , which will actually depend on small parameters δ1 , δ2 , δ3 that are fractional powers of 2−k (which is why we shall from then on denote these measure by dνδ , in place of dνk ), we shall perform yet another Littlewood-Paley decomposition, this time with respect to the variable x3 . This will allow us to restrict to subdomains on which |φ| ∼ 2−2j , j ≥ j0 . Then, depending on the size of 22j δ3 , we shall have to distinguish three different situations, and the size condition on |φ| will help us to obtain important additional information on the support of these further localized measures dνδ,j . Since the decay of the Fourier transforms of these measures is strongly nonisotropic, in a last step we shall perform a dyadic decomposition in each of the frequency variables ξ1 , ξ2 , and ξ3 dual to x1 , x2 , and x3 , which will lead to complex measures dνjλ into which the measure dνδ,j will be decomposed, with dyadic numbers λi = 2ki , i = 1, 2, 3, forming the vector λ = (λ1 , λ2 , λ3 ). Following Tomas’ T ∗ T -argument, we shall then estimate the norms of the operators Tjλ
of convolution with ν
λ , as operators from Lp to Lp and, finally, control the sum j

of the norms of these operators over all dyadic λ. This program will necessitate the distinction of various subcases. In the end, a few cases in which we cannot sum these operator norms will remain open, and we shall collect all these remaining cases in Proposition 4.2. In all these cases we will have that φ is of type A and m = 2. The treatment of these remaining open cases in the course of the proof of Proposition 4.2 will require even more refined methods, in particular, certain interpolation arguments, in order to capture the endpoint p = pc and in some cases a deeper

58

CHAPTER 4

analysis based on further frequency localizations in terms of the distance to certain “Airy cones.” All this will be carried out in Chapter 5.

4.1 PRELIMINARY REDUCTIONS BY MEANS OF LITTLEWOODPALEY DECOMPOSITIONS In a first step, we shall follow the arguments from the preceding chapter and decompose the measure µρ1 dyadically by means of the dilations associated to the principal weight κ. Applying subsequent rescalings, we may then reduce ourselves by means of Littlewood-Paley theory to proving a uniform estimate analogous to estimate (3.7) for the renormalized measures νk , k ∈ N, which are now given by |κ|k

νk , f  := 2

µk , f ◦

δ2ek 

 =

x1 ≥0

f (x, φ k (x)) η(δ2−k x)χ (x)ρ1 (x1 , x2 ) dx.

The functions φ k are again defined by (3.3). Observe that x1 ∼ 1 ∼ |x2 | in the support of the integrand. Recall also from (1.11) that φ(x1 , x2 ) = φ a (x1 , x2 − ψ(x1 )), where according to (1.9) we may write ψ(x1 ) = x1m ω(x1 ),

(m ≥ 2),

with a smooth function ω satisfying ω(0) = 0. What we then need to prove is that for ε > 0 sufficiently small, there are constants Cε > 0 and k0 ∈ N such that for every k ≥ k0  1/2 |f
|2 dνk ≤ Cε f Lpc (R3 ) , f ∈ S(R3 ), (4.1) with a constant Cε not depending on k. In order to prove this estimate, observe that by (3.3) and (2.23), our rescaled phase function φ k can be written in the form φ k (x) = φ(x, δ), with ˜ 1 , x2 , δ1 , δ2 )(x2 − x1m ω(δ1 x1 ))2 + δ3 x1n β(δ1 x1 ), φ(x, δ) := b(x

(4.2)

where we have put δ = (δ1 , δ2 , δ3 ) := (2−κ1 k , 2−κ2 k , 2−(nκ1 −1)k ). Notice that δ consists of small parameters that tend to 0 as k tends to infinity. Moreover, b˜ is a smooth function in all its variables, given by  b(δ1 x1 , δ2 x2 ), for φ of type A, ˜ 1 , x2 , δ1 , δ2 ) := (4.3) b(x x b (δ x , δ x ) + δ 2m−1 x 2 b (δ x ), for φ of type D. 1 1 1 1 2 2 2 2 2 1 2 Here we have used that δ2 = δ1m (cf. Remark 2.12). We shall indeed ignore the particular dependence of δ on k and regard φ(x, δ) as a smooth function of x and δ

59

SURFACES WITH LINEAR HEIGHT BELOW 2

for abitrary (sufficiently small) δj ≥ 0. Note that δ3 := 0 when φ is of type A∞ or D∞ . Notice also that ˜ 1 , x2 , 0, 0)| ∼ 1. |b(x It is thus easily seen by means of a partition of unity argument that it will suffice to prove the following proposition in order to verify (4.1). Proposition 4.1. Let φ(x, δ) be as in (4.2). Then, for every point v = (v1 , v2 ) such that v1 ∼ 1 and v2 = v1m ω(0), there exists a neighborhood V of v in H + such that for every cutoff function η ∈ D(V ), the measure νδ given by  νδ , f  := f (x, φ(x, δ)) η(x1 , x2 ) dx satisfies a restriction estimate 1/2  |f
|2 dνδ ≤ Cη f Lpc (R3 ) ,

f ∈ S(R3 ),

(4.4)

provided δ is sufficiently small, with a constant Cη that depends only on the C l -norm of η, for some l ∈ N. In order to prove this proposition, we shall perform yet another dyadic decomposition, this time with respect to the x3 -variable, in order to restrict ourselves to level ranges of the function φ(x, δ). A straightforward modification of the Littlewood-Paley argument at the end of Chapter 3 then allows us to reduce the proof to establishing uniform restriction estimates for the following family of measures:  (4.5) νδ,j , f  := f (x, φ(x, δ)) χ1 (22j φ(x, δ))η(x1 , x2 ) dx. Here, χ1 ∈ D(R) is again a suitable fixed, nonnegative smooth bump function supported in, say, the set (−2, − 12 ) ∪ ( 12 , 2) such that χ1 ≡ 1 in a neighborhood of the points −1 and 1. Notice that νδ,j is supported where |φ(x, δ)| ∼ 2−2j . That is, in place of (4.4), it will be sufficient to prove an analogous uniform estimate 

|f
|2 dνδ,j

1/2

≤ Cη f Lpc (R3 ) ,

f ∈ S(R3 ),

(4.6)

for all j ∈ N sufficiently large, say, j ≥ j0 , where the constant Cη depends neither on δ nor on j. Notice that by (4.2),  2 ˜ 1 , x2 , δ1 , δ2 ) x2 − x1m ω(δ1 x1 ) + 22j δ3 x1n β(δ1 x1 ), 22j φ(x, δ) = 22j b(x In order to verify (4.6), we shall, therefore, distinguish three cases, depending on the size of 22j δ3 . 4.1.1 The situation where 22j δ 3
1 Observe first that if j is sufficiently large, then by (4.2) and since x1 ∼ 1, νδ,j = 0 ˜ δ1 , δ2 ) and β(0) have opposite signs. So, let us, for instance, assume unless b(v,

60

CHAPTER 4

˜ 1 , x2 , δ1 , δ2 ) > 0 and β(δ1 x1 ) < 0 on the support of η. Then β˜ := −β > 0, that b(x and we may rewrite ˜ 1 , x2 , δ1 , δ2 )(x2 − x1m ω(δ1 x1 ))2 − 22j δ3 x1n β(δ ˜ 1 x1 ). 22j φ(x, δ) = 22j b(x We introduce new coordinates y by putting y1 := x1 and y2 := 22j φ(x1 , x2 , δ). Solving for x2 , one easily finds that    ˜ 1 y1 ), δ1 , δ2 x2 = b˜1 y1 , 2−2j y2 + δ3 y1n β(δ  ˜ 1 y1 ) + y1m ω(δ1 y1 ) , × 2−2j y2 + δ3 y1n β(δ ˜ Moreover, by the support where b˜1 is smooth and has properties similar to b. 2j properties of the amplitude χ1 (2 φ(x, δ))η(x1 , x2 ), we see that for the new coordinates we also have y1 ∼ 1 ∼ y2 , and changing to the coordinates (y1 , y2 ), we may rewrite  2−2j f (y1 , φ(y, δ, j ), 2−2j y2 ) a(y, δ, j ) χ1 (y1 )χ1 (y2 ) dy, νδ,j , f  = √ δ3 with a cutoff function χ1 as before, as well as where a(y, δ, j ) is smooth in y and δ, with C l -norms uniformly bounded in δ and j, and where the new phase function φ(x, δ, j ) is given by    ˜ 1 x1 ), δ1 , δ2 φ(x, δ, j ) := b˜1 x1 , 2−2j x2 + δ3 x1n β(δ  ˜ 1 x1 ) + x1m ω(δ1 x1 ) . × 2−2j x2 + δ3 x1n β(δ We have renamed the variable y to become x here, since if we define the normalized measure ν˜ δ,j by  ˜νδ,j , f  := f (x1 , φ(x, δ, j ), x2 ) a(x, δ, j ) χ1 (x1 )χ1 (x2 ) dx, (4.7) then the restriction estimate (4.6) for the measure νδ,j is equivalent to the following restriction estimate for the measure ν˜ δ,j : 

|f
|2 d ν˜ δ,j ≤ Cη

2j (1−2/p
) c f 2 δ3 2 Lpc (R3 ) ,

f ∈ S(R3 ),

(4.8)

for all j ∈ N sufficiently large, say j ≥ j0 , where the constant Cη depends neither on δ nor on j. Formula (4.7) shows that the Fourier transform of the measure ν˜ δ,j can be expressed as an oscillatory integral  (4.9) ν

˜ δ,j (ξ ) = e−i(x,δ,j,ξ ) a(x, δ, j ) χ1 (x1 )χ1 (x2 ) dx, where the complete phase function  is given by (x, δ, j, ξ ) := ξ2 φ(x, δ, j ) + ξ3 x2 + ξ1 x1 .

61

SURFACES WITH LINEAR HEIGHT BELOW 2

In order to establish the restriction estimates (4.8), we shall finally perform dyadic frequency decompositions of the measure ν˜ δ,j in each of the three coordinates. To this end, we again fix a suitable smooth cutoff function χ1 ≥ 0 on R supported in (−2, − 12 ) ∪ ( 21 , 2) such that the functions χk (t) := χ1 (21−k t), k ∈ N \ {0}, in combination with a suitable smooth function χ0 supported in (−1, 1), form a partition of unity, that is, ∞ 

χk (t) = 1

for all t ∈ R.

k=0

For every multi-index k = (k1 , k2 , k3 ) ∈ N3 , we let χk (ξ ) := χk1 (ξ1 )χk2 (ξ2 )χk3 (ξ3 ) and, finally, define the smooth functions νk,j by

ν

˜ δ,j (ξ ). k,j (ξ ) := χk (ξ )ν In order to simplify the notation, we have suppressed here the dependency of this smooth function on the small parameters given by δ. We then find that  ν˜ δ,j = νk,j , (4.10) k∈N3

in the sense of distributions. To simplify the subsequent discussion, we shall concentrate on those measures νk,j for which none of its ki components are zero, since the remaining cases where for instance ki is zero can be dealt with in the same way as the corresponding cases where ki ≥ 1 is small. Now, if 1 ≤ λi = 2ki −1 , i = 1, 2, 3, are dyadic numbers, we shall accordingly write νjλ in place of νk,j , that is, ξ  ξ  ξ  1 2 3

(4.11) χ1 χ1 ν˜ δ,j (ξ ). ν
jλ (ξ ) = χ1 λ1 λ2 λ3 Note that |ξi | ∼ λi , Moreover, by (4.7), νjλ (x) = λ1 λ2 λ3

on supp ν
jλ .

(4.12)

 χˇ 1 (λ1 (x1 − y1 )) χˇ 1 (λ2 (x2 − φ(y, δ, j )))

×χˇ 1 (λ3 (x2 − y2 )) a(y, δ, j ) χ1 (y1 )χ1 (y2 ) dy,

(4.13)

where fˇ denotes the inverse Fourier transform of f. We begin by estimating the Fourier transform of νjλ . To this end, we first integrate in x1 in (4.7) and then in x2 , assuming that (4.12) holds true. We shall concentrate on those νjλ for which λ1 ∼ λ2 ∼ δ3 22j λ3 . (4.14) In all other cases, the phase  in (4.9) has no critical point on the support of the amplitude, and we obtain much faster Fourier decay estimates by repeated integrations

62

CHAPTER 4

by parts in x1 , respectively, x2 , and the corresponding terms can be considered as error terms. Observe also that 2 ∂ −3/2 −4j ∂x 2 (x, δ, j, ξ ) ∼ λ2 δ3 2 2 on the support of the amplitude. We therefore distinguish two subcases. 3/2

Case 1. 1 ≤ λ1
δ 3 24j . In this case we cannot gain from the integration in x2 , but, by applying van der Corput’s lemma of order M = 2 (or the method of stationary phase) in x1 , we obtain ν
jλ ∞
λ1

−1/2

.

(4.15)

3/2

Case 2. λ1
δ 3 24j . Then, by first applying the method of stationary phase to the integration in x1 and subsequently applying the classical van der Corput Lemma 2.1 of order M = 2 to the integration in x2 , we obtain ν
jλ ∞
λ1

−1/2

−3/2 −4j −1/2

(λ2 δ3

2


δ3 22j λ−1 1 . 3/4

)

(4.16)

Next, from (4.13), by making use of the first and the third factor of the integrand, we trivially obtain the following estimate for the L∞ -norm of νjλ : νjλ ∞
λ2 ∼ λ1

(4.17)

in Case 1 as well as in Case 2. All these estimates are uniform in δ for δ sufficiently small. For each of the measures νjλ , we can now obtain suitable restriction estimates by applying the usual approach. Let us denote by Tδ,j the convolution operator Tδ,j : ϕ  → ϕ ∗ ν

˜ δ,j , and, similarly, by Tjλ the convolution operator Tjλ : ϕ  → ϕ ∗ ν
jλ . Formally, by (4.10), Tδ,j decomposes as Tδ,j =



k

Tj2

(4.18)

k∈N3

if 2k represents the vector 2k := (2k1 , 2k2 , 2k3 ) (with a suitably modified definition k of Tj2 when one of the components ki is zero). If we denote by T p→q the norm of = ν
λ and T λ = T as an operator from Lp to Lq , then clearly T λ j

1→∞

j ∞

νjλ ∞ . The estimates (4.15)–(4.17) thus yield the following bounds:  3/2 λ1−1/2 , if 1 ≤ λ1
δ3 24j , λ Tj 1→∞
δ 3/4 22j λ−1 , if λ  δ 3/2 24j , 1 1 3 3

j

2→2

63

SURFACES WITH LINEAR HEIGHT BELOW 2

and Tjλ 2→2
λ1 . Interpolating the estimates (4.16) and (4.17), and defining the critical interpolation parameter θ = θc by 1/pc
= (1 − θ )/∞ + θ/2 = θ/2, that is, 2 , pc


θ := we find that Tjλ pc →pc



 (3θ−1)/2  , λ1

if 1 ≤ λ1
δ3 24j ,

 δ 3(1−θ)/4 22(1−θ)j λ2θ−1 ,

if λ1  δ3 24j ,

3/2

(4.19) 3/2

1

3

where according to Remark 2.12,  m+1    3m + 1 , θ=    m+1 , 3m + 2

if φ is of type A, (4.20) if φ is of type D,

since pc
= 2d + 2. Observe that in particular, 1 3

< θ ≤ 37 ,

(4.21)

and θ = 37 if and only if m = 2 and φ is of type A. The latter case will turn out to be the most difficult one. Observe next that the main contributions to the series (4.18) come from those √ dyadic λ = 2k for which λ1 ∼ λ2 ∼ δ3 22j λ3 . Under these relations, for λ1 given, λ2 and λ3 may only vary in a finite set whose cardinality is bounded by a fixed number. This shows that, up to an easily bounded error term, 3/2

δ3 24j

Tδ,j pc →pc





(3θ−1)/2

λ1



+

λ1 =2

3(1−θ)/4 2(1−θ)j (2θ−1) 2 λ1 .

δ3

3/2

λ1 >δ3 24j

Here, and in the sequel, summation over λ1 , λ2 , and so on, will always mean that we are summing over dyadic numbers λ1 , λ2 , and so on, only. Now, by (4.21), 2θ − 1 < 0 and 0 < 3θ − 1 ≤ 1, which yields 3(3θ−1)/4 (3θ−1)2j

Tδ,j pc →pc

δ3

2

.

Recall that, by the standard T ∗ T -argument of Tomas, applied to the restriction operator Rf := fˆd ν˜ δ,j , we have  |f
|2 d ν˜ δ,j ≤ (2π )−3 Tδ,j pc →pc
f 2pc , and thus we need to prove that 3(3θ−1)/4 (3θ−1)2j

δ3

2


≤ C δ3 22j (1−2/pc )

64

CHAPTER 4

in order to verify that the restriction estimate (4.8) holds true for p = pc = 2d + 2. However, since 2/pc
= θ, the previous estimate is equivalent to (5−9θ)/4

22j (4θ−2) ≤ Cδ3

.

But, since 2 δ3  1 and 2θ − 1 < 0, we see that 22j (4θ−2) ≤ C δ32−4θ , and therefore we have to verify only that 2 − 4θ ≥ (5 − 9θ )/4, that is, 7θ ≤ 3, which is true according to (4.21). We thus have verified the restriction estimate (4.6) in this subcase. 2j

4.2 RESTRICTION ESTIMATES FOR NORMALIZED RESCALED MEASURES WHEN 22j δ 3
1 There remains the case 22j δ3 ≤ C, where C is a fixed, possibly large constant. In this situation, we perform the change of variables (x1 , x2 )  → (x1 , x2 + x1m ω(δ1 x1 )) and subsequently scale in x2 by the factor 2−j . This allows us to rewrite the measure νδ,j given by (4.5) as    νδ,j , f  = 2−j f x1 , 2−j x2 + x1m ω(δ1 x1 ), 2−2j φ a (x, δ, j ) a(x, δ, j ) dx, where ˜ 1 , 2−j x2 + x1m ω(δ1 x1 ), δ1 , δ2 )x22 + 22j δ3 x1n β(δ1 x1 ) φ a (x, δ, j ) := b(x

(4.22)

and a(x, δ, j ) := χ1 (φ a (x, δ, j )) η(x1 , 2−j x2 + x1m ω(δ1 x1 )). Let us introduce here the normalized measures ν˜ δ,j given by  ˜νδ,j , f  := f (x1 , 2−j x2 + x1m ω(δ1 x1 ), φ a (x, δ, j )) a(x, δ, j ) dx.

(4.23)

(4.24)

Then, it is easy to see by means of a scaling of the variable x3 by the factor 2−2j that the restriction estimate (4.6) for the measure νδ,j is equivalent to the following restriction estimate for the measure ν˜ δ,j : 

|f
|2 d ν˜ δ,j ≤ Cη 2(1−4/pc )j f 2Lpc (R3 ) = Cη 2(1−2θ)j f 2Lpc (R3 ) ,


S

f ∈ S(R3 ), (4.25)

for all j ∈ N sufficiently large, say j ≥ j0 , where the constant Cη depends neither on δ nor on j. In order to prove (4.25), we again distinguish two subcases. 4.2.1 The situation where 22j δ3  1 ˜ 1 , 0, 0, 0)x 2 , Notice that here the phase φ a (x, δ, j ) is a small perturbation of b(v 2 ˜ 1 , 0, 0, 0) ∼ 1. This shows also that in the new coordinates appearing in where b(v

65

SURFACES WITH LINEAR HEIGHT BELOW 2

(4.24), we have x1 ∼ 1 ∼ |x2 | on the support of the amplitude a, which in turn implies ∂ a φ (x, δ, j ) ∼ 1. ∂x2 We can thus write ν

˜ δ,j (ξ ) =



(4.26)

e−i(x,δ,j,ξ ) a(x, δ, j )χ1 (x1 )χ1 (x2 ) dx,

(4.27)

where the complete phase function  is now given by (x, δ, j, ξ ) := ξ3 φ a (x, δ, j ) + 2−j ξ2 x2 + ξ2 x1m ω(δ1 x1 ) + ξ1 x1 ,

(4.28)

a

with φ given by (4.22), and where χ1 has similar properties as before. As in the previous subcase, we perform dyadic frequency decompositions of the measure ν˜ δ,j by means of the functions in (4.10) and define the measure νjλ as in (4.11). Then in the present situation we have  λ νj (x) = λ1 λ2 λ3 χˇ 1 (λ1 (x1 − y1 )) χˇ 1 (λ2 (x2 − 2−j y2 − y1m ω(δ1 y1 ))) ×χˇ 1 (λ3 (x3 − φ a (y, δ, j ))) a(y, δ, j ) χ1 (y1 )χ1 (y2 ) dy.

(4.29)

We begin by estimating the Fourier transform of νjλ . To this end, we first integrate in x2 in (4.27) and then in x1 . We may assume that (4.12) holds true. Then the phase function  has no critical point in x2 unless λ3 ∼ 2−j λ2 ; similarly, if we assume that λ3 ∼ 2−j λ2 , then there is no critical point with respect to x1 unless λ2 ∼ λ1 . We shall, therefore, concentrate on those νjλ for which λ1 ∼ λ2

2−j λ2 ∼ λ3 .

and

(4.30)

In all other cases, we obtain much faster Fourier decay estimates by repeated integrations by parts, so that the corresponding terms can be considered as error terms. Case 1. 1 ≤ λ1 ≤ 2j . In this case the phase function has essentially no oscillation in the x2 variable. But, by applying van der Corput’s lemma of order M = 2 (or the method of stationary phase) in x1 , we obtain in view of (4.30) that ν
jλ ∞
λ1

−1/2

.

(4.31)

Case 2. λ1 > 2j . Observe that in this case, our assumptions imply that δ3 22j λ3  λ3  λ2 if j ≥ j0  1. Moreover, depending on the signs of the ξi , we may have no critical point or exactly one nondegenerate critical point with respect to each of the variables x2 and x1 . So, integrating by parts, respectively applying the method of stationary phase in the presence of a critical point, first in x2 and then in x1 , we obtain −1/2 −1/2 λ ∼ 2j/2 λ−1 . (4.32) ν
λ
λ j ∞

3

1

1

Next, we estimate the L∞ -norm of νjλ . To this end, notice that (4.26) shows that we may change coordinates in (4.29) by putting (z 1 , z 2 ) := (y1 , φ a (y1 , y2 , δ, j )).

66

CHAPTER 4

Since the Jacobian of this coordinate change is of order 1, we thus obtain that  λ ˜ δ, j ) dz 1 dz 2 ; |νj (x)|
λ1 λ2 λ3 χˇ 1 (λ1 (x1 − z 1 )) χˇ 1 (λ3 (x3 − z 2 )) a(z, hence νjλ ∞
λ2 ∼ λ1

(4.33)

in Case 1 as well as in Case 2. For the operators Tjλ that appear in this subcase, the estimates (4.31)–(4.33) thus yield the following bounds:  λ1−1/2 , if 1 ≤ λ1 ≤ 2j , Tjλ 1→∞
2j/2 λ−1 , if λ > 2j , 1 1 and Tjλ 2→2
λ1 . Interpolating these estimates, we find that  λ1(3θ−1)/2 , if 1 ≤ λ1 ≤ 2j , Tjλ pc →pc

2[(1−θ)/2]j λ2θ−1 , if λ > 2j , 1 1

(4.34)

where θ is again given by (4.20). Now, in view of (4.30), the main contribution to the series (4.18) comes here from those dyadic λ = 2k for which λ1 ∼ λ2 and 2−j λ2 ∼ λ3 . Thus, up to an easily bounded error term, the operator Tδ,j arising in this subcase can be estimated by j

Tδ,j pc →pc



2 

(3θ−1)/2

λ1

λ1 =2

+

∞ 

2[(1−θ)/2]j λ12θ−1
2[(3θ−1)/2]j ≤ 2(1−2θ)j ,

λ1 =2j +1

since, by (4.21) we have 2θ − 1 < 0 and (3θ − 1)/2 ≤ 1 − 2θ. This verifies the restriction estimate (4.25) and thus concludes the proof of Proposition 4.1 in this subcase as well. 4.2.2 The situation where 22j δ3 ∼ 1 Notice that in this situation we can no longer conclude that x2 ∼ 1 on the support of the amplitude a(x, δ, j ), only that |x2 |
1, whereas it is still the case that x1 ∼ 1. Also observe that here the cases A∞ and D∞ are excluded, since in these cases δ3 = 0. Putting σ := 22j δ3

and

˜ 1 , 2−j x2 + x1m ω(δ1 x1 ), δ1 , δ2 ), b (x, δ, j ) := b(x

we may rewrite the complete phase in (4.28) as (x, δ, j, ξ ) = ξ1 x1 + ξ2 x1m ω(δ1 x1 ) + ξ3 σ x1n β(δ1 x1 ) + 2−j ξ2 x2 + ξ3 b (x, δ, j ) x22 ,

(4.35)

67

SURFACES WITH LINEAR HEIGHT BELOW 2

where σ ∼ 1 and |b (x, δ, j )| ∼ 1, and in place of (4.29) we obtain  νjλ (x) = λ1 λ2 λ3 χˇ 1 (λ1 (x1 − y1 )) χˇ 1 (λ2 (x2 − 2−j y2 − y1m ω(δ1 y1 ))) ×χˇ 1 (λ3 (x3 − b (y, δ, j ) y22 − σy1n β(δ1 y1 ))) × a(y, δ, j )χ1 (y1 )χ0 (y2 ) dy.

(4.36)

Here, we have suppressed the dependence of νjλ on the parameter σ in order to simplify the notation. Also observe that we then may drop the parameter δ3 from the definition of δ, that is, we may assume that δ = (δ1 , δ2 ), since only σ depends on δ3 . Recall from (4.23) that a(y, δ, j ) is supported 1 ∼ 1 and |y2 |
1.   where y −1/2 Observe that by Lemma 2.1(b), we have χˇ 1 λ3 (c − t 2 ) dt ≤ Cλ3 , with a constant C that is independent of c. Thus, making use of the localizations given for the integration in y2 from the third factor, respectively, second factor, in the integrand of (4.36) and then for the integration in y1 by the first factor, it is easy to see that νjλ ∞
min{λ2 λ3 , 2j λ3 } = λ3 min{λ2 , 2j λ3 } . 1/2

1/2

1/2

(4.37)

In order to estimate ν
jλ (ξ ), we may again assume that (4.12) holds true. In the oscillatory integral defining ν
jλ (ξ ), we shall first perform the integration in x1 . If one of the quantities λ1 , λ2 , or λ3 is much bigger than the other two, we see that we have no critical point in x1 on the support of the amplitude, so that the corresponding terms can again be viewed as error terms. Let us therefore assume that all three λi s are of comparable size or that two of them are of comparable size and the third one is much smaller. We shall begin with the latter situation and distinguish various possibilities. Case 1. λ1 ∼ λ3 and λ2  λ1 . In this case, we apply the method of stationary phase to the integration in x1 and, subsequently, van der Corput’s estimate to the x2 -integration and obtain ν
jλ ∞
λ1

−1/2 −1/2 λ3

∼ λ−1 1 .

1.1. The subcase where λ2 ≤ 2j λ1 . Then, by (4.37), νjλ ∞
λ2 λ1 , and we obtain by interpolation, in a similar way as before, that 1/2

1/2

(3θ−2)/2 θ λ2 .

Tjλ pc →pc

λ1

Here, 3θ−2 < 0 because of (4.21). Notice next that if 2j λ1 ≤ λ1 , that is, 2 1/2 2j if λ1 ≥ 2 , then by our assumptions λ2 ≤ 2j λ1 , and if λ1 < 22j , then I of the we may use that λ2 ≤ λ1 . We thus find that the contribution Tδ,j operators Tjλ , with λ satisfying the assumptions of this subcase, to Tδ,j can 1/2

68

CHAPTER 4

be estimated by λ1 2   2j

I Tδ,j pc →pc





(3θ−2)/2 θ λ1 λ2

+

λ1 =2 λ2 =2 2 

λ1 ∞ 

2j




(5θ−2)/2

λ1

λ1 =2

+ λ1

∞ 

2j λ1

=22j +1

λ2 =2

1/2



(3θ−2)/2 θ λ2

λ1

2θj λ12θ−1 .

=22j +1

But, we have seen that 2θ − 1 < 0, so that I pc →pc

max{j, 2(5θ−2)j }. Tδ,j I Now, if 5θ − 2 > 0, then, again because of (4.21), Tδ,j pc →pc

2(5θ−2)j ≤ I (1−2θ)j (1−2θ)j 2 . And, if 5θ − 2 ≤ 0, then Tδ,j pc →pc

j
2 , that is, I Tδ,j pc →pc

2(1−2θ)j .

(4.38)

1.2. The subcase where λ2 > 2j λ1 . Then, by (4.37), νjλ ∞
2j λ1 , and we obtain by interpolation, in a similar way as before, that 1/2

Tjλ pc →pc

2θj λ12θ−1 . Observing that we have 2j λ1 < λ2 ≤ λ1 and then also λ1 > 22j , we see II of the operators Tjλ , with λ satisfying the assumpthat the contribution Tδ,j tions in this subcase, to Tδ,j can be estimated by 1/2

II Tδ,j pc →pc

2θj

∞ 



λ12θ−1
2θj

λ1 =22j 2j λ1/2 0 is sufficiently small. In particular, σjλ2 ,λ3 (x) is again given by the expression (4.36), only with the first factor χˇ 1 (λ1 (x1 − y1 )) in the integrand replaced by χˇ 0 (λ2 (x1 − y1 )) and λ1 replaced by λ2 . Thus we obtain the same type of estimates, λ2 ,λ3 σ
∞
λ−1 j 2 ,

σjλ2 ,λ3 ∞
λ2 min{λ2 , 2j }. 1/2

(4.40)

λ2 ,λ3 . Denote by Tjλ2 ,λ3 the operator of convolution with σ
j

2.1. The subcase where λ2 ≤ 22j . Then we have σjλ2 ,λ3 ∞
λ2 , and interpolating this with the first estimate in (4.40), we obtain 3/2

Tjλ2 ,λ3 pc →pc

λ2θ−1 λ2

3θ/2

(5θ−2)/2

= λ2

.

III We thus find that the contribution Tδ,j of the operators Tjλ , with λ satisfying the assumptions of this subcase to Tδ,j , can be estimated by 2  2j

III pc →pc
Tδ,j




(5θ−2)/2

λ2

.

λ2 =2

Arguing in a similar way as in Subcase 1.1, this implies that III pc →pc

2(1−2θ)j . Tδ,j

(4.41)

2.2. The subcase where λ2 > 22j . Then, by (4.40), σjλ2 ,λ3 ∞
2j λ2 , and we obtain by interpolation, in a similar way as before, that Tjλ2 ,λ3 pc →pc

λ2θ−1 2θj λθ2 = 2θj λ22θ−1 , where, according to (4.21), 2θ − 1 < 0. We thus find that the contribution IV of the operators Tjλ , with λ satisfying the assumptions of this subcase, Tδ,j to Tδ,j can be estimated by ∞ 

IV pc →pc

2θj Tδ,j

λ2

λ22θ−1
2(5θ−2)j .

=22j

As before, this implies that IV pc →pc

2(1−2θ)j . Tδ,j

(4.42)

Case 3. λ1 ∼ λ2 and λ3  λ1 . Notice that the phase  has no critical point with respect to x2 when 2−j λ2  λ3 , so we shall concentrate on the case where λ2
2j λ3 . Then we can estimate ν
jλ in the same way as in the previous cases and obtain ν
jλ ∞
λ1

−1/2 −1/2 λ3 .

(4.43)

70

CHAPTER 4

3.1. The subcase where λ3  λ1 2−j . Then (2−j λ1 )2
λ3  λ1 , and hence we may assume that λ1 ≤ 22j ; from (4.37) and the previous estimate for ν
jλ , we obtain by interpolation 1/2

(3θ−1)/2 (2θ−1)/2 λ3 .

Tjλ pc →pc

λ1

V of the operators Tjλ , with λ satisfying We thus find that the contribution Tδ,j the assumptions of this subcase, to the operator Tδ,j can be estimated by 2 



2j

V Tδ,j pc →pc



2  2j

(3θ−1)/2 (2θ−1)/2 λ3

λ1


2(1−2θ)j

λ1 =2 (2−j λ1 )2
λ3 λ1

(7θ−3)/2

λ1

λ1 =2

(recall that 2θ − 1 < 0, according to (4.21)). If θ < 37 , this implies the desired estimate V pc →pc

2(1−2θ)j Tδ,j

However, if θ = the estimate

3 , 7

( if θ < 37 ).

(4.44)

that is, if φ is of finite type A and m = 2, we get only V pc →pc

j 2(1−2θ)j . Tδ,j

(4.45)

In order to improve on this estimate, we shall have to apply a complex interpolation argument. There will be a few more cases that require such an interpolation argument, and we shall collect them all in Proposition 4.2 (see also Section 5.3). We also remark that   (3θ−1)/2 (2θ−1)/2 λ1 λ3
2[(3θ−1)/2]j
2(1−2θ)j , 2≤λ1
2j (2−j λ1 )2
λ3 λ1

so that we need only to control the terms with λ1  2j . 3.2. The subcase where λ3  λ1 2−j . Then we have 1/2

λ3  min{λ1 , (2−j λ1 )2 }, which implies that necessarily λ1  2j , and interpolation yields in this case that −(1−θ)/2 (3θ−1)/2 λ3 .

λ pc →pc

2θj λ1 Tδ,j

First, assume that λ1 > 22j . Then λ1 = min{λ1 , (2−j λ1 )2 }, so that we V I,1 the sum of the operators Tjλ , with λ shall use λ3  λ1 . Denoting by Tδ,j satisfying the assumptions of this subcase and λ1 > 22j , and recalling that 3θ − 1 > 0 and 2θ − 1 < 0, we see that V I,1 pc →pc

2θj Tδ,j

λ1 ∞   λ1

=22j

λ3 =2

−(1−θ)/2 (3θ−1)/2 λ3

λ1


2(5θ−2)j ≤ 2(1−2θ)j . (4.46)

71

SURFACES WITH LINEAR HEIGHT BELOW 2

There remains the case where 2j  λ1 ≤ 22j . Then λ3  (2−j λ1 )2 . V I,2 the sum of the operators Tjλ , with λ satisfying the Denoting by Tδ,j assumptions of this subcase and 2j  λ1 ≤ 22j , and recalling that 3θ − 1 > 0 and 2θ − 1 < 0, we see that

θj


2

λ1 =2j 2 

−j

(2 λ1 ) 2  2j

V I,2 pc →pc
Tδ,j

2

−(1−θ)/2 (3θ−1)/2 λ3

λ1


2(1−2θ)j

λ3 =2

2j

×

(7θ−3)/2

λ1


2(1−2θ)j ,

(4.47)

λ1 =2

provided θ < 37 . If θ = 37 , we pick up an additional factor j as in (4.45): V I,2 Tδ,j pc →pc

j 2j/7 = j 2(1−2θ)j .

(4.48)

In order to improve on this estimate, we shall have to again apply a complex interpolation argument (cf. Section 5.3). What is left is Case 4. Case 4. λ1 ∼ λ2 ∼ λ3 . Here, we can first apply the method of stationary phase to the integration in x2 . This produces a phase function in x1 , which is of the form φ1 (x1 ) = ξ1 x1 + ξ2 (ω(0)x1m + error) + ξ3 (σβ(0)x1n + error), with small error terms of order O(|δ| + 2−j ). We assume again that (4.12) holds true. Then φ1 has a singularity of Airy type, which implies that the oscillatory integral with phase φ1 that we have arrived at decays with order O(|λ|−1/3 ). Indeed, we have n ≥ 2m + 1 and m ≥ 2, and since x1 ∼ 1, it is easy to see by studying the linear (j ) system of equations yj = φ1 (x1 ), j = 1, 2, 3, that there exist constants 0 < c1 ≤ c2 that do not depend on ξ and x1 ∼ 1 such that c1 |ξ | ≤

3 

(j )

|φ1 (x1 )| ≤ c2 |ξ |.

j =1

Hence, our claim follows from Lemma 2.1. We thus find that ν
jλ ∞
λ1

−1/2 −1/3 λ1

−5/6

= λ1

.

4.1. The subcase where λ1 > 22j . Then, by (4.37) νjλ ∞
2j λ1 , and we obtain (11θ−5)/6

Tjλ pc →pc

2θj λ1

.

The estimates in (4.21) show that 11θ − 5 < 0, which implies that the conV II tribution Tδ,j of the operators Tjλ , with λ satisfying the assumptions of this

72

CHAPTER 4

subcase, to Tδ,j can again be estimated by V II pc →pc

Tδ,j

∞ 

(11θ−5)/6

2θj λ1


2(14θ−5)j/3
2(1−2θ)j ,

(4.49)

λ1 =22j

provided that θ ≤ 25 . According to (4.20), this is true, with the only exception being the case where φ is of type A and m = 2. , so that Observe also that if m = 2, then θ = 37 and pc
= 14 3 −1/21 λ 3j/7 Tj pc →pc

2 λ1 , and  −1/21 23j/7 λ1
2j/7 = 2(1−2θ)j . λ1 >26j

This leaves open the sum over the terms with λ1 ≤ 26j , in the case where φ is of type A and m = 2. 4.2. The subcase where λ1 ≤ 22j . Then, by (4.37) νjλ ∞
λ1 , and we obtain (14θ−5)/6 . Tjλ pc →pc

λ1 3/2

V III We thus find that the contribution Tδ,j of the operators Tjλ , with λ satisfying the assumptions of this subcase, to Tδ,j can be estimated by 2  2j

V III Tδ,j pc →pc





(14θ−5)/6

λ1

.

λ1 =2

If 14θ − 5 ≤ 0, then we immediately obtain the desired estimate V III pc →pc

j
2(1−2θ)j , so assume that 14θ − 5 > 0. Then Tδ,j V III Tδ,j pc →pc

2[(14θ−5)/3]j , and arguing as before (compare (4.49)), we see that V III pc →pc

2(1−2θ)j , Tδ,j

(4.50)

unless φ is of type A and m = 2. But, recall that the case A∞ was excluded here, so that φ is of type An−1 , with finite n ≥ 5 (compare Proposition 2.11). The estimates (4.38)–(4.44), (4.46)–(4.47), (4.49), and (4.50) show that estimate (4.25) also holds true in the situation of this section, which completes the proof of Proposition 4.1, with the exception of the case where φ is of type An−1 , with finite n ≥ 5 and m = 2, in which we still need to improve on the estimates (4.45) and (4.48) in Subcases 3.1 and 3.2; moreover, we need to find stronger estimates for the cases where λ1 ∼ λ2 ∼ λ3 when λ1 ≤ 26j . Observe also that in Case 3, we have that λ1 ∼ λ2 , and thus we may assume that λ2 = 2K λ1 , where K is from a finite set of integers. This allows us to assume that λ2 = 2K λ1 for a given, fixed integer K, and for the sake of simplicity, we shall even assume that K = 0, so that λ1 = λ2 (the other cases can be treated in exactly the same way). In a similar way, we may and shall assume that λ1 = λ2 = λ3 in Case 4. Thus, in order to complete the proof of Proposition 4.1, and hence that of Theorem 1.14 when h lin (φ) < 2, what remains to be proven is the following.

73

SURFACES WITH LINEAR HEIGHT BELOW 2

Proposition 4.2. Assume that φ is of type An−1 , with m = 2 and finite n ≥ 5, so and θ := 2/pc
= 37 . Then the following hold true, provided j, M ∈ N that pc
= 14 3 are sufficiently large and δ is sufficiently small: (a) Let −M 2 λ1

2  2j

V νδ,j

:= λ1

=2M+j

λ3

=(2−M−j λ

νj(λ1 ,λ1 ,λ3 ) , 1

)2

V V the convolution operator ϕ  → ϕ ∗ ν

and denote by Tδ,j δ,j . Then V Tδ,j 14/11→14/3 ≤ C 2j/7 .

(4.51)

(b) Let 2 

2 (2−M−j λ1 )

λ1 =2M+j

λ3 =2

2j

VI νδ,j

:=

νj(λ1 ,λ1 ,λ3 ) ,

VI VI and denote by Tδ,j the convolution operator ϕ  → ϕ ∗ ν

δ,j . Then VI Tδ,j 14/11→14/3 ≤ C 2j/7 .

(4.52)

(c) Let 2  6j

V II := νδ,j

νj(λ1 ,λ1 ,λ1 )

λ1 =2 V II V II and denote by Tδ,j the convolution operator ϕ  → ϕ ∗ ν

δ,j . Then V II Tδ,j 14/11→14/3 ≤ C 2j/7 .

(4.53)

Here, the constant C depends neither on δ nor on j. The proof of Proposition 4.2 will be given in the next chapter. Remark 4.3. Recall that in the situation of Subsection 4.2.2, we had replaced the small parameter δ3 by the parameter σ = 22j δ3 ∼ 1. If φ is now of type An−1 and m = 2, it will often be convenient in the sequel to augment our former vector δ = (δ1 , δ2 ) by the parameter δ0 := 2−j  1, that is, we redefine δ to become δ := (δ0 , δ1 , δ2 ). In this way, we may replace the parameter j ∈ N by the small parameter δ0 . Observe that according to (4.3) and (4.35), we may then rewrite, in (4.36),  ba (δ1 y1 , δ0 δ2 y2 ), if φ is of type A, b (y, δ1 , δ2 , j ) = b0 (y, δ) := δ −1 ba (δ y , δ δ y ), if φ is of type D , 1 1 0 2 2 1 where ba (y1 , y2 ) := b(y1 , y2 + y1m ω(y1 )) expresses b in adapted coordinates.

74

CHAPTER 4

Then, by (4.36), we may write      λ λ νj (x) =: νδ (x) = λ1 λ2 λ3 χˇ 1 λ1 (x1 − y1 ) χˇ 1 λ2 (x2 − δ0 y2 − y1m ω(δ1 y1 ))   ×χˇ 1 λ3 (x3 − b0 (y, δ) y22 − σy1n β(δ1 y1 )) η(y, δ) dy, (4.54) where η ∈ C ∞ (R2 × R3 ) is supported if y1 ∼ 1 and |y2 |
1 (and, say, |δ| ≤ 1) and where χ1 is a smooth cutoff function supported near 1. Notice that the measure νδλ indeed also depends on σ ∼ 1, but we shall suppress this dependency in order to simplify the notation.

Chapter Five Improved Estimates by Means of Airy-Type Analysis We next turn to the proof of Proposition 4.2. Recall from the discussion in the preV VI and Tδ,j appearing in parts (a) and (b) of vious chapter that for the operators Tδ,j  that proposition, we had already established the conjectured Lp → Lp estimates, with only the exception of the endpoint p = pc . It may, therefore, not come as a surprise that this endpoint result can be established by means of Stein’s interpolation theorem for analytic families of operators (see [SW71]), as will be shown in Sections 5.2 and 5.3. Indeed, we shall construct analytic families of complex measure µζ , for ζ in the complex strip  given by 0 ≤ Re ζ ≤ 1, by introducing V VI and νδ,j , respectively complex coefficients in the sums defining the measures νδ,j (our approach is somewhat different from more classical ones, for instance, in the work by Stein and Tomas, but it is better adapted to the dyadic frequency decompositions that we have performed and appears to be even more elementary). These coefficients will be chosen as exponentials of suitable affine-linear expression in ζ V VI , respectively, µθc = νδ,j . As it will turn in such a way that, in particular, µθc = νδ,j out, the main problem will then consist in establishing suitable uniform bounds for the measure µζ when ζ lies on the right boundary line of  (compare estimates (5.49) and (5.57)). The arguments required to prove these uniform estimates for the functions µ1+it , t ∈ R, will turn out to be rather involved. In particular, we shall have to invoke our uniform estimates for oscillatory sums from Lemmas 2.7 and 2.9, in combination with suitable estimates for particular classes of integrals of sublevel type. We shall see several further instances of such kind of arguments in the last two chapters, in which we shall deal with phase functions for which the function φ(x1 , x2 ) = (x2 − x12 )3 + x1n can be viewed as a prototype. These phase functions will turn out to generate the highest degree of complexity among all, and thus the corresponding arguments in Chapters 7 and 8 will become even more involved. As for part (c) in Proposition 4.2, recall from our discussion in Chapter 4 that VI , but we even have we not only missed the endpoint estimate for the operator Tδ,j to close a large gap between the desired estimate and the actual estimate given by (4.49), in the case where φ is of type A and m = 2. The reason for this gap lies in the fact that the Fourier transform of the measure νδλ can be expressed as a one-dimensional integral of Airy type (after an easy application of the method of stationary phase in the second variable), whose decay rate will be smallest along a certain “Airy cone.” We shall, therefore, perform yet another dyadic frequency decomposition into a region close to the given Airy cone and dyadic regions in which the distance to the cone is of a fixed order of size. The corresponding measures will be denoted

76

CHAPTER 5

λ λ by νδ,Ai and νδ,l (compare (5.12)). Here, Lemma 2.2 will become important. Such a type of decomposition is familiar from other contexts, for instance, from work by Greenleaf and Seeger [GS99] on oscillatory integrals with folding canonical relations. A major problem will then consist in deriving enough information, in particular λ λ and νδ,l from knowledge of their Fourier on the L∞ -norms, of the functions νδ,Ai λ λ transforms, in particular, from the support properties of ν
and ν (compare δ,Ai

δ,l

estimates (5.15) and (5.24)). This will necessitate further decompositions of the measures µλδ,l in Subsection 5.2.2. The estimates that we can establish in this way  will then allow us to verify the conjectured Lp → Lp estimates for the operator V II Tδ,j , except for the endpoint p = pc . In order to capture also the endpoint, we shall again have to apply a complex interpolation argument of a similar kind as described before. This will be done in Subsection 5.2. We shall begin with the discussion of part (c) in Proposition 4.2, since this case causes even more challenges than the discussion of parts (a) and (b) (which will be treated in the last Section 5.2).

5.1 AIRY-TYPE DECOMPOSITIONS REQUIRED FOR PROPOSITION 4.2(C) In order to prove estimate (4.52) in Proposition 4.2, we recall that σ ∼ 1 and that we are assuming that 2 ≤ λ1 = λ2 = λ3 ≤ 26j . In order to simplify the notation, we shall in the sequel denote by λ the common value of λ1 = λ2 = λ3 and put s1 :=

ξ1 ξ2 ξ3 , s2 := , s3 := , ξ3 ξ3 λ

(5.1)

so that |s1 | ∼ |s2 | ∼ |s3 | ∼ 1 and ξ = λs3 (s1 , s2 , 1). In view of the special role that s3 will play, we shall write s := (s1 , s2 , s3 ),

s  := (s1 , s2 ).

Correspondingly, we shall rewrite the complete phase in (4.35) as ˜ δ, σ, s1 , s2 ),

(x, δ1 , δ2 , j, ξ ) = λs3 (x, where ˜

(x, δ, σ, s1 , s2 ) := s1 x1 + s2 x12 ω(δ1 x1 ) + σ x1n β(δ1 x1 ) + δ0 s2 x2 + x22 b0 (x, δ),

(5.2)

with b0 (x, δ) defined as in Remark 4.3. Recall also that ω(0)  = 0, β(0)  = 0, and b0 (x, 0) = b(0, 0)  = 0.

77

IMPROVED ESTIMATES

According to (4.54), we then have νjλ (ξ ) = νδλ (ξ ) = χ1 (s1 s3 )χ1 (s2 s3 )χ1 (s3 )



˜

e−iλs3 (x,δ,σ,s1 ,s2 ) a(x, ˜ δ) dx,

where the amplitude a(x, ˜ δ) := a(x, δ)χ1 (x1 )χ0 (x2 ) is a smooth function of x supported where x1 ∼ 1 and |x2 |
1 and whose derivatives are uniformly bounded with respect to the parameters δ. Moreover, if Tδλ denotes the convolution operator Tδλ ϕ := ϕ ∗ νδλ , then we see that the estimate (4.52) can be rewritten as      −1/7 Tδλ  ≤ C δ0 .  2≤λ≤δ0−6

14/11→14/3

(5.3)

We shall need to understand the precise behavior of νδλ (ξ ). To this end, consider the integration with respect to x2 in the corresponding integral. Notice that there always is a critical point x2c with respect to x2 . Writing x2 = δ0 s2 t and applying the implicit function theorem to t, we find that x2c = δ0 s2 Y2 (δ1 x1 , δ2 , δ0 s2 ),

(5.4)

where Y2 is smooth and of size |Y2 | ∼ 1. Notice also that Y2 (0, 0, 0) = −1/(2b(0, 0)) when δ = 0. We let ˜ 1 , x2c , σ, s1 , s2 ) = (x ˜ 1 , δ0 s2 Y2 (δ1 x1 , δ2 , δ0 s2 ), σ, s1 , s2 ). (x1 , δ, σ, s1 , s2 ) := (x (5.5) Applying the method of stationary phase with parameters to the x2 -integration (see, e.g., [S93]) and ignoring the region away from the critical point x2c , which leads to better estimates by means of integrations by parts, we find that we may assume that   νδλ (ξ ) = λ−1/2 χ1 (s1 s3 )χ1 (s2 s3 )χ1 (s3 ) e−iλs3 (y1 ,δ,σ,s ) a0 (y1 , s, δ; λ) χ1 (y1 ) dy1 , (5.6) where χ1 is a smooth cutoff function supported, say, in the interval [ 21 , 2]. Moreover, a0 (y1 , s, δ; λ) is smooth and uniformly a classical symbol of order 0 with respect to λ. By this we mean that it is a classical symbol of order zero for every given parameter (here these are y1 , s1 , s2 , s3 and δ), and the constants in the symbol estimates are uniformly controlled for these parameters. It will be ∂ a0 (y1 , s, δ; λ) is even a symbol of important to observe that this implies that ∂λ order −2 with respect to λ, uniformly in y1 , s, δ (the latter property will become relevant later!). We shall need more precise information on the phase . Indeed, in the subsequent lemmata, we shall establish two different presentations of , both of which will become relevant. Lemma 5.1. For |x1 |
1, we may write (x1 , δ, σ, s1 , s2 ) = s1 x1 + s2 x12 ω(δ1 x1 ) + σ x1n β(δ1 x1 ) + (δ0 s2 )2 Y3 (δ1 x1 , δ2 , δ0 s2 ), where Y3 is smooth and Y3 (δ1 x1 , δ2 , δ0 s2 ) = c0 + O(|δ|), with c0 := −1/4b (0, 0)  = 0.

78

CHAPTER 5

Proof. We have (x1 , δ, σ, s1 , s2 ) = s1 x1 +s2 x12 ω(δ1 x1 )+σ x1n β(δ1 x1 )+δ0 s2 x2c +(x2c )2 b0 (x1 , x2c , δ), so that, by definition, Y3 (δ1 x1 , δ2 , δ0 s2 ) := Y2 (δ1 x1 , δ2 , δ0 s2 ) + Y2 (δ1 x1 , δ2 , δ0 s2 )2 b0 (x1 , x2c , δ), where for δ = 0 we have Y3 (0, 0, 0) = Y2 (0, 0, 0) + Y2 (0, 0, 0)2 b0 (0, 0, 0) = −

1  = 0, 4b(0, 0)

because Y2 (0, 0, 0) = −1/(2b(0, 0)).

Q.E.D.

Next, we shall verify that  has indeed a singularity of Airy type with respect to the variable x1 . To this end, let us first consider the case where δ = 0. Then (x1 , 0, σ, s1 , s2 ) := s1 x1 + s2 x12 ω(0) + σ x1n β(0), and depending again on the signs of s2 ω(0) and β(0), the first derivative (with respect to x1 )   (x1 , 0, σ, s1 , s2 ) = s1 + 2s2 ω(0)x1 + nσβ(0)x1n−1 may have a critical point, or not. If not,  will have at worst nondegenerate critical points, and this case can be treated again by the method of stationary phase, respectively, integrations by parts. We shall therefore concentrate on the case where   does have a critical point x1c , which will then be given explicitly by  1/(n−2) 2ω(0) . s2 x1c = x1c (0, σ, s2 ) := − n(n − 1)σβ(0) Let us assume that s2 > 0 (the case where it is negative can be treated similarly). By scaling in x1 , we may and shall assume for simplicity that −

2ω(0) =1 n(n − 1)σβ(0)

(and s2 ∼ 1).

(5.7)

, and |  (x1c , 0, σ, s1 , s2 )| ∼ 1. Thus, the implicit Then x1c (0, σ, s2 ) = s2 function theorem shows that for δ sufficiently small, there is a unique critical point x1c = x1c (δ, σ, s2 ) of   depending smoothly on δ, σ and s2 , that is, 1/(n−2)

  (x1c (δ, σ, s2 ), δ, σ, s1 , s2 ) = 0. Lemma 5.2. The phase  given by (5.5) can be developed locally around the critical point x1c of   in the form (x1c (δ, σ, s2 ) + y1 , δ, σ, s1 , s2 )=B0 (s  , δ, σ ) − B1 (s  , δ, σ )y1 +B3 (s2 , δ, σ, y1 )y13 , where B0 , B1 and B3 are smooth functions and where |B3 (s2 , δ, σ, y1 )| ∼ 1, and indeed (n−3)/(n−2)

B3 (s2 , δ, σ, 0) = s2

G4 (s2 , δ, σ ),

79

IMPROVED ESTIMATES

where G4 is smooth and satisfies G4 (s2 , 0, σ ) =

n(n − 1)(n − 2) σβ(0). 6

Moreover, we may write  1/(n−2) c  G1 (s2 , δ, σ ),   x1 (δ, σ, s2 ) = s2  1/(n−2) n/(n−2) B0 (s  , δ, σ ) = s1 s2 G1 (s2 , δ, σ ) − s2 G2 (s2 , δ, σ ),     B (s  , δ, σ ) = −s + s (n−1)/(n−2) G (s , δ, σ ), 1 1 3 2 2 with smooth functions G1 , G2 , and G3   G1 (s2 , 0, σ )     G2 (s2 , 0, σ )      G (s , 0, σ ) 3 2

(5.8)

satisfying = 1, n2 − n − 2 σβ(0), 2 = n(n − 2)σβ(0).

(5.9)

=

Notice that all the numbers in (5.9) are nonzero, since we assume n ≥ 5. Moreover, if we put G5 := G1 G3 − G2 , then we have G3 (s2 , 0, σ )  = 0 and

G5 (s2 , 0, σ ) =

n2 − 3n + 2 σβ(0)  = 0. 2

(5.10)

Proof. The first statements in (5.8) and (5.9) are obvious. Next, by (5.5) and (5.4) we have B0 (s  , δ, σ ) = (x1c (δ, σ, s2 ), δ, σ, s1 , s2 ) = s1 s2

1/(n−2)

G1 (s2 , δ, σ )

n/(n−2) (G1 (s2 , δ, σ )2 ω(δ1 x1c ) + σ G1 (s2 , δ, σ )n β(δ1 x1c ) +s2 (n−4)/(n−2) Y3 (δ1 x1c , δ2 , δ0 s2 )), +δ02 s2

where x1c is given by the first identity in (5.8). In combination with (5.7), we thus obtain the second identity in (5.8) and the third in (5.9) because s2 ∼ 1. Similarly, −B1 (s  , δ, σ ) =   (x1c (δ, σ, s2 ), δ, σ, s1 , s2 ) = s1 + 2s2 x1c ω(δ1 x1c ) + nσ (x1c )n−1 β(δ1 x1c ) + O(|δ|), which in view of (5.7) easily implies the last identities in (5.8) and (5.9). Finally, when y1 = 0, then 6B3 (s2 , δ, σ, 0) =   (x1c (δ, σ, s2 ), δ, σ, s1 , s2 ) = n(n − 1)(n − 2)σβ(0)(x1c )n−3 + O(|δ|), which shows that |B3 (s2 , δ, σ, y1 )| ∼ 1 for |y1 | sufficiently small. The proof of (5.10) is straightforward. Q.E.D.

80

CHAPTER 5

Translating the coordinate y1 in (5.6) by x1c , Lemma 5.2 then allows us to rewrite (5.6) in the following form:  νδλ (ξ ) = λ−1/2 χ1 (s1 s3 )χ1 (s2 s3 )χ1 (s3 ) e−iλs3 B0 (s ,δ,σ )  3  × e−iλs3 (B3 (s2 ,δ,σ,y1 )y1 −B1 (s ,δ,σ )y1 ) a0 (y1 , s, δ; λ) χ0 (y1 ) dy1 .

(5.11)

Here, χ0 is a smooth cutoff function supported in sufficiently small neighborhood of the origin, and a0 (y1 , s, δ; λ) is again a smooth function (possible different from the one in (5.6)), which is uniformly a classical symbol of order 0 with respect to λ. We shall now make use of Lemma 2.2, with B = 3. Let us apply this lemma and the subsequent remark to the oscillatory integral in (5.11). Putting u := B1 (s, δ, σ ), in view of this lemma we shall decompose the frequency support of νδλ furthermore into the domain where λ2/3 |B1 (s, δ, σ )|
1 (this is essentially a conic region in ξ space (cf. (5.1)), which will be called the “region near the Airy cone,” defined by B1 = 0), and the remaining domain into the conic regions where (2−l λ)2/3 |B1 (s, δ, σ )| ∼ 1, for M0 ≤ 2l ≤ Mλ1 , where M0 , M1 ∈ N are sufficiently large. Notice also that the Airy cone is given by the equation B1 = 0, that is, by (n−1)/(n−2)

s1 = s2

G3 (s2 , δ, σ )

(recall here from (5.1) that the sj are functions sj (ξ ) of the frequency variables ξ, which are homogeneous of degree 0). More precisely, we choose smooth cutoff functions χ0 and χ1 such that χ0 = 1 on a sufficiently large neighborhood of the origin and χ1 (t) is supported where |t| ∼ 1

λ −2l/3 χ t) = 1 on R \ {0} and define the functions νδ,Ai and l∈Z 1 (2 λ and νδ,l by 2/3 λ B1 (s  , δ, σ )) νδλ (ξ ), ν
δ,Ai (ξ ) := χ0 (λ −l 2/3 λ B1 (s  , δ, σ )) νδλ (ξ ), ν δ,l (ξ ) := χ1 ((2 λ)

so that



λ + νδλ = νδ,Ai

M0 ≤ 2l ≤

λ , M1

λ νδ,l .

(5.12)

M0 ≤2l ≤λ/M1 λ λ Denote by Tδ,Ai and Tδ,l the convolution operators λ λ Tδ,Ai ϕ := ϕ ∗ ν
δ,Ai ,

λ λ Tδ,l ϕ := ϕ ∗ ν δ,l .

Since δ0 = 2−j , we note that in order to prove (5.3) and thus Proposition 4.2, it , then will suffice to prove the following estimate: if pc := 14 13  2≤λ≤δ0−6

λ

Tδ,Ai

pc →pc

        λ + T δ,l   M0 ≤2l ≤λ/M1 2≤λ≤δ−6  0

−1/7

≤ C δ0

pc →pc

provided δ is sufficiently small and M0 , M1 ∈ N are sufficiently large.

,

(5.13)

81

IMPROVED ESTIMATES λ 5.1.1 Estimation of Tδ,Ai

We first consider the region near the Airy cone and prove the following. Lemma 5.3. There are constants C1 , C2 such that −5/6 λ

ν
, δ,Ai ∞ ≤ C1 λ

(5.14)

λ

∞ ≤ C2 λ7/6 ,

νδ,Ai

(5.15)

uniformly in σ and δ, provided λ is sufficiently large and δ is sufficiently small. Notice that by interpolation (again with θ = 37 ), these estimates imply that λ

Tδ,Ai

pc →pc
(λ−5/6 )4/7 (λ7/6 )3/7 = λ1/42 ,

so that



−1/7

λ

Tδ,Ai

pc →pc
δ0

,

(5.16)

2≤λ≤δ0−6

which is exactly the estimate that we need (cf. (5.13)). Let us turn to the proof of Lemma 5.3. The first estimate (5.14) is immediate from (5.11) and Lemma 2.2 (with M = 3). In order to prove the second estimate, observe first that by Lemma 2.2(a) and the subsequent remark, we may write  3  χ0 (λ2/3 B1 (s, δ, σ )) e−iλs3 (B3 (s2 ,δ,σ,y1 )y1 −B1 (s ,δ,σ )y1 ) a0 (y1 , s, δ; λ) χ0 (y1 ) dy1 = λ−1/3 χ0 (λ2/3 B1 (s  , δ, σ )) g(λ2/3 B1 (s  , δ, σ ), λ, δ, σ, s), where g is a smooth function whose derivates of any order are uniformly bounded on its natural domain. λ , (5.11) and this identity yield Applying the Fourier inversion formula to νδ,Ai that  λ λ−1/2 λ−1/3 χ0 (λ2/3 B1 (s  , δ, σ )) χ1 (s1 s3 )χ1 (s2 s3 )χ1 (s3 ) eiξ ·x νδ,Ai (x) = 

×e−iλs3 B0 (s ,δ,σ ) g(λ2/3 B1 (s  , δ, σ ), ξ3 , δ, σ, s) dξ. We again change coordinates from ξ = (ξ1 , ξ2 , ξ3 ) to (s1 , s2 , s3 ) according to (5.1). We then find that   λ 13/6 e−iλs3 (B0 (s ,δ,σ )−s1 x1 −s2 x2 −x3 ) χ0 (λ2/3 B1 (s  , δ, σ )) νδ,Ai (x) = λ   (5.17) × g λ2/3 B1 (s  , δ, σ ), λ, δ, σ, s χ˜ 1 (s) ds1 ds2 ds3 , with a smooth function g, and where χ˜ 1 (s) := χ1 (s1 s3 )χ1 (s2 s3 )χ1 (s3 ) s32 localizes to a region where sj ∼ 1, j = 1, 2, 3.

82

CHAPTER 5

Observe first that when |x|  1, then we easily obtain by means of integration by parts that λ |νδ,Ai (x)| ≤ CN λ−N ,

N ∈ N, if |x|  1.

Indeed, when |x1 |  1, then we integrate by parts repeatedly in s1 to see this, and a similar argument applies when |x2 |  1, where we use the s2 -integration. Observe that in each step, we gain a factor λ−1 and lose at most λ2/3 . Finally, when |x1 | + |x2 |
1 and |x3 |  1, then we can integrate by parts in s3 in order to establish this estimate. We may therefore assume in the sequel that |x|
1. Then we perform yet another change of coordinates, passing from s  = (s1 , s2 ) to (z, s2 ), where z := λ2/3 B1 (s  , δ, σ ). Applying (5.8), we find that (n−1)/(n−2)

2

z = λ 3 (−s1 + s2

G3 (s2 , δ, σ ))

so that s1 = s2(n−1)(n−2) G3 (s2 , δ, σ ) − λ−2/3 z.

(5.18)

In combination with (5.8), we obtain that B0 (s, δ, σ ) = −λ−2/3 z s2

1/(n−2)

n/n−2

G1 (s2 , δ, σ ) + s2

G5 (s2 , δ, σ ).

(5.19)

We may thus rewrite  λ 3/2 e−iλs3 (z,s2 ,x1 ,δ,σ ) νδ,Ai (x) = λ   (n−1)/(n−2) × g z, λ, δ, σ, s2 G3 (s2 , δ, σ ) − λ−2/3 z, s2 , s3   (n−1)/(n−2) G3 (s2 , δ, σ ) − λ−2/3 z, s2 χ0 (z) dz ds2 ds3 , (5.20) × χ˜ 1 s2 where n/(n−2)

(z, s2 , x1 , δ, σ ) := s2

G5 (s2 , δ, σ ) − s2(n−1)(n−2) G3 (s2 , δ, σ )x1 − s2 x2 − x3

+ λ−2/3 z (x1 − s2

1/(n−2)

G1 (s2 , δ, σ )).

(5.21)

Recall that n ≥ 5 and, from (5.10), that G3 (s2 , δ, σ )  = 0 and G5 (s2 , δ, σ )  = 0 when δ = 0. Moreover, the exponents n/(n − 2), (n − 1)/(n − 2), and 1 of s2 , which appear in (regarding the last term in (5.21) as an error term), are all different. Recall also that we assume |x|
1. It is then easily seen that this implies that, when δ = 0, 3 

|∂sj2 (z, s2 , x1 , δ, σ )| ∼ 1

for every s2 ∼ 1,

j =1

uniformly in z and σ. The same type of estimate then remains valid for δ sufficiently small. We may thus apply the van der Corput type Lemma 2.1 to the s2 -integration in (5.20), which in combination with Fubini’s theorem, yields λ

νδ,Ai

∞ ≤ Cλ3/2 λ−1/3

and, hence, (5.15). This concludes the proof of Lemma 5.3.

83

IMPROVED ESTIMATES

5.1.2 Estimation of T λδ,l We next regard the region away from the Airy cone. The study of this region will require substantially more refined techniques. Let us first note that by (5.6) and Fourier inversion we have  λ χ1 (s1 s3 )χ1 (s2 s3 )χ1 (s3 ) χ1 ((2−l λ)2/3 B1 (s  , δ, σ )) (x) = λ3 λ−1/2 νδ,l × e−iλs3 ((y1 ,δ,σ,s1 ,s2 )−s1 x1 −s2 x2 −x3 ) a(y1 , δ, σ, s; λ) χ1 (y1 ) dy1 ds, (5.22) where the amplitude a has properties similar to those of a0 . In order to indicate the problems that we have to face here, let us state (yet without proof) an analogue to Lemma 5.3, which we believe gives essentially optimal estimates (the proof will be established later in the course of the proof of the even more refined estimates of the next section). Lemma 5.4. There is a constant C so that λ

≤ C2−l/6 λ−5/6 ,

ν δ,l ∞

(5.23)



λ

νδ,l

∞ ≤ C min λ7/6 2l/3 ,



λ , δ0

(5.24)

uniformly in σ and δ, provided δ is sufficiently small. In order to apply this lemma, let us write λ = 2r , r ∈ N. Then, according to (5.23), we have −(5r+l)/6 λ ,

ν δ,l ∞
2

For k ∈ N, we therefore define νδ,k :=



(5.25)

r

2 νδ,l ,

Ik

where Ik := {(r, l) ∈ N2 : 5r + l = k, 2r ≤ δ0−6 } (if Ik = ∅, then by definition νδ,k := 0). Then    λ νδ,l = νδ,k , (5.26) k∈N

M0 ≤2l ≤λ/M1 2≤λ≤δ0−6

and we have the following consequence of Lemma 5.4. Corollary 5.5. There is a constant C so that


νδ,k ∞ ≤ C 2−k/6 ;

(5.27) −1/3

νδ,k ∞ ≤ C 22k/9 δ0

,

(5.28)

uniformly in σ and δ, provided δ is sufficiently small. Proof. The first estimate (5.27) follows immediately from (5.25) because the sup2r ports of the functions {ν δ,l }r,l are essentially disjoint.

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CHAPTER 5

Next, we decompose Ik = Ik1 ∪ Ik2 , where Ik1 := {(r, l) ∈ N2 : 5r + l = k, 2r+2l ≤ δ0−6 }, Ik2 := {(r, l) ∈ N2 : 5r + l = k, δ0−6 < 2r+2l , 2r ≤ δ0−6 }. Notice that according to (5.24), for (r, l) ∈ Ik1 we have λ7/6 2l/3 ≤ δλ0 ; hence,

νδλ ∞
27r/6 2l/3 = 22k/9 2(r+2l)/18 , whereas for (r, l) ∈ Ik2 we have νδλ ∞
2r /δ0 = (22k/9 /δ0 )2−(r+2l)/9 , so that

νδ,k ∞ ≤ C 2 9 k 2



2(r+2l)/18 +

(r,l)∈Ik1

22k/9  −(r+2l)/9 2 . δ0 2 (r,l)∈Ik

Comparing the latter sums with one-dimensional geometric series and using that 2r+2l ≤ δ0−6 in the first sum and 2r+2l > δ0−6 in the second sum, we obtain (5.28). Q.E.D. Let us denote by Tδ,k the convolution operator ϕ  → ϕ ∗ ν
δ,k . Interpolating the estimates in the preceding lemma, again with parameter θc := 37 , we obtain −1/7

Tδ,k pc →pc
δ0

−1/7

uniformly in k, whereas for 1 ≤ p < pc we get Tδ,k p→p
2−εk δ0 ε > 0 that depends on p, so that by (5.12), (5.16), and (5.26),     −1 −1   Tδλ 
δ0 7 +

Tδ,k p→p
δ0 7 .   2≤λ≤δ0−6

p→p

for some

k∈N

We thus barely fail to establish the estimate (5.13) at the critical exponent p = pc . In order to prove the estimate (5.13) also at the endpoint p = pc , we need to apply an interpolation argument. As mentioned in Section 2.4, in the majority of situations we shall make use of Stein’s interpolation theorem for analytic families of operators, but in a few cases we can alternatively apply the real interpolation result of Proposition 2.6, which is based on ideas of Bak and Seeger [BS11] and whose assumptions are much easier to verify. Nevertheless, we shall also encounter several situations in which we don’t know how to adapt the methods from [BS11] but that still can by treated by means of complex interpolation. The latter applies also to the proof of the endpoint estimate in Proposition 4.2(c). Indeed, what seems to prevent the application of the real interpolation method is that the (complex) measures νδ,k arise from the positive measure νδ by means of spectral localizations to certain frequency regions, that is, νδ,k = νδ ∗ ψδ,k , and the obstacle in applying the method from [BS11] is that there is no uniform bound for the L1 -norms of the functions ψδ,k as k tends to infinity. The proofs, based on complex interpolation, are technically involved, and our arguments outlined in the next section can be viewed as prototypical for other proofs of the same kind appearing in later chapters.

85

IMPROVED ESTIMATES

5.2 THE ENDPOINT IN PROPOSITION 4.2(C): COMPLEX INTERPOLATION We keep the notation of the previous section. According to (5.11) and Lemma 5.2, we may write (recalling that ξ = λs3 (s1 , s2 , 1))  − 12 λ ν χ1 ((2−l λ)2/3 B1 (s  , δ, σ ))χ˜ 1 (s) e−iλs3 B0 (s ,δ,σ ) J (λ, s, δ, σ ), (5.29) δ,l (ξ ) := λ

where we recall that χ˜ 1 localizes to a region where sj ∼ 1, j = 1, 2, 3, and where  ˜ J (λ, s, δ, σ ) := e−iλs3 0 (y1 ,δ,σ,s1 ,s2 ) a0 (y1 , s, δ; λ) χ0 (y1 ) dy1 with ˜ 0 (y1 , δ, σ, s1 , s2 ) := B3 (s2 , δ, σ, y1 )y13 − B1 (s  , δ, σ )y1 .  Since B1 is of size (2l /λ)2/3 , we scale by the factor (2l /λ)1/3 in the integral defining J (λ, s, δ, σ ) by putting y1 = (2l /λ)1/3 u1 and obtain  l  l −1 1/3 e−is3 2 0 (u1 ,s ,δ,λ,l) a0 ((2l λ−1 )1/3 u1 , s, δ, λ) J (λ, s, δ, σ ) = (2 λ ) × χ0 ((2l λ−1 )1/3 u1 ) du1 with 0 (u1 , s  , δ, λ, l) := B3 (s2 , δ, σ, (2l λ−1 )1/3 u1 ) u31 − (2l λ−1 )−2/3 B1 (s  , δ, σ ) u1 · Observe that the coefficients of u1 and of u31 in 0 are both of size 1, so that 0 will have no critical point with respect to u1 unless |u1 | ∼ 1. We may therefore choose a smooth cutoff function χ1 ∈ C0∞ (R) supported away from 0 so that 0 has no critical point outside the support of χ1 , and decompose J := J (λ, s, δ, σ ) = J1 + J∞ , where J1 = J1 (λ, s, δ, σ ) is given by  l  l −1 1/3 e−is3 2 0 (u1 ,s ,δ,λ,l) a0 ((2l λ−1 )1/3 u1 , s, δ, λ) J1 := (2 λ ) × χ0 ((2l λ−1 )1/3 u1 ) χ1 (u1 ) du1 . Accordingly, in view of (5.29) we may decompose λ λ λ νδ,l = νl,1 + νl,∞ ,

where the summands are defined by  −1/2 λ ν χ1 ((2−l λ)2/3 B1 (s  , δ, σ ))χ˜ 1 (s) e−iλs3 B0 (s ,δ,σ ) J1 (λ, s, δ, σ ), l,1 (ξ ) := λ  −1/2 λ ν
χ1 ((2−l λ)2/3 B1 (s  , δ, σ ))χ˜ 1 (s) e−iλs3 B0 (s ,δ,σ ) J∞ (λ, s, δ, σ ) l,∞ (ξ ) := λ

(we have dropped the dependence on δ in order to simplify the notation).

86

CHAPTER 5

λ Let us first consider the contribution given by the νl,∞ : By means of integration by parts, we easily obtain that for every N ∈ N, we have |J∞ |
(2l λ−1 )1/3 2−lN ; hence, −1/2 l −1 1/3 −lN λ (2 λ ) 2

ν
l, ∞ ∞
λ

∀N ∈ N.

(5.30)

Next, we may assume that we have chosen χ˜ 1 so that the Fourier inversion formula λ (x) reads for νl,∞  λ 3 λ eiλs3 (s1 x1 +s2 x2 +x3 ) ν
νl,∞ (x) = λ l, ∞ (ξ ) ds R3

(with ξ = λs3 (s1 , s2 , 1)). We then use the change of variables from s  = (s1 , s2 ) to (z, s2 ), where now z := (2−l λ)2/3 B1 (s  , δ, σ ), and find that (compare (5.18)) (n−1)/(n−2)

s1 = s2

G3 (s2 , δ, σ ) − (2−l λ)−2/3 z,

(5.31)

and, in particular (compare (5.19)), B0 (s, δ, σ ) = −(2−l λ)−2/3 z s2

1/(n−2)

n/(n−2)

G1 (s2 , δ, σ ) + s2

G5 (s2 , δ, σ ). (5.32)

λ Notice that 2l /λ ≤ 1/M1  1. And, if we plug in the previous formula for ν
l,∞ and λ write νl,∞ (x) as an oscillatory integral with respect to the variables u1 , z, s2 , s3 , we see that the complete phase is of the form n/(n−2)

−λs3 (s2

(n−1)/(n−2)

G5 (s2 , δ, σ ) − x1 s2 l −1

G3 (s2 , δ, σ )

−s2 x2 − x3 + O(2 λ (1 + |u1 | ))), 3

where according to (5.10), |G5 | ∼ 1. Observe that the localization given by the function χ0 in the definition of J (λ, s, δ, σ ) implies that 2l λ−1 |u1 |3  1. Again, first applying N integrations by parts with respect to u1 and then van der Corput’s lemma (with M = 3) for the integration in s2 , also taking into account the Jacobian of our change of coordinates to z, we see that

νl,λ ∞ ∞
λ3 λ−1/2 (2l λ−1 )1/3 2−lN (2−l λ)−2/3 λ−1/3 = λ7/6 2−l(N−1) . Interpolating between this estimate and (5.30), with θc = 37 , we see that the conλ λ , which maps ϕ to ϕ ∗ ν
volution operator Tl,∞ l, ∞ , can be estimated by

Tl,λ∞ pc →pc
λ(−5/6)(4/7)+(7/6)(3/7) 2−l = λ1/42 2−l , if we choose N = 2. This implies the desired estimate   −1/7

Tl,λ∞ pc →pc
δ0 . M0 ≤2l ≤λ/M1 2≤λ≤δ0−6

87

IMPROVED ESTIMATES λ 5.2.1 The operators Tl,1

λ We now turn to the investigation of the convolution operator Tl,1 , which maps ϕ to  λ ϕ ∗ ν . According to (5.13), what we need to prove is that the operator l,1

T1 :=





λ Tl,1

M0 ≤2l ≤λ/M1 2≤λ≤δ0−6

satisfies −1/7

T1 pc →pc
δ0

,

(5.33)

with a bound that is independent of δ and σ. Now, if the phase 0 has no critical point on the support of χ1 , then we can λ as we estimate J1 in the same way as J∞ before and can handle the operators Tl,1 λ did for the Tl,∞ . Let us therefore assume in the sequel that 0 does have a critical point uc1 ∈ supp χ1 , so that |u1 | ∼ 1. Applying the method of stationary phase, we then get |J1 |
(2l λ−1 )1/3 2−l/2 ; hence, −1/2 l −1 1/3 −l/2 λ (2 λ ) 2 = λ−5/6 2−l/6 = 2−k/6 ,

ν l,1 ∞
λ

(5.34)

where we have used the same abbreviations, λ := 2r , k = k(r, l) := 5r + l, as in the previous section. In view of this estimate, we define for ζ in the complex strip  := {ζ ∈ C : 0 ≤ Re ζ ≤ 1} the following analytic family of measures   r ζ /3 2[k(r,l)(3−7ζ )]/18 νl,2 1 , µζ (x) := γ (ζ ) δ0 M0 ≤2l ≤2r /M1 2≤2r ≤δ0−6

where γ (ζ ) :=

27(ζ −1)/2 − 1 , 2−2 − 1

ζ . Observe that for ζ = θc = 37 , and denote by Tζ the operator of convolution with µ 1/7 we have Tθc = δ0 T1 , so that by Stein’s interpolation theorem [SW71], (5.33) will follow if we can prove the following estimates on the boundaries of the strip  :

Tit 1→∞ ≤ C and T1+it 2→2 ≤ C, where the constant C is independent of t ∈ R and the parameters δ, σ (provided δ is sufficiently small). Equivalently, we shall prove that

 µit ∞ ≤ C

∀t ∈ R,

(5.35)

µ1+it ∞ ≤ C

∀t ∈ R.

(5.36)

2 Since the supports of the functions {ν l,1 } are almost disjoint for l, r in the given range, we see that the first estimate (5.35) is an immediate consequence of (5.34). r

88

CHAPTER 5

The main problem will consist in estimating µ1+it ∞ . To this end, observe that, again by Fourier inversion, we have (with ξ = λs3 (s1 , s2 , 1))  λ λ (x) = λ3 eiλs3 (s1 x1 +s2 x2 +x3 ) ν νl,1 l,1 (ξ ) ds. R3

Using once again the change of variables from s1 to z, so that z = (2−l λ)2/3 (n−1)/(n−2) B1 (s  , δ, σ ) and s1 = s2 G3 (s2 , δ, σ ) − (2−l λ)−2/3 z, we find that (compare (5.31), (5.32))    λ 3/2 l e−is3 1 (x,u1 ,z,s2 ,δ,λ,l) a (2l λ−1 )1/3 u1 , z, s2 , s3 , δ; λ νl,1 (x) = λ 2 × χ1 (u1 )χ1 (z)χ1 (s2 )χ1 (s3 ) du1 dz ds2 ds3 ,

(5.37)

where 1 = 1 (x, u1 , z, s2 , δ, λ, l) is given by

1 := 2l (B3 (s2 , δ, σ, (2l λ−1 )1/3 u1 ) u31 − z u1 ) n/(n−2)

+λ(s2

G5 (s2 , δ, σ ) − s2(n−1)(n−2) G3 (s2 , δ, σ ) x1 − s2 x2 − x3 )

+λ(2l λ−1 )2/3 z (x1 − s2

1/(n−2)

G1 (s2 , δ, σ )).

(5.38)

Moreover, a(v, u1 , z, s2 , s3 , δ; λ) is a smooth function that is uniformly a classical symbol of order 0 with respect to λ. Notice that, in order to simplify the notation, here we have suppressed the dependence on σ, which we shall also do in the sequel. 5.2.2 Preliminary reductions Assume now, first, that |x|  1. If |x1 |  |(x2 , x3 )|, then we easily see by means of integrations by parts in (5.37) with respect to the variables s2 or s3 that |νl,1 (x)|
λ−N for every N ∈ N, and if |x1 |  |(x2 , x3 )|, then we easily obtain |νl,1 (x)|
(λ(2l λ−1 )2/3 )−N , by means of integrations by parts in z. Since 2l ≤ λ, it follows easily that there are constants A ≥ 1 and C such that sup|x|≥A supt∈R |µ1+it (x)| ≤ C, uniformly in δ and σ. From now on we shall therefore assume that |x| ≤ A. For such x fixed, we decompose the support of χ1 (s2 ) into the subset LI I of all s2 such that ε(2l λ−1 )1/3 < |x1 − s2

1/(n−2)

G1 (s2 , δ, σ )|
0 will be a sufficiently small fixed number. If we restrict the set of integration in (5.37) to these subsets with respect to the λ λ λ λ , νl,I variable s2 , we obtain corresponding measures νl,I I and νl,I I I into which νl,1 decomposes, that is, λ λ λ λ = νl,I + νl,I νl,1 I + νl,I I I .

Observe also that |λ(2l λ−1 )2/3 (x1 − s2 1/(n−2) |x1 − s2 G1 (s2 , δ, σ )| ≥ (2l λ−1 )1/3 .

1/(n−2)

G1 (s2 , δ, σ ))| ≥ 2l if and only if

89

IMPROVED ESTIMATES

Thus, if s2 ∈ LI , the last term in (5.38) becomes dominant as a function of z, provided we choose ε sufficiently small. Consequently, the phase has no critical point as a function of z, and applying N integrations by parts in z, we may estimate λ (x)|
λ3/2 2l |νl,I



ds2

 {s2

:λ1/3 22l/3


λ3/2 2l

 λ1/3 22l/3 |v|≥C2l

1/(n−2)

λ1/3 22l/3 |x1 − s2

1/(n−2) |x1 −s2 G1 (s2 ,δ,σ )|≥C2l , |s2 |∼1}

G1 (s2 , δ, σ )|

N

dv (λ1/3 22l/3 |v|)N


λ3/2 2l (λ1/3 22l/3 )−1 2(1−N)l = λ7/6 2(4/3−N)l . Similarly, if s2 ∈ LI I I , the first term in (5.38) becomes dominant as a function of z, and thus N integrations by parts in z and the fact that the s2 -integral is restricted 1 to a set of size (2l λ−1 ) 3 yield the same estimate: λ 3/2 l −Nl l −1 1/3 |νl,I 2 2 (2 λ ) = λ7/6 2(4/3−N)l . I I (x)|
λ

This implies the desired estimate    (1+it)/3 γ (1 + it) δ0



  2r 2r 2k(r,l)(3−7(1+it))/18 (νl,I + νl,I I I )(x)

M0 ≤2l ≤2r /M1 2≤2r ≤δ0−6



1/3


δ0

λ λ λ−10/9 2−2l/9 (|νl,I (x)| + |νl,I I I (x)|)

M0 ≤2l ≤λ/M1 , 1λM02 δ0−6

Assume next that 2−l 22m/3 |B˜ 1 (x, δ, σ )|
δ02 2m/3 , but 2m−2l |B˜ 0 (x, δ, σ )|  Then an integration by parts in s3 shows that

δ02 2m/3 .

|fm,x (2l )|
(2m−2l |B˜ 0 (x, δ, σ )|)−1 , so that we can argue in the same way is in the preceding subcase to see that the I,2 contribution of the corresponding fm,x (2l ) to µI1+it (x) is again uniformly bounded with respect to t, x, δ, and σ. We may thus assume that 22m/3 2−l |B˜ 1 (x, δ, σ )|
δ02 2m/3 and 2m−2l |B˜ 0 (x, δ, σ )|
δ02 2m/3 . Then we may rewrite fm,x (2l ) in (5.43) as  2 m/3 l fm,x (2l ) = e−is3 δ0 2 5 (x,v,δ,m,2 ) a(2l−m/3 , v, x1 , s3 , δ; 2m−2l , 2l ) × χ˜ 1 (v)χ1 (s3 ) dv ds3 ,

95

IMPROVED ESTIMATES

where

5 := B˜ 2 (x, δ0 2l−m/3 v, δ, σ ) v 2 + δ0−2 2m/3−l B˜ 1 (x, δ, σ )v + 22m/3−2l δ0−2 B˜ 0 (x, δ, σ). ˜ 5 (x, v, δ, m, 2l )|
1. Observe also that here |

Let us first consider those l for which 22m/3 2−l |B˜ 1 (x, δ, σ )|  δ02 2m/3 . Then the coefficient of 5 of the linear term in v is small, so that we may change variables in the integral from v to 5 (x, v, δ, m, v) (as a new variable), which then easily shows that fm,x (2l ) is of the form fm,x (2l ) = F (δ02 2m/3 ; 22m/3−2l δ0−2 B˜ 0 (x, δ, σ ), δ0−2 2m/3−l B˜ 1 (x, δ, σ ), 2l−m/3 , δ; 2m−2l , 2l ), where F is a smooth function that is a Schwartz function with respect to the first variable, whose Schwartz norms are all uniformly bounded with respect to the other variables. Moreover, F is uniformly a classical symbol of order 0 in both of the last two variables. More precisely, we may estimate   α α β ∂ 1 ∂ 2 ∂ F (z 1 ; z 2 , z 3 , z 4 , z 5 ; µ1 , µ2 ) ≤ CN,α ,α ,β (1 + |z 1 |)−N |µ1 |−α1 |µ2 |−α2 , 1 2 µ1 µ2 z for every N ∈ N. This clearly implies that |fm,x (2l )|
(δ02 2m/3 )−N for every N ∈ N. However, such an estimate is not sufficient in order to control the summation in l. We therefore isolate the leading homogeneous term of order 0 of F with respect to the last two variables, which gives a smooth function h(δ02 2m/3 ; 22m/3−2l δ0−2 B˜ 0 (x, δ, σ ), δ0−2 2m/3−l B˜ 1 (x, δ, σ ), 2l−m/3 , δ) of its variables, and the remainder terms, which clearly can be estimated by a constant times (δ02 2m/3 )−N ((2m−2l )−1 + 2−l ). The second factor allows us to sum in l, and then the first factor (choosing N = 1) leads again to an estimate of the form (5.46) for the contribution by the remainder terms. In order to control the main term given by the function h, we shall again apply Lemma 2.7. Let us here define a cuboid Q by the following set of restrictions, for suitable R1 , ε2 > 0 : |y1 | = |2−2l 22m/3 δ0−2 B˜ 0 (x, δ, σ )| ≤ R1 , |y3 | = |2l 2−m/3 | ≤ ε0 ,

|y2 | = |2−l δ0−2 2m/3 B˜ 1 (x, δ, σ )| ≤ ε2 ,

|y4 | = |2−l δ03 2m/2 | ≤ 1

(the last condition stems for the additional summation restriction in the definition I,2 (x)), and let us define Hm,δ (y1 , . . . , y4 ) := h(δ02 2m/3 ; y1 , y2 , y3 , δ). Then of µI1+it (choosing N = 1),

Hm,δ C 1 (Q) ≤ C(δ02 2m/3 )−1 , I,2 (x) and thus Lemma 2.7 implies that the sum over the ls in the definition of µI1+it can be estimated by C(δ02 2m/3 )−1 , so that the remaining sum in m can again be estimated by the expression in (5.46). This concludes the discussion of this subcase.

96

CHAPTER 5

We are thus eventually reduced to those ls for which 22m/3 2−l |B˜ 1 (x, δ, σ )| ∼  1 and 2m−2l |B˜ 0 (x, δ, σ )|
δ02 2m/3 . Assume more precisely that we consider here pairs (m, l) for which 1 2 m/3 ≤ 22m/3 2−l |B˜ 1 (x, δ, σ )| ≤ Aδ02 2m/3 , (5.47) δ 2 A 0 where A  1 is a fixed constant. In this situation, the phase 5 will have only nondegenerate critical points of size 1 as a function of v or else none. The latter case can be treated as before, so assume that we do have a critical point v c such that |v c | ∼ 1. Then we may apply the method of stationary phase in v in (5.43), which leads to the following estimate for fm,x (2l ): δ02 2m/3

|fm,x (2l )|
(δ02 2m/3 )−1/2 . But, given m, (5.47) means that we are summing over at most log A2 different ls, and thus the contribution of those fm,x (2l ) that we are considering here to the sum I,2 (x) can be estimated by forming µI1+it  1/3 2m/18 (δ02 2m/3 )−1/2
1. C log A2 δ0 2m >M02 δ0−6

Combining this estimate with the previous ones, we see that we can bound I,2 (x)| ≤ C, with a constant C that is independent of t, x, δ and σ . This |µI1+it concludes the proof of the estimate (5.39), hence of (5.36) and (5.33), finally of (4.53), and consequently of Proposition 4.2(c).

5.3 PROOF OF PROPOSITION 4.2(A), (B): COMPLEX INTERPOLATION For the proofs of parts (a) and (b) of Proposition 4.2 we shall make use of similar interpolation schemes. A crucial result for part (a) will also be Lemma 2.9 on oscillatory double sums. 5.3.1 Estimate (4.51) in Proposition 4.2(a) Recall that δ0 = 2−j , that 2  2j

V νδ,j

=

−M 2 λ1

νj(λ1 ,λ1 ,λ3 )

λ1 =2M+j λ3 =(2−M−j λ1 )2

(in this notation, summation is always meant to be over dyadic λj s), and that, by −1/2 −1/2 (4.43), νjλ ∞
λ1 λ3 . We therefore define an analytic family of measures for ζ in the strip  = {ζ ∈ C : 0 ≤ Re ζ ≤ 1} by ζ /3

µζ (x) := γ (ζ ) δ0

2j 

−M+k 1

k1

2(3−7ζ )k1 /6 2(3−7ζ )k3 /6 νj(2

,2k1 ,2k3 )

,

k1 =M+j k3 =−2M+2k1 −2j

where γ (ζ ) is an entire function that will serve a similar role as the function γ (z) in Subsection 5.2.1. Its precise definition will be given later (based on Remark 2.10). It will again be uniformly bounded on , such that γ (θc ) = γ ( 73 ) = 1.

97

IMPROVED ESTIMATES

By Tζ we denote the operator of convolution with µ ζ . Observe that for ζ = 1/7 V V ; hence, Tθc = 2−j/7 Tδ,j , so that, again by Stein’s θc = 37 , we have µθc = δ0 νδ,j interpolation theorem, (4.51) will follow if we can prove the following estimates on the boundaries of the strip  :

 µit ∞ ≤ C

∀t ∈ R,

(5.48)

µ1+it ∞ ≤ C

∀t ∈ R.

(5.49)

As before, the first estimate (5.48) is an immediate consequence of the estimates (4.43), so let us concentrate on (5.49), that is, assume that ζ = 1 + it, with t ∈ R. We then have to prove that there is a constant C such that |µ1+it (x)| ≤ C,

(5.50)

where C is independent of t, x, δ, and σ. Let us introduce the measures µλ1 ,λ3 given by µλ1 ,λ3 (x) := (λ1 λ3 )−2/3 νj(λ1 ,λ1 ,λ3 ) (x), which allow us to rewrite −2

µ1+it (x) = γ (1 +

(1+it)/3 it) δ0

δ0 

−M 2 λ1

λ1 =2M δ0−1

λ3 =2−2M (δ0 λ1 )2

(λ1 λ3 )−7it/6 µλ1 ,λ3 (x). (5.51)

Notice that according to Remark 4.3,  4/3 1/3 χˇ 1 (λ1 (x1 − y1 )) χˇ 1 (λ1 (x2 − δ0 y2 − y12 ω(δ1 y1 ))) µλ1 ,λ3 (x) = λ1 λ3 ×χˇ 1 (λ3 (x3 − b0 (y, δ) y22 − σy1n β(δ1 y1 ))) η(y, δ) dy, where η is supported where y1 ∼ 1 and |y2 |
1. Assume first that |x|  1. Since χˇ 1 is rapidly decreasing, after scaling in y1 by the factor 1/λ1 , we then easily see 1/3 for every N ∈ N. Since 2j
λ1 ≤ 22j and that |µλ1 ,λ3 (x)| ≤ CN λ1 λ−N 3 (2−j λ1 )2
λ3  λ1 in the sum defining µ1+it (x), this easily implies (5.50). From now on, we may and shall therefore assume that |x|
1. 1/2 By means of the change of variables y1  → x1 − y1 /λ1 , y2  → y2 /λ3 and Taylor expansion around x1 we may rewrite 1/3 −1/6

µλ1 ,λ3 (x) = λ1 λ3 with

µ˜ λ1 ,λ3 (x),

 µ˜ λ1 ,λ3 (x) :=

χˇ 1 (y1 )Fδ (λ1 , λ3 , x, y1 , y2 ) dy1 dy2 ,

(5.52)

where −1/2

Fδ (λ1 , λ3 , x, y1 , y2 ) := η(x1 − λ−1 1 y1 , λ3 ×χˇ 1 (A −

y22 b0 (x1

y2 , δ) χˇ 1 (D − Ey2 + r1 (y1 )) −1/2

− λ−1 1 y1 , λ3

y2 , δ) + λ3 λ−1 1 r2 (y1 )).

98

CHAPTER 5

Here, the quantities A = A(x, λ3 , δ), D = D(x, λ1 , δ) and E = E(λ1 , λ3 , δ), given by A := λ3 QA (x), with

D =: λ1 QD (x),

−1/2

E := δ0 λ1 λ3

QA (x) := x3 − σ x1n β(δ1 x1 ), QD (x) := x2 − x12 ω(δ1 x1 ), (5.53)

do not depend on y1 , y2 , and ri (y1 ) = ri (y1 ; λ−1 1 , x1 , δ), i = 1, 2, are smooth funcand x ) satisfying estimates of the form tions of y1 (and λ−1 1 1    l     ∂ −1  ri (y1 ; λ1 , x1 , δ) ≤ Cl |y1 |l+1 for every l ≥ 1. |ri (y1 )| ≤ C|y1 |,  −1  ∂(λ1 )  (5.54) Notice that we may here assume that |y1 |
λ1 , because of our assumption |x|
1 and the support properties of η. It will also be important to observe that E = −1/2 ≤ 2M/2 for the index set of λ1 , λ3 over which we sum in (5.51). δ0 λ1 λ3 In order to verify (5.50), given x, we shall split the sum in (5.51) into three parts, according to whether |A(x, λ3 , δ)|  1, or |A(x, λ3 , δ)|
1 and |D(x, λ1 , δ)|  1, or |A(x, λ3 , δ)|
1 and |D(x, λ1 , δ)|
1. 1. The part where |A|  1. Denote by µ11+it (x) the contribution to µ1+it (x) by the terms for which |A(x, λ3 , δ)| > K, where K  1 is a large constant. We claim that for every  > 0, |µ˜ λ1 ,λ3 (x)|
|A|−1/2 ,

if |A| = |A(x, λ3 , δ)| > K,

(5.55)

provided K is sufficiently large. This estimate will imply the right kind of estimate, 

1/3

|µ11+it (x)|
δ0



{λ3 :1≤λ3 ≤δ0−2 , λ3 |QA (x)|≥K} λ1 ≤δ0−1 λ1/2 3






{λ3 :1≤λ3 ≤δ0−2 , λ3 |QA (x)|≥K}

1/3 −1/6

λ1 λ3 (λ3 |QA (x)|)1/2−

1 1
1/2− , (λ3 |QA (x)|)1/2− K

since we are summing over dyadic λ3 s. In order to verify (5.55), observe first that if we apply the van der Corput–type estimate in Lemma 2.1(b) (with M = 2) to the integration in y2 (making use of the last factor of Fδ ), we obtain  |Fδ (λ1 , λ3 , x, y1 , y2 )| dy2 ≤ C, where the constant C is independent of y1 , x, λ, and δ (recall that |b0 | ∼ 1!). Let ε > 0. It follows in particular that the contribution of the region where |y1 |  |A|ε to µ˜ λ1 ,λ3 can be estimated by the right-hand side of (5.55) because of the rapidly decreasing factor χˇ 1 (y1 ) in the double integral defining µ˜ λ1 ,λ3 (x). Let us thus consider the part of µ˜ λ1 ,λ3 (x) given by integrating over the region where |y1 | ≤ C|A|ε , where C is a fixed positive number. Here, according to (5.54), we have |r2 (y1 )|
|A|ε , and hence |A + λ3 λ−1 1 r2 (y1 )| ∼ |A| if we choose, for instance, ε < 12 and K sufficiently large.

99

IMPROVED ESTIMATES

Then an easy estimation for the y2 -integration leads to      −1/2 −1 y , λ y , δ) + λ λ r (y )) χˇ 1 (A − y22 b0 (x1 − λ−1  dy2
|A|−1/2 , 1 2 3 2 1 3 1 1 and integrating subsequently in y1 over the region |y1 | ≤ C|A|ε , we again arrive at the right-hand side of (5.55). 2. The part where |A|
1 and |D|  1. Denote by µ21+it (x) the contribution to µ1+it (x) by the terms for which |A(x, λ3 , δ)| ≤ K and |D(x, λ1 , δ)| > K. We claim that here 1 (5.56) , if |D| = |D(x, λ1 , δ)| > K, |µ˜ λ1 ,λ3 (x)|
|D| provided K is sufficiently large. It is again easy to see that this estimate will imply the right kind of estimate for |µ21+it (x)| (just interchange the roles of A and D and of λ1 and λ3 in the arguments of the previous situation). In order to prove (5.56), consider first the contribution to µ˜ λ1 ,λ3 (x) given by the integration over the region where |y1 | ≥ C|D|ε , where C is a fixed positive number. Arguing in the same way as in the previous situation, we find that this part can be estimated by the right-hand side of (5.56). Next, we consider the contribution to µ˜ λ1 ,λ3 (x) given by the integration over the region where |y1 | < C|D|ε and |y2 |  C|D|ε . According to (5.54), we then have that |rj (y1 )|
|D|ε , j = 1, 2, so that we may assume that |A + λ3 λ−1 1 r2 (y1 )|  |D|ε ; hence −1/2

|A − y22 b0 (x1 − λ−1 1 y1 , λ3

2ε y2 , δ) + λ3 λ−1 1 r2 (y1 )|  |D| .

This easily implies that this part of µ˜ λ1 ,λ3 (x) can also be estimated by the right-hand side of (5.56). What remains is the contribution by the region where |y1 | < C|D|ε and |y2 | < C|D|ε (with C sufficiently large, but fixed). Since E
1, we here have that |D − Ey2 + r1 (y1 )|  |D|, and again we see that we can estimate using the right-hand side of (5.56). 3. The part where |A|
1 and |D|
1. Finally, denote by µ31+it (x) the contribution to µ1+it (x) by those terms for which |A(x, λ3 , δ)| ≤ K and |D(x, λ1 , δ)| ≤ K. In this case, it is easily seen from formula (5.52) and (5.54) that −1/2

µ˜ λ1 ,λ3 (x) = J˜(A, D, E, λ−1 1 , λ3

, λ3 λ−1 1 ),

where J˜ is a smooth function of all its bounded variables; hence −1/2

δ0 µλ1 ,λ3 (x) = E 1/3 J (A, D, E 1/3 , λ−1 1 , λ3 1/3

, λ3 λ−1 1 ),

where again J is a smooth function. Let us write λ1 = 2m1 , λ3 = 2m2 , with m1 , m2 ∈ N. In combination with (5.51) −it/3 µ1+it (x) can be written in the form (2.19), with we then see that δ0 7 7 (α1 , α2 ) := (− 6 , − 6 ) and M1 = δ0−2 , M2 := 2−M δ0−2 . The cuboid Q is defined by

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the following set of restrictions: |y1 | = |λ3 QA (x)| ≤ K,

|y2 | = λ1 |QD (x)| ≤ K,

1/3 −1/6 1/3 δ0

|y3 | = |E 1/3 | = λ1 λ3 −1/2

|y5 | = λ3

≤ 2M/3 ,

|y4 | = λ−1 1 ≤ 1,

−M |y6 | = |λ−1 , 1 λ3 | ≤ 2

≤ 1,

2 M |y7 | = |λ21 λ−1 3 δ0 | ≤ 2 ,

−1 −M |y8 | = |λ−1 . 1 δ0 | ≤ 2

The first three conditions arise from our assumptions |A|
1, |D|
1, and |E|
1, and the last three arise from the restrictions on the summation indices in (5.51). Moreover, for the function H in Lemma 2.9, we my choose H (y1 , . . . , y8 ) := y3 J (y1 , . . . , y6 ). The corresponding vectors (β1k , β2k ) are given by (0, 1), (1, 0), ( 13 , − 16 ), (−1, 0), (0, − 12 ), (2, −1), (−1, 1) and (−1, 0). Therefore, if we choose for γ (ζ ) the corresponding function γ3/7 (ζ ) of Remark 2.10, then Lemma 2.9 shows that indeed µ31+it (x) also satisfies the estimate (5.50). This concludes the proof of Proposition 4.2 (a). 5.3.2 Estimate (4.52) in Proposition 4.2(b) Recall that δ0 = 2−j , that 2 

2 (2−M−j λ1 )

λ1 =2M+j

λ3 =2

2j

VI νδ,j

:=

νj(λ1 ,λ1 ,λ3 ) ,

−1/2 −1/2 and that, by (4.43), νjλ ∞
λ1 λ3 . We therefore define an analytic family of measures for ζ in the strip  = {ζ ∈ C : 0 ≤ Re ζ ≤ 1} by

µζ (x) :=

ζ /3 γ (ζ ) δ0

2j 

−2M+2k1 −2j

k1 =M+j

k3 =1



k1

2(3−7ζ )k1 /6 2(3−7ζ )k3 /6 νj(2

,2k1 ,2k3 )

,

where we shall put 27(1−z)/2 − 1 . 3 By Tζ we denote the operator of convolution with µ ζ . Observe that for ζ = θc = 37 , 1/7 V I VI we have µθc = δ0 νδ,j , hence, Tθc = 2−j/7 Tδ,j , so that, arguing exactly as in the preceding subsection by means of Stein’s interpolation theorem, (4.52) will follow if we can prove that there is constant C such that γ (ζ ) :=

|µ1+it (x)| ≤ C,

(5.57)

where C is independent of t, x, δ, and σ. As before, we introduce the measures µλ1 ,λ3 given by µλ1 ,λ3 (x) := (λ1 λ3 )−2/3 νj(λ1 ,λ1 ,λ3 ) (x), which allow us to rewrite 2 

2 (2−M−j λ1 )

λ1 =2M+j

λ3 =2

2j

µ1+it (x) = γ (1 +

(1+it)/3 it) δ0

(λ1 λ3 )−7it/6 µλ1 ,λ3 (x).

(5.58)

101

IMPROVED ESTIMATES

Recall also that according to Remark 4.3,  4/3 1/3 χˇ 1 (λ1 (x1 − y1 )) χˇ 1 (λ1 (x2 − δ0 y2 − y12 ω(δ1 y1 ))) µλ1 ,λ3 (x) = λ1 λ3 ×χˇ 1 (λ3 (x3 − b0 (y, δ) y22 − σy1n β(δ1 y1 )) η(y, δ) dy, where η is supported where y1 ∼ 1 and |y2 |
1. Assume first that |x|  1. If ≤ (λ1 λ2 )−N/2 |x1 |  1 or |x2 |  1, this easily implies that |µλ1 ,λ3 (x)| ≤ CN λ−N 1 1/2 for every N ∈ N because λ1  λ3 . Thus (5.57) follows immediately. and And, if |x3 |  1, we may estimate the last factor in the integrand by CN λ−N 3 4/3 1/3−N −1 −2/3 1/3−N then easily obtain that |µλ1 ,λ3 (x)| ≤ CN λ1 λ3 λ1 (λ1 δ0 )−1 = 2j λ1 λ3 . j 1/2 Summing first over all λ1  2 λ3 and then over λ3 , we find that |µ1+it (x)|
1/3 δ0 2j/3
1. From now on, we may and shall therefore assume that |x|
1. By means of the change of variables y1  → x1 − y1 /λ1 , y2  → y2 /(δ0 λ1 ) we rewrite −2/3 1/3 λ3

µλ1 ,λ3 (x) = δ0−1 λ1 with

 µ˜ λ1 ,λ3 (x) :=

µ˜ λ1 ,λ3 (x),

χˇ 1 (y1 )F˜δ (λ1 , λ3 , x, y1 , y2 ) dy1 dy2 ,

(5.59)

where −1 −1 ˇ 1 (D − y2 + r1 (y1 )) F˜δ (λ1 , λ3 , x, y1 , y2 ) := η(x1 − λ−1 1 y1 , δ0 λ1 y2 , δ) χ −1 −1 −1 ×χˇ 1 (A + y22 E b0 (x1 − λ−1 1 y1 , δ0 λ1 y2 , δ) + λ3 λ1 r2 (y1 )).

The quantities A := λ3 QA (x),

D := λ1 QD (x),

with QA (x) := x3 − σ x1n β(δ1 x1 ),

E :=

λ3 , (δ0 λ1 )2

QD (x) := x2 − x12 ω(δ1 x1 ), (5.60)

appearing here again do not depend on y1 , y2 , and the functions ri (y1 ) are as before (i.e., they are indeed smooth functions of y1 , λ−1 1 , x1 , and δ, and again satisfy estimates of the form (5.54)). Notice that here we also have that λ3 /λ1  1. Recall also that we may assume that |y1 |
λ1 because of our assumption |x|
1 and the support properties of η, and that δ0−1 λ−1 1  1. Observe finally that our summation conditions imply that E  1. Notice also that the first factor χˇ 1 (y1 ) in (5.59) in combination with the second factor of Fδ clearly allow for a uniform estimate  1/2 2/3 λ3 1/3 |µ˜ λ1 ,λ3 (x)|
1; hence, δ0 |µλ1 ,λ3 (x)|
. δ0 λ 1 However, these estimate are not quite sufficient in order to prove estimate (5.57), so we need to improve on them. The second estimate suggests to introducing new

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dyadic summation variables λ0 , λ4 in place of λ1 , λ3 so that λ0 λ4 λ3 = λ24 and λ1 = , δ0

(5.61) −2/3

1/3

so in these new variables we would have δ0 |µλ1 ,λ3 (x)|
λ0 . More precisely, recalling that λ3 = 2k3 , we decompose the summation over k3 in (5.58) into two arithmetic progressions by writing k3 = 2k4 + i, with i ∈ {0, 1} fixed for each of these progressions. Since all of these sums can be treated in essentially the same way, let us assume for simplicity that i = 0, so that k3 = 2k4 . Letting λ4 := 2k4 and λ0 := 2k0 and writing k1 := k0 + k4 + j, we indeed obtain (5.61). Replacing without loss of generality the sum over the dyadic λ3 in (5.58) by the sum over the corresponding arithmetic progression with i = 0, it is also easy to check that the summation restrictions 2M+j ≤ λ1 ≤ 22j and 2 ≤ λ3 ≤ (2−M−j λ1 )2 are equivalent to the conditions 2M ≤ λ0 ≤ (2δ0 )−1 ,

2 ≤ λ4 ≤ (δ0 λ0 )−1 .

We may thus estimate in (5.58) |µ1+it (x)| ≤

−1 (2δ 0) 



−2/3  λ0 γ (1

+ it)

−1 (δ 0 λ0 )

−7it/2

λ4

λ4 =2

λ0 =2M

  µ˜ λ0 λ4 /δ0 ,λ24 (x).

For λ0 and x fixed, we let fλ0 ,x (λ4 ) := µ˜ λ0 λ4 /δ0 ,λ24 (x), ρt,λ0 (x) := γ (1 + it)

−1 (δ 0 λ0 )

−7it/2

λ4

fλ0 ,x (λ4 ).

λ4 =2

The previous estimate shows that in order to verify (5.57), it will suffice to prove the following uniform estimate: there exist constants C > 0 and  ≥ 0 with  < 23 , so that for all x with |x|
1 and δ sufficiently small, we have |ρt,λ0 (x)| ≤ Cλ0

for

2M ≤ λ0 ≤ (2δ0 )−1 .

In order to prove this, observe that by (5.59),  fλ0 ,x (λ4 ) = χˇ 1 (y1 )Fδ (λ0 , λ4 , x, y1 , y2 ) dy1 dy2 ,

(5.62)

(5.63)

where Fδ (λ0 , λ4 , x, y1 , y2 ) := η(x1 − δ0 (λ0 λ4 )−1 y1 , (λ0 λ4 )−1 y2 , δ) χˇ 1 (D − y2 + r1 (y1 )) × χˇ 1 (A + y22 E b0 (x1 − δ0 (λ0 λ4 )−1 y1 , (λ0 λ4 )−1 y2 , δ) + δ0 λ4 λ−1 0 r2 (y1 )) and A = A(x, λ4 , δ) = λ24 QA (x),

D = D(x, λ0 , λ4 , δ) =

E = E(λ0 ) = with

QA (x) := x3 − σ x1n β(δ1 x1 ),

λ0 λ4 QD (x), δ0

1 , λ20 QD (x) := x2 − x12 ω(δ1 x1 ). (5.64)

103

IMPROVED ESTIMATES

Given x and λ0 , we shall split the summation in λ4 into three parts, according to whether |D|  1, |D|
1 and |A|  1 or |D|
1 and |A|
1. 1 (x) the contribution to ρt,λ0 (x) by the 1. The part where |D|  1. Denote by ρt,λ 0 terms for which |D|  1. We first consider the contribution to fλ0 ,x (λ4 ) given by integrating in (5.63) over the region where |y1 |  |D|ε (where ε > 0 is assumed to be sufficiently small). Here, the rapidly decaying first factor χˇ 1 (y1 ) in (5.59) leads to an improved estimate of this contribution of the order |D|−N for every N ∈ N, which allows us to sum over the dyadic λ4 for which λ4 |λ0 QD (x)/δ0 | = |D|  1, and the contribution to ρt,λ0 (x) is of order O(1), which is stronger than what is needed in (5.62). We may therefore restrict ourselves in the sequel to the region where |y1 |  |D|ε . Observe that, because of (5.54), this implies in particular that |ri (y1 )|  |D|ε , i = 1, 2. By looking at the second factor in Fδ , we see that the contribution by the regions where, in addition, |y2 | < |D|/2 or |y2 | > 3|D|/2 is again of 1 (x) are again adthe order |D|−N for every N ∈ N, and their contributions to ρt,λ 0 missible. What remains is the region where |y1 |  |D|ε and |D|/2 ≤ |y2 | ≤ 3|D|/2. In addition, we may assume that y2 and D have the same sign, since otherwise we can estimate as before. Let us therefore assume, for example, that D > 0 and that D/2 ≤ y2 ≤ 3D/2. The change of variables y2  → Dy2 then allows us to rewrite the corresponding contribution to fλ0 ,x (λ4 ) as   ˜ χˇ 1 (y1 )F˜δ (λ0 , λ4 , x, y1 , y2 ) dy2 dy1 , fλ0 ,x (λ4 ) := D |y1 ||D|ε

1/2≤y2 ≤3/2

where here F˜δ (λ0 , λ4 , x, y1 , y2 ) := η(x1 − δ0 (λ0 λ4 )−1 y1 , (λ0 λ4 )−1 Dy2 , δ) ×χˇ 1 (D − Dy2 + r1 (y1 )) × χˇ 1 (A + y22 ED 2 b0 (x1 − δ0 (λ0 λ4 )−1 y1 , (λ0 λ4 )−1 Dy2 , δ) + δ0 λ4 λ−1 0 r2 (y1 )). Recall also that |b0 | ∼ 1, and notice that, according to Remark 4.3, |∂y2 b0 |
δ0 δ2  1 if φ is of type A, and similarly |∂y2 b0 |
δ0 δ2 /δ1 = δ0 δ1m−1  1 if φ is of type D. In combination with the localization given by η, this shows that, given y1 , we may change variables from y2 to z := y22 ED 2 b0 (x1 − δ0 (λ0 λ4 )−1 y1 , (λ0 λ4 )−1 Dy2 , δ) and use the last factor of F˜δ in order to estimate the integral in y2 (respectively, z) by C|ED 2 |−1 . Subsequently, we may estimate the integration with respect to y1 by means of the factor χˇ 1 (y1 ) and find that |f˜λ0 ,x (λ4 )| ≤ C

D 1 =C . 2 |ED | |ED|

Interpolating this with the trivial estimate |f˜λ0 ,x (λ4 )| ≤ C leads to |f˜λ0 ,x (λ4 )| ≤ C

1 = Cλ0 |D|−/2 , |ED|/2

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CHAPTER 5

where we chose  > 0 so that  < 23 . The factor |D|−/2 then allows us to sum 1 in λ4 , and we see that altogether we arrive at the estimate |ρt,λ (x)| ≤ Cλ0 . This 0 completes the proof of estimate (5.62) in this first case. 2 (x) the contribution to 2. The part where |D|
1 and |A|  1. Denote by ρt,λ 0 ρt,λ0 (x) by the terms for which |D|
1 and |A|  1. Arguing in a similar way as in the previous case, only with D replaced by A, we see that we may restrict ourselves to the regions where |y1 |
|A|ε and |y2 |
|A|ε (where ε > 0 is any fixed, positive constant). In the remaining regions, we can gain a factor CN |A|−N in the estimate of fλ0 ,x (λ4 ) in a trivial way. But, if |y1 |
|A|ε and |y2 |
|A|ε and if ε > 0 is sufficiently small, then     r (y )   |A|, A + y22 E b0 (x1 − δ0 (λ0 λ4 )−1 y1 , (λ0 λ4 )−1 y2 , δ) + δ0 λ4 λ−1 2 1 0

and thus we obtain an estimate of the same kind, that is, |fλ0 ,x (λ4 )| ≤ CN |A|−N

for every N ∈ N.

Summing over all dyadic λ4 such that λ24 |QA (x)| = |A|  1, this implies 2 (x)| ≤ C. |ρt,λ 0 3 3. The part where |D|
1 and |A|
1. Denote by ρt,λ (x) the contribution to 0 ρt,λ0 (x) by the terms for which |D| ≤ K and |A| ≤ K, where K > 0 is a suf3 (x) can again be estimated by means ficiently large constant. Observe that ρt,λ 0 of Lemma 2.7. Indeed, the cuboid Q will here be defined by means of the con−1 −1 −1 −M−1 , w2 := λ3 /λ1 = ditions |D| ≤ K, |A| ≤ K and w1 := λ−1 1 δ0 = λ4 λ0 ≤ 2 −2M , λ4 (δ0 λ0 ) ≤ 1 (compare also the properties of the functions λ4 (δ0 /λ0 ) ≤ 2 ri (y1 )), and if we define M := 1/(δ0 λ0 ), α := − 72 and

Hx,δ (A, D, E, w1 , w2 )  := χˇ 1 (y1 )η(x1 − δ0 w1 y1 , w2 y2 , δ)χˇ 1 (D − y2 + r1 (y1 ; δ0 w1 , x1 , δ)) × χˇ 1 (A + y22 E b0 (x1 − δ0 w1 y1 , w1 y2 , δ) + w2 r2 (y1 ; δ0 w1 , x1 , δ)) dy1 dy2 , 3 then (5.63) shows that fλ0 ,x (λ4 ) = Hx,δ (A, D, E, w1 , w2 ), and γ (1 + it)−1 ρt,λ (x) 0 is an oscillatory sum of the form (2.15) (with summation index l := k4 ). Moreover, one easily checks that

Hx,δ C 1 (Q) ≤ C, with a constant C that does not depend on x and δ. Applying Lemma 2.7, we 3 therefore obtain the estimate |ρt,λ (x)| ≤ C. This completes the proof of estimate 0 (5.62) and hence also the proof of Proposition 4.2 (b).

Chapter Six The Case When hlin (φ) ≥ 2: Preparatory Results We now turn to the case where h lin (φ) ≥ 2. In this case, our strategy will be as follows. We first look at every edge γl of the Newton polyhedron N (φ a ) of φ (when written in the adapted coordinates y given by (1.10), (1.11)), which lies completely within the closed half-space where t2 ≥ t1 . As explained in Chapter 1, each such edge is associated with a unique weight κ l . Following a basic idea from [PS97], we then decompose the domain (3.1), whose contribution we still need to control and which is of the form |y2 | ≤ εy1m in the adapted coordinates y, into subdomains of the form Dla := {(y1 , y2 ) ∈ H + : εl y1al < |y2 | ≤ Nl y1al } and intermediate domains a

Ela := {(y1 , y2 ) ∈ H + : Nl+1 y1 l+1 < |y2 | ≤ εl y1al }, with εl > 0 small and Nl > 0 large. Note that the domain Dla is homogeneous with respect to the dilations associated to the weight κ l , whereas the domains Ela should be viewed as “transition domains” between two different homogeneities. These domains will cover all of the domain (3.1), with the exception of a domain a , which will be associated with the principal face π(φ a ) in a certain sense and Dpr whose study will create the most serious problems. In Section 6.2 we shall deal with the Fourier restriction estimates for the pieces of the hypersurface S associated with the transition domains El (corresponding to the domains Ela in our original coordinates x), by means of decompositions into bidyadic rectangles and rescaling arguments, which will eventually allow us to reduce these estimates to surfaces and corresponding measures supported over squares of dimension 1 × 1 (in the coordinates y). The localizations to these bidyadic rectangles, which will indeed correspond to “curved” bidyadic boxes in our original coordinates x, will again be based on Littlewood-Paley theory, applied to the variables x1 and x3 . Here, Condition (R) will be needed once more. After all these reductions to measures supported over squares of dimension 1 × 1, it will turn out that the corresponding hypersurfaces have nonvanishing curvature with respect to the first variable x1 , so that we can apply Greenleaf’s restriction estimate, in fact for every p such that p  ≥ 6. Since pc ≥ 2d + 2 ≥ 6, this allows us to cover, in particular, the range where p  ≥ pc , and after undoing the rescalings that we have performed along the way, we shall verify that the Fourier restriction estimates that

106

CHAPTER 6

we obtain in this way for the contributions by the curved bidyadic boxes do sum absolutely. Notice that here the condition d = h lin (φ) ≥ 2 is sufficient. Next, in Section 6.3, we shall turn to the contributions by the domains Dl (i.e., the domains in our original coordinates x that correspond the domains Dla in the adapted coordinates y). For a better understanding of these domains, it may be useful to notice that for analytic φ, the domain Dla will contain all the (real) roots of φ a belonging to the cluster [l] if we choose εl sufficiently small and Nl sufficiently large—this is immediate from our discussions in Chapter 1. For the study of the contributions of these domains, we shall first perform a dyadic decomposition, followed by a rescaling, by means of the dilations associated to the weight κ l . Again, we will be able to reduce to Fourier restriction estimates for “normalized measures” supported over squares of dimension 1 × 1, contained in the set where y1 ∼ 1. By covering such a unit square by a finite number of small squares, we may even reduce it to small squares of dimension ε  × ε . Assume that a given small square of this type is centered at a point v = (1, c0 ). Then we shall look at the κ l -principal part φκal of φ a and distinguish between three cases: the case where ∂2 φκal (v)  = 0 (Case 1), the case where ∂2 φκal (v) = 0 and ∂1 φκal (v)  = 0 (Case 2), and the case where ∂2 φκal (v) = 0 and ∂1 φκal (v) = 0 (Case 3). Each of these cases will be treated in a different way. Case 1 turns out to be the easiest one, and it can be treated in a way that does not require the condition h lin (φ) ≥ 2. The other two cases can be dealt with in a comparatively easy way, again by means of Greenleaf’s restriction theorem. The reason for this is that under the assumption that the edge γl lies within the closed half plane bounded from below by the bisectrix, one can show that there is a good control of the multiplicity B − 1 of any real root of ∂2 φκal in terms of the Newton distance d (compare (6.20) and (6.23)). We note that our arguments show that in Case 2 and Case 3, we do then get the desired Fourier restriction estimates for the normalized measures whenever p ≥ 2d + 2. Such a strong control of the multiplicities B − 1 will no longer exist for the a in our original coordiremaining domain Dpr corresponding to the domain Dpr nate x. In our study of the domain Dpr in Section 6.5, we shall therefore devise a stopping-time algorithm that will allow us to decompose the domain Dpr into further subdomains of type D(l) and E(l) , where each of the domains D(l) will be homogeneous in suitable “modified adapted” coordinates, and the domains E(l) can again be viewed as “transition domains.” In the analytic case, when we are looking at our original coordinates x, our algorithm basically allows us in some sense to “zoom” into small, hornlike neighborhoods of every real root belonging to the cluster of roots whose leading part is given by the principal root jet ψ and thus “resolve” any possible branching of roots, so that in the end no further branching will take place. The new transition domains E(l) can be treated in the same way as in Section 6.2. As for the domains D(l) , we shall again distinguish three cases, analogous to our approach in Section 6.3. However, our algorithm will be built in such a way that we will not stop whenever we arrive at Case 3, so that in the end only the Cases 1 and 2 will have to be dealt with. Case 1 can be treated as before, and thus what remains is Case 2. Now, under the assumption that h lin (φ) ≥ 5, it turns out that Case 2 can be dealt with

FIRST RESULTS WHEN hlin (φ) ≥ 2

107

by means of a fibration of the given piece of surface into a family of curves, in combination with Drury’s Fourier restriction theorem for curves with nonvanishing torsion (cf. Theorem 2 in [Dru85]). This will conclude our proof of Proposition 3.3 for that case. Regretfully, that method breaks down when h lin (φ) < 5, so that in the subsequent two chapters we shall devise an alternative approach for dealing with Case 2.

6.1 THE FIRST DOMAIN DECOMPOSITION We begin by recalling that we are assuming that the original coordinates x are linearly adapted but not adapted to φ, so that we may assume from now on that d = h lin ≥ 2. Notice that this implies that h = h(φ) > 2. Moreover, based on Varchenko’s algorithm, we can locally find an adapted coordinate system y1 = x1 , y2 = x2 − ψ(x1 ) for the function φ near the origin. In these coordinates, φ is given by φ a (y) := φ(y1 , y2 + ψ(y1 )) (cf. (1.10), (1.11)). Observe next that the domain (3.1) in which the function ρ1 is supported and that we still need to control in Proposition 3.3 can also be described as the subset of ∩ H + on which |x2 − ψ(x1 )| ≤ εx1m . We also recall from Chapter 1 that the vertices of the Newton polyhedron N (φ a ) l = 0, . . . , n, so that the Newton polyof φ a are assumed to be the points (Al , Bl ),
hedron N (φ a ) is the convex hull of the set l ((Al , Bl ) + R2+ ), where Al−1 < Al for every l ≥ 1. Moreover, Ll := {(t1 , t2 ) ∈ R2 : κ1l t1 + κ2l t2 = 1} denotes the line passing through the points (Al−1 , Bl−1 ) and (Al , Bl ), and al = κ2l /κ1l . The al can be identified as the distinct leading exponents of the set of all roots of φ a in the case where φ a is analytic (cf. Remark 1.8), and the cluster of those roots whose leading exponent in their Puiseux series expansion is given by al is associated with the edge γl = [(Al−1 , Bl−1 ), (Al , Bl )] of N (φ a ). Following essentially the discussion in Section 8.2 of [IKM10], we choose the integer l0 ≥ 1 in such a way that 0 =: a0 < · · · < al0 −1 ≤ m < al0 < · · · < al < al+1 < · · · < an < an+1 := ∞. Then the vertex (Al0 −1 , Bl0 −1 ) lies strictly above the bisectrix, that is, Al0 −1 0 are small and the Nl > 0 are large parameters, which will be fixed later. We remark that the domain El0 −1 can be written like El , with l = l0 − 1, if we replace, with some slight abuse of notation, al0 −1 by m and κl0 −1 by κ. We shall make use of this unified way of describing El in the sequel. In our to avoid possible confusion, we note that here the superscript a will always refer to the representation of the given domain in adapted coordinates and is not related to the number a defined in (6.1). What will remain after removing these domains is a domain of the form  {(x1 , x2 ) ∈ H + : |x2 − ψ(x1 )| ≤ N x1a } in Case (a); (6.2) Dpr := {(x1 , x2 ) ∈ H + : |x2 − ψ(x1 )| ≤ εx1a }, in all other cases,

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where N is sufficiently large and ε is sufficiently small. Notice here that if al0 = ∞, then we must be in Case (c1), with a = m, and so the domain |x2 − ψ(x1 )| ≤ εx1m does agree with Dpr . In Cases (c1) and (c2), we shall furthermore regard the domains Elpr −1 := Dpr = {(x1 , x2 ) ∈ H + : |x2 − ψ(x1 )| ≤ εx1a } as “generalized” transition domains. Notice that in Case (c2), this domain will cover the domain in (3.1), since here a = m, so that the proof of Proposition 3.3 will be complete once we have handled all these transition domains in the next section. In a similar way, the discussion of Case (c1) will be complete once we have handled the domains El and Dl . This will eventually reduce our problem to studying the domain Dpr in Cases (a) and (b). In order to simplify the exposition, in the sequel we shall always regard the domains that will appear as subdomains of the half plane H + on which x1 > 0, usually without further mentioning.

6.2 RESTRICTION ESTIMATES IN THE TRANSITION DOMAINS E l WHEN hlin (φ) ≥ 2 Following a standard approach, we would like to study the contributions of the domains El by means of a decomposition of the corresponding y-domains Ela into dyadic rectangles. These rectangles correspond to a kind of “curved boxes” in the original coordinates x, so that we cannot achieve the localization to them by means of Littlewood-Paley decompositions in the variables x1 and x2 . However, the following lemma shows that this localization can, nevertheless, be induced by means of Littlewood-Paley decompositions in the variables x1 and x3 . We shall formulate this lemma for a general smooth, finite-type function with

(0, 0) = 0 and ∇ (0, 0) = 0 in place of φ a , since it will be applied not only to φ a . However, we shall keep the notation introduced for φ a , denoting, for instance, by (Al , Bl ), l = 0, . . . , n, the vertices of the Newton polyhedron of , by κ l the weight associated to the edge γl = [(Al−1 , Bl−1 ), (Al , Bl )], and so on. Lemma 6.1. For l ≥ l0 , let [(Al−1 , Bl−1 ), (Al , Bl )] and [(Al , Bl ), (Al+1 , Bl+1 )] be two subsequent compact edges of N ( ), with common vertex (Al , Bl ), and associated weights κ l and κ l+1 . Recall also that al = κ2l /κ1l < al+1 = κ2l+1 /κ1l+1 . For a given M > 0 and δ > 0 sufficiently small, consider the domain a

E a := {(y1 , y2 ) : 0 < y1 < δ, 2M y1 l+1 < |y2 | ≤ 2−M y1al }. Then the following holds true: (a) There is a constant C > 0 such that  

(y) = cAl ,Bl y1Al y2Bl 1 + O(δ C + 2−M )

on E a ,

(6.3)

where cAl ,Bl denotes the Taylor coefficient of corresponding to (Al , Bl ). More precisely, (y) = cAl ,Bl y1Al y2Bl (1 + g(y)), where |g (β) (y)| ≤ Cβ (δ C + 2−M ) −β −β |y1 1 y2 2 | for every multi-index β ∈ N2 .

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(b) For M, j ∈ N sufficiently large, the following conditions are equivalent: (i) y1 ∼ 2−j , (y1 , y2 ) ∈ E a , and 2Al j +Bl k (y) ∼ 1; (ii) y1 ∼ 2−j , y2 ∼ 2−k and al j + M ≤ k ≤ al+1 j − M. Moreover, if we set φj,k (x) := 2Al j +Bl k (2−j x1 , 2−k x2 ), then under the previous conditions we have that φj,k (x) = cAl ,Bl x1Al x2Bl (1 + O(2−Cj + 2−M )) on the set where x1 ∼ 1, |x2 | ∼ 1, in the sense of the C ∞ -topology. The statements in (a) and (b) remain valid also in the case where l = l0 − 1. Proof. When is analytic, these results have essentially been proven in Section 8.3 of [IKM10], at least implicitly. We shall here give an elementary proof that also works for smooth functions . We begin with the case where l > l0 . Notice first that (b) is an immediate consequence of (a). In order to prove (a), let us denote by N the Taylor polynomial of degree N of centered at the origin. Since ( − N )(y1 , y2 ) = O(|y1 |N + |y2 |N ), it is easily seen that y1−Al y2−Bl ( − N )(y1 , y2 ) = O(2−Bl M ) on E a , provided N is sufficiently large and δ is small. It therefore suffices to prove (6.3) for

N in place of . α1 α2 If (y1 , y2 ) ∼ ∞ α1 ,α2 =0 cα1 ,α2 y1 y2 is the Taylor series of centered at the origin, then we decompose the polynomial N as N = P + + P − , where

cα1 ,α2 y1α1 y2α2 , P + (y1 , y2 ) := α1 +α2 ≤N,α2 >Bl −

P (y1 , y2 ) :=



cα1 ,α2 y1α1 y2α2 .

α1 +α2 ≤N,α2 ≤Bl

Let (α1 , α2 ) be one of the multi-indices appearing in P − and assume it is different from (Al , Bl ). Let (y1 , y2 ) ∈ E a , and assume, for notational convenience, that y2 > 0. Since clearly Al , Bl > 0, we have  α2 −Bl y1α1 y2α2 α +a α −(A +a B ) α1 −Al α2 −Bl α1 −Al M al+1 = y y ≤ y y = 2(α2 −Bl )M y1 1 l+1 2 l l+1 l . 2 1 2 1 1 A l Bl y1 y2 It is easy to see that Al + al+1 Bl = Al+1 + al+1 Bl+1 , so that y1α1 y2α2

y1Al y2Bl

α +al+1 α2 −(Al+1 +al+1 Bl+1 )

≤ 2(α2 −Bl )M y1 1

.

(6.4)

But, since γl+1 is an edge of N ( ), we have that κ1l+1 α1 + κ2l+1 α2 ≥ 1, that is, α1 + al+1 α2 ≥ (κ1l+1 )−1 , whereas Al+1 + al+1 Bl+1 = (κ1l+1 )−1 . Thus, (6.4) implies that y1α1 y2α2 ≤ 2(α2 −Bl )M y1Al y2Bl , so that y1α1 y2α2 ≤ 2−M y1Al y2Bl when α2 < Bl . And, when α2 = Bl , then (α1 , α2 ) lies in the interior of N ( ), so that α1 + al+1 α2 − (Al+1 + al+1 Bl+1 ) > 0, hence y1α1 y2α2 ≤ δ C y1Al y2Bl for some positive constant C. The estimates of the derivatives of g(y) = (y)/cAl ,Bl y1Al y2Bl − 1 follow in a very similar way. The terms in P + can be estimated analogously, making use here of the estimates y2 ≤ 2−M y1al and κ1l α1 + κ2l α2 ≥ 1. This proves (a).

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Finally, if l = l0 , exactly the same arguments work if we redefine al0 −1 to be m Q.E.D. and κl0 −1 to be κ since κ2 /κ1 = m. A similar result also applies to the generalized transition domains Elpr −1 arising in Cases (c1) and (c2), provided we can factor the root y2 = 0 to its given order, which applies in particular when is real analytic (some easy examples show that it may be false otherwise). Recall that in these cases, the principal face of N (φ a ) is an unbounded half line with left endpoint (A, B). More generally, we have the following result. Lemma 6.2. Assume that (A, B) is a vertex of N ( ) such that the unbounded horizontal half line with left endpoint (A, B) is a face of N ( ), and assume in addition that factors as (y1 , y2 ) = y2B ϒ(y1 , y2 ), with a smooth function ϒ. Moreover, let Lκ := {(t1 , t2 ) ∈ R2 : κ1 t1 + κ2 t2 = 1} be a nonhorizontal supporting line for N ( ) (i.e., κ1 > 0) passing through (A, B), and let a := κ2 /κ1 . We then let E a := {(y1 , y2 ) : 0 < y1 < δ, |y2 | ≤ 2−M y1a }. Then the following hold true: (a) There is a constant C > 0 such that  

(y) = cA,B y1A y2B 1 + O(δ C + 2−M )

on E a ,

where cA,B denotes the Taylor coefficient of corresponding to (A, B). More pre−β −β cisely, (y) = cA,B y1A y2B (1 + g(y)), where |g (β) (y)| ≤ Cβ (δ C + 2−M )|y1 1 y2 2 | 2 for every multi-index β ∈ N . (b) For M, j ∈ N sufficiently large, the following conditions are equivalent: (i) y1 ∼ 2−j , (y1 , y2 ) ∈ E a , and 2Aj +Bk (y) ∼ 1; (ii) y1 ∼ 2−j , y2 ∼ 2−k , and aj + M ≤ k. Moreover, if we set φj,k (x) := 2Aj +Bk (2−j x1 , 2−k x2 ), then under the previous conditions we have that φj,k (x) = cA,B x1A x2B (1 + O(2−Cj + 2−M )) on the set where x1 ∼ 1, |x2 | ∼ 1, in the sense of the C ∞ -topology. Proof. It suffices again to prove (a). By our assumption, (y1 , y2 ) = y2B ϒ(y1 , y2 ), so that (y)/y1A y2B = ϒ(y)/y1A . Approximating ϒ by its Taylor polynomial of sufficiently high degree, we again see that we may reduce to the case where ϒ, hence , is a polynomial. Then let (α1 , α2 ) be any point different from (A, B) in its Taylor support. Since α2 ≥ B, assuming again that y2 > 0, we see that  α2 −B y1α1 y2α2 = y1α1 −A y2α2 −B ≤ y1α1 −A 2−M y1a = 2−(α2 −B)M y1α1 +aα2 −(A+aB) . A B y1 y2 Moreover, clearly α1 + aα2 ≥ A + aB, and α1 + aα2 > A + aB when α2 = B. We can thus argue in a very similar way as in the proof of Lemma 6.1 to finish the proof. Q.E.D.

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Let us now fix l ∈ {l0 − 1, . . . , lpr − 1}, and consider the corresponding (generalized) transition domain El from Section 6.1, which can be written as a

El = {(x1 , x2 ) : N x1 l+1 < |x2 − ψ(x1 )| ≤ εx1al }, where, with some slight abuse of notation, we have again redefined al0 −1 := m and al let alpr := ∞ in Cases (c1) and (c2), so that x1 pr := 0, by definition. Following [IKM10], we shall localize to the domain El by means of a cutoff function  x − ψ(x )   x − ψ(x )  2 1 2 1 ) (1 − χ , τl (x1 , x2 ) := χ0 0 a εx1al N x1 l+1 with ε = εl and N = Nl and where χ0 ∈ C0∞ (R) is again supported in [−1, 1] and χ0 ≡ 1 on [− 12 , 12 ] (actually, χ0 may depend on l). In Case (c), when l = lpr − 1 and alpr = ∞, the second factor has to be interpreted as 1, that is,  x − ψ(x )  2 1 τlpr −1 (x1 , x2 ) = χ0 . εx1a Recall that φ is assumed to satisfy Condition (R). Proposition 6.3. Let l ∈ {l0 − 1, . . . , lpr − 1}. Then, if ε > 0 is chosen sufficiently small and N > 0 sufficiently large, 1/2  |f|2 dµτl ≤ Cp f Lp (R3 ) , f ∈ S(R3 ), S

whenever p ≥ pc .

Proof. Consider partitions of unity j χj (s) = 1 and k χ˜ j,k (s) = 1 on R \ {0} with χ , χ˜ ∈ C0∞ (R) supported in [−2, − 12 ] ∪ [ 21 , 2] (respectively, [−2Bl , −2−Bl ] ∪ [2−Bl , 2Bl ]), where χj (s) := χ (2j s) and, for j fixed, χ˜ j,k (s) := χ (2Al j +Bl k s), and let χj,k (x1 , x2 , x3 ) := χj (x1 )χ˜ j,k (x3 ) = χ (2j x1 )χ˜ (2Al j +Bl k x3 ) ,

j, k ∈ Z.

Notice here that Bl > Bl+1 ≥ 0. We next let µj,k := χj,k µτl and assume that µ has sufficiently small support near the origin. Then clearly µj,k = 0 unless j ≥ j0 , where j0 > 0 is a large number, which we shall choose in a suitable way later. But then, according to Lemma 6.1, we may assume in addition that al j + M ≤ k ≤ al+1 j − M,

(6.5) M

−M

where M is a large number. Indeed, we may choose N := 2 and ε := 2 , and then Lemma 6.1 (b) shows that µj,k = 0 for all pairs (j, k) not satisfying (6.5). Notice that this also implies that k ≥ k0 for some large number k0 . Observe also that the measure µj,k is supported over a “curved box” given by x1 ∼ 2−j and |x2 − ψ(x1 )| ∼ 2−k . This shows that the localization that we have achieved by means of the cutoff function χj,k is very similar to the localization that we could have imposed by means of the cutoff function χ (2j x1 )χ (2k (x2 − ψ(x1 ))). Then, applying again Littlewood-Paley theory, now in the variables x1 and x3 , and interpolating with the trivial L1 → L∞ estimate for the Fourier transform, we see that in order to prove Proposition 6.3, it suffices to prove uniform restriction

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estimates for the measures µj,k at the critical exponent, that is,

|f|2 dµj,k ≤ Cf 2Lpc (R3 ) , when (j, k) satisfies (6.5) and j ≥ j0 , (6.6) S

provided M and j0 are chosen sufficiently large. We introduce the normalized measures νj,k given by

 νj,k , f  := f x1 , 2mj −k x2 + x1m ω(2−j x1 ), φj, k (x1 , x2 ) aj, k (x) dx, where

   −j −k −j al j +M−k x2 aj, k (x) = η 2 x1 , 2 x2 + ψ(2 x1 ) χ0 2 x1al    x2 ×(1 − χ0 ) 2al+1 j −M−k al+1 χ (x1 )χ˜ φj, k (x1 , x2 ) . x1

Here, according to Lemma 6.1, the functions φj,k satisfy φj, k (x1 , x2 ) = cx1Al x2Bl + O(2−M )

in

C∞

on domains where x1 ∼ 1, |x2 | ∼ 1, and the amplitude aj,k in the preceding integral is supported in such a domain. Observe that

 (6.7) µj,k , f  = 2−j −k f 2−j y1 , 2−mj y2 , 2−(Al j +Bl k) y3 dνj,k (y), which follows easily by means of a change to adapted coordinates in the integral defining the measure µj,k and scaling in x1 by the factor 2−j and in x2 by the factor 2−k . We observe that the measure νj,k is supported on the surface given by Sj,k := {(x1 , 2mj −k x2 + x1m ω(2−j x1 ), φj, k (x1 , x2 )) : x1 ∼ 1 ∼ x2 }, which is a small perturbation of the limiting surface S∞ := {(x1 , x1m ω(0), cx1Al x2Bl ) : x1 ∼ 1 ∼ x2 } since mj − k ≤ al j − k ≤ −M because of (6.5). Notice also that |∂(cx1Al x2Bl )/ ∂x2 | ∼ 1 since Bl ≥ 1. This shows that S∞ , and hence also Sj,k (for j and M sufficiently large), is a smooth hypersurface with one nonvanishing principal curvature (with respect to x1 ) of size ∼ 1. This implies that | νj,k (ξ )| ≤ C(1 + |ξ |)−1/2 uniformly in j and k. Moreover, the total variations of the measures νj,k are uniformly bounded, that is, supj,k νj,k 1 < ∞. We may thus apply again Greenleaf’s Theorem 1.1 in order to prove that a uniform estimate

(6.8) |f|2 dνj,k ≤ C f 2Lp (R3 )

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holds true whenever p ≥ 6, with a constant C that is independent of j, k. Since pc ≥ 2d + 2 ≥ 6, this holds in particular for p = pc . Rescaling this estimate by means of (6.7), this implies that

 |f|2 dµj,k ≤ C2−j −k+2[(m+1+Al )j +Bl k]/pc f 2Lpc (R3 ) . (6.9) But, we may write k in the form k = θ al j + (1 − θ )al+1 j + M˜ with 0 ≤ θ ≤ 1 ˜ ≤ M, and then and |M| −j − k + 2

  m + 1 + Al + al Bl (m + 1 + Al )j + Bl k = −j θ 1 + a − 2 l pc pc

   m + 1 + Al + al+1 Bl Bl ˜ + −1 + 2  M. −j (1 − θ ) 1 + al+1 − 2 pc pc 

Recall next that by the definitions of the notion of r-height and of the critical exponent pc , we have pc ≥ 2(hl + 1) whenever l ≥ l0 . And, (1.17) shows that hl + 1 =

1 + (1 + m)κ1l m + 1 + 1/κ1l . = |κ l | 1 + al

(6.10)

Moreover, we have seen in the proof of Lemma 6.1 that Al + al Bl = 1/κ1l , so that 2(hl + 1) = 2

m + 1 + Al + al Bl . 1 + al

We thus find that 1 + al − 2(m + 1 + Al + al Bl )/pc ≥ 0. Arguing in a similar way for l + 1 in place of l, by using that pc ≥ 2(hl+1 + 1) and Al + al+1 Bl = 1/κ1l+1 , we also see that 1 + al+1 − 2(m + 1 + Al + al+1 Bl )/pc ≥ 0. Consequently, the exponent on the right-hand side of the estimate (6.9) is uniformly bounded from above, which verifies the claimed estimate (6.6). Assume next that l = l0 − 1. Observe that in this case, by following Varchenko’s algorithm, one observes that the left endpoint (Al0 −1 , Bl0 −1 ) of the edge [(Al0 −1 , Bl0 −1 ), (Al0 , Bl0 )] of the Newton polyhedron of φ a belongs also to the Newton polyhedron of φ and lies on the principal line L = Lκ of N (φ), whose slope is the reciprocal of κ2 /κ1 = m. Thus, if we formally replace hl0 −1 by d in the previous argument (compare also Remark 1.9(a)), it is easily seen that the previous argument works in exactly the same way. What remains to be considered are the generalized transition domains Elpr −1 in Cases (c1) and (c2). Observe that in this case, our Condition (R) allows us to assume that := φ a satisfies the factorization hypothesis of Lemma 6.2, possibly after modifying ψ by adding a suitable flat function. We may therefore argue in a similar way as before, by applying Lemma 6.2 in place of Lemma 6.1, and obtain the estimate

 |f|2 dµj,k ≤ C2−j −k+2[(m+1+A)j +Bk]/pc f 2Lpc (R3 ) , (6.11) S

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where here B = h is the height of φ and where now we may assume only that al j + M ≤ k. Observe next that −1 + 2B/pc ≤ 0 since, by (1.18), we have pc = 2(hr + 1) ≥ 2h = 2B. We may thus estimate the exponent in (6.11) by −j − k + 2

(m + 1 + A)j + Bk pc

   m + 1 + A + aB 2B  ≤ −j a + 1 − 2 + −1+  M  pc pc   m + 1 + A + aB a+1  . pc − 2 ≤ −j  pc a+1 And, in Case (c1), arguing as before we see that 2(m + 1 + A + aB)/(a + 1) = 2(hlpr + 1) ≤ pc . Finally, in Case (c2), we have m = a. Moreover, the point (A, B) lies on the principal line L of N (φ), so that κ1 A + κ2 B = 1, that is, A + aB = 1/κ1 . This shows that   m + 1 + A + aB 1 2 =2 1+ = 2(1 + d) ≤ pc . a+1 κ 1 + κ2 We thus see that the uniform estimate (6.6) is valid also for the generalized transition domains. Q.E.D. 6.3 RESTRICTION ESTIMATES IN THE DOMAINS D l , l < l pr , WHEN hlin (φ) ≥ 2 We shall now consider the domains Dl , l = l0 , . . . , lpr − 1, which are homogeneous in the adapted coordinates. Recall that for such l we have al > m. Again following [IKM10] we can localize to these domains by means of cutoff functions  x − ψ(x )   x − ψ(x )  2 1 2 1 − χ0 , l = l0 , . . . , lpr − 1, ρl (x1 , x2 ) := χ0 al N x1 εx1al with ε = εl and N = Nl and where χ0 is as in the previous section. Recall that such domains do appear only in Cases (a), (b) and (c1). Proposition 6.4. Let h lin (φ) ≥ 2, and assume that l < lpr . Then, if ε > 0 is chosen sufficiently small and N > 0 sufficiently large, 1/2  |f|2 dµρl ≤ Cp f Lp (R3 ) , f ∈ S(R3 ), S

whenever p  ≥ pc . Proof. Similarly to the proof of Proposition 3.1, we denote by {δr }r>0 the dilations l l associated to the weight κ l , that is, δr y := (r κ1 y1 , r κ2 y2 ), where by y we again denote our adapted coordinates. Recall that the κ l -principal part φκal of φ a is homogeneous of degree one with respect to these dilations and that we are interested in

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a κ l -homogeneous (for small dilations) domain of the form Dla = {(y1 , y2 ) : 0 < y1 < δ, εy1al < |y2 | ≤ Ny1al } with respect to the y-coordinates, where δ > 0 can still be chosen as small as we please. We shall prove that, given any real number c0 with ε ≤ |c0 | ≤ N, there is some ε > 0 such that the desired restriction estimate holds true on the domain D(c0 ) in x-coordinates corresponding to the homogeneous domain D a (c0 ) := {(y1 , y2 ) : 0 < y1 < δ, |y2 − c0 y1al | ≤ ε y1al } in y-coordinates. Since we can cover Dla by a finite number of such narrow domains, this will imply Proposition 6.4. We can essentially localize to a domain D(c0 ) by means of a cutoff function  x − ψ(x ) − c x al  2 1 0 1 . ρ(c0 ) (x1 , x2 ) := χ0 ε x1al Let us again fix a suitable smooth cutoff function χ ≥ 0 on R2 supported in an annulus A := {x ∈ R2 : 12 ≤ |y| ≤ R} such that the functions χka := χ ◦ δ2k form a partition of unity. In the original coordinates x, these correspond to the functions χk (x) := χka (x1 , x2 − ψ(x1 )). We then decompose the measure µρ(c0 ) dyadically as

µρ(c0 ) = µk , (6.12) k≥k0 χk ρ(c0 )

where µk := µ . Notice that by choosing the support of η sufficiently small, we can choose k0 ∈ N as large as we need. It is also important to observe that this decomposition can essentially be achieved by means of a dyadic decomposition with respect to the variable x1 , which again allows us to apply Littlewood-Paley theory! Moreover, changing to adapted coordinates in the integral defining µk and scaling by δ2−k , we find that

 l l l l −k|κ l | f 2−κ1 k x1 , 2−κ2 k x2 + 2−mκ1 k x1m ω(2−κ1 k x1 ), 2−k φk (x) µk , f  = 2 × η(δ2−k x)χ (x) χ0

 x − c x al  2 0 1 dx, ε x1al

where φk (x) := 2k φ a (δ2−k x) = φκal (x) + error terms of order O(2−δk )

(6.13)

with respect to the C ∞ topology (and δ > 0). We consider the corresponding normalized measure νk given by

  l l l ˜ dx, νk , f  := f x1 , 2(mκ1 −κ2 )k x2 + x1m ω(2−κ1 k x1 ), φk (x) η(x) with amplitude η(x) ˜ := η(δ2−k x)χ (x)χ0 ((x2 − c0 x1al )/(ε x1al )). Observe that the support of the integrand is contained in the thin neighborhood U (v) := A ∩ {(x1 , x2 ) : |x2 − c0 x1al | ≤ 2ε x1al }

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of v = v(c0 ) := (1, c0 ) and that the measure νk is supported on the hypersurface Sk := {gk (x1 , x2 )  l l l := x1 , 2(mκ1 −κ2 )k x2 + x1m ω(2−κ1 k x1 ), φk (x1 , x2 ) : (x1 , x2 ) ∈ U (v)}, which, for k sufficiently large, is a small perturbation of the limiting variety  S∞ := {g∞ (x1 , x2 ) := x1 , ω(0)x1m , φκal (x) : (x1 , x2 ) ∈ U (v)} since mκ1l − κ2l < al κ1l − κ2l = 0 and since φ k tends to φκal because of (6.13). The corresponding limiting measure will be denoted by ν∞ . By Littlewood-Paley theory (applied to the variable x1 ) and interpolation, in order to prove the desired restriction estimates for the measure µρ(c0 ) , it suffices again to prove uniform restriction estimates for the measures µk , that is, 1/2  2  |f | dµk ≤ C f Lpc , (6.14) with a constant C not depending on k ≥ k0 . We shall obtain these by first proving restriction estimates for the measures νk . Indeed, we shall prove that for ε  sufficiently small, the estimate  1/2 |f|2 dνk ≤ Cf Lpc (6.15) holds true, with a constant C which does not depend on k. Then, after rescaling, estimate (6.15) implies the following estimate for µk :  1/2 l l  |f|2 dµk ≤ C 2−k(|κ |/2−(κ1 (1+m)+1)/pc ) f Lpc . (6.16) But, by (1.17) (respectively, (6.10)) we have that   |κ l | κ1l (1 + m) + 1 |κ l | 2(hl + 1) = − 1− , 2 pc 2 pc where, by definition, pc ≥ 2(hl + 1). This shows that the exponent on the righthand side of (6.16) is less than or equal to zero, which verifies (6.14). We turn to the proof of (6.15). Recall that v = (1, c0 ). Depending on the behavior of φκal near v, we shall distinguish among three cases. Case 1. ∂2 φκal (v)
= 0. This assumption implies that we may use y2 := φκal (x1 , x2 ) in place of x2 as a new coordinate for S∞ (which thus is a hypersurface, too) and then also for Sk , in place of x2 , provided ε is chosen small enough and k, sufficiently large. Since x1 ∼ 1 on U (v), this then shows that Sk is a hypersurface with one nonvanishing principal curvature. Therefore, we can again apply Theorem 1.1 and obtain that for p ≥ 6 and k sufficiently large, the estimate  1/2 |f|2 dνk ≤ Cp f Lp (6.17) holds true, with a constant Cp that does not depend on k. This applies in particular to p = pc since pc ≥ 2d + 2 ≥ 6, and we obtain (6.15).

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Assume next that ∂2 φκal (v) = 0. Then v = (1, c0 ) is a real root of ∂2 φκal of multiplicity, say, B − 1 ≥ 1, so that a Taylor expansion with respect to x2 around c0 and homogeneity show that ˜ 1 , x2 ), ∂2 φκal (x1 , x2 ) = (x2 − c0 x1al )B−1 Q(x ˜ is a κ l -homogenous fractionally smooth function (polynomial in x2 ) such where Q ˜ that Q(v)  = 0. Integrating in x2 and again making use of the κ l -homogeneity of φκal , we find that 1/κ1l

φκal (x1 , x2 ) = (x2 − c0 x1al )B x2D Q(x1 , x2 ) + c1 x1

,

(6.18)

where B ≥ 2, D ∈ N, and c1 ∈ R and where Q is again a κ l -homogenous fractionally smooth function satisfying Q(1, c0 )  = 0 and Q(1, 0)  = 0 (recall that c0  = 0). On the other hand, since l < lpr , the edge γl = [(Al−1 , Bl−1 ), (Al , Bl )] lies above the bisectrix (but notice that if l = lpr − 1, then in Case (b) its right endpoint will lie on the bisectrix). If we then factor φκal (x1 , x2 ) = x2Bl Q0 (x1 , x2 ), where Q0 (x1 , x2 ) is a κ l -homogenous fractionally smooth function such that Q0 (1, 0)  = 0, then this implies that Bl ≥ h > d, where again d = d(φ). Moreover, since al > m ≥ 2, so that the edge γl is less steep than the line L (which intersects the bisectrix at (d, d)), we have 1/κ2 > 1/κ2l . We claim that this implies that d , (6.19) 2 where degx2 denotes the degree with respect to the variable x2 . Indeed, since φκal is κ l -homogenous of degree one, we have (Bl + degx2 Q0 )κ2l ≤ 1; hence degx2 Q0
2 since B ≥ 2. B
2B − 1, so that θ−

1 1 1−B 1 < − = 14 , then we obtain

(4θ−1)/3 θ−1/B 7θ/3−1/B−1/3 Tkλ0 ,λ3 p→p
λ3 λ3 = λ3 . {λ0 ≥1:λ0
λ3 }

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But, 7 1 1 7 1 1 1 2 B 2 − B − 3/2 θ− − < − − =− 1, so that κ2l < 1. Observe next that the mapping F : (x1 , c)  → (x1 , cx1al ) provides local smooth coordinates (x1 , c) near v = (1, c0 ), since the Jacobian JF of F at the point (1, c0 ) is given by JF (1, c0 ) = 1. We may, therefore, fiber the variety S∞ into the family of curves γc (x1 ) := g∞ (F (x1 , c)) = (x1 , ω(0)x1m , φκal (F (x1 , c)),

c ∈ V (c0 ),

where V (c0 ) is a sufficiently small neighborhood of c0 , provided ε is chosen suf1/κ l ficiently small. But, (6.26) implies that the curve γc0 (x1 ) = (x1 , ω(0)x1m , c1 x1 1 ) has nonvanishing torsion near v1 , since v1  = 0, and so the same is true for the curves γc when c is sufficiently close to c0 . If, in a similar way, we fiber the surface Sk into the family of curves γck (x1 ) := gk (F (x1 , c)),

c ∈ V (c0 ),

then for k sufficiently large and V (c0 ) sufficiently small, these curves will have nonvanishing torsion uniformly bounded from above and below, and the measure νk will decompose into the direct integral



f (γck (x1 )) η(x ˜ 1 , c) dx1 dc = f dc dc, νk , f  = V (c0 )

W (v1 )

FIRST RESULTS WHEN hlin (φ) ≥ 2

123

where η˜ is a smooth function with compact support in W (v1 ) × V (c0 ) and W (v1 ) is a sufficiently small neighborhood of v1 . Here, dc is a measure which has a smooth density with respect to the arc length measure on the curve γck . We may thus apply Drury’s Fourier restriction theorem for curves with nonvanishing torsion (cf. Theorem 2 in [Dru85]) to the measures dc and obtain 1/2  |fˆ|2 dc ≤ Cp f Lp (R3 ) , W (v1 )



provided p > 7 and 2 ≤ p /6, that is, if p  ≥ 12. The constant Cp will then be independent of c provided the neighborhoods V (c0 ) and W (v1 ) are sufficiently small and k is sufficiently large. But, if h lin ≥ 5, then we do have pc ≥ 2(h lin + 1) ≥ 12, so that we also obtain estimate (6.15) in this way. 6.4 RESTRICTION ESTIMATES IN THE DOMAIN Dpr WHEN hlin (φ) ≥ 5 What remains to be studied is the piece of the surface S corresponding to the domain Dpr , in Cases (a) and (b), that is,  {(x1 , x2 ) : |x2 − ψ(x1 )| ≤ N x1a }, in Case (a), Dpr := in Case (b), {(x1 , x2 ) : |x2 − ψ(x1 )| ≤ εx1a }, where N > 0 can be any given number in Case (a) and where ε may be assumed to be sufficiently small in Case (b) (cf. (6.2)). Recall that this domain is associated to the edge γlpr of N (φ a ), whose (modulus of) slope is given by 1/a (cf. (6.1)). Our goal will to prove the following. Proposition 6.7. Assume that h lin (φ) ≥ 5 and that we are in Case (a) or (b). Then for any given N > 0 in Case (a), respectively, every sufficiently small ε in Case (b), we have 1/2  |f|2 dµρlpr ≤ Cp f Lp (R3 ) , f ∈ S(R3 ), Dpr



whenever p ≥ pc . Within the domain Dpr , the upper bound B < d/2 for the multiplicity B of real roots that we used in our discussion of Case 3 in the previous section, as well as the condition B < d/2 + 1 that we used in our discussion of Case 2, will in general no longer be satisfied. Even the weaker condition B < hr (φ)/2, which would still suffice for the previous argument in Case 3, may fail, as the following examples shows. Examples 6.8. (a) The first example, φ(x1 , x2 ) := (x2 − x12 − x13 )(x2 − x12 − x14 )3 , deals with Case 3. Here, φpr (x1 , x2 ) = (x2 − x12 )4 , the multiplicity of the root x12 satisfies 4 > d(φ) = 83 , so that the coordinates (x1 , x2 ) are not adapted to φ.

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Adapted coordinates are given by y1 := x1 , y2 := x2 − x12 , and in these coordinates φ is given by φ a (y1 , y2 ) = (y2 − y13 )(y2 − y14 )3 . N (φ a ) has three vertices, (A0 , B0 ) := (0, 4), (A1 , B1 ) := (3, 3), and (A2 , B2 ) := (0, 15), with corresponding edges γ1 := [(0, 4), (3, 3)] and γ2 := [(3, 3), (0, 15)] 1 1 1 4 , 4 ) and κ 2 := ( 15 , 15 ). Moreover, one easily and associated weights κ 1 := ( 12 11 13 computes by means of (1.17) that h1 = 4 and h2 = 5 . We thus see that hr (φ) = . The multiplicity of the root r1 (y1 ) := y13 associated to the first edge γ1 lyh1 = 11 4 ing above the bisectrix is 1 < 83 /2 and thus satisfies the condition (6.23), whereas the root r2 (y1 ) := y14 of multiplicity B = 3 associated with the edge γ2 below the . bisectrix does not even satisfy B < hr (φ), since 3 > 11 4 (b) The second example, φ(x1 , x2 ) := (x2 − x12 − x13 )(x2 − x12 − x14 )3 + x120 , dealing with Case 2, is a small modification of the previous one. Here, arguments similar to those in the first example show that φ a (y1 , y2 ) = (y2 − y13 )(y2 − y14 )3 + y120 . Again d = d(φ) = 83 , N (φ a ) has the same edges as before, and hr = hr (φ) = . The multiplicity of the root r2 (y1 ) = y14 of ∂2 φκalpr associated to the edge γ2 h1 = 11 4 lying below the bisectrix is given by B − 1 = 2, whereas d/2 + 1 = 73 < 3 = B. Following our approach from the previous section, we shall consider small, κ lpr homogeneous neighborhoods of a given point v = (1, c0 ) and distinguish between the following three cases: Case 1. ∂2 φκalpr (v)  = 0. Case 2. ∂2 φκalpr (v) = 0 and ∂1 φκalpr (v)  = 0. Case 3. ∇φκalpr (v) = 0. In Case 1 we can argue as in the corresponding case in Section 6.3, since our arguments in that case did not make use of the assumption l > lpr . Observe next that if we are in Case 3, then by homogeneity we shall also have φκalpr (v) = 0, so that v is a root of φκalpr of order two or higher. Notice that in the analytic setting, multiple roots of φκalpr will correspond to possibly branching roots of φ a . In order to overcome the problem of an insufficient control of the multiplicity of this root that we saw in Example 6.8(a), we shall perform a suitable change of coordinates, which will allow for a localization to small, horn-shaped neighborhoods of branches of these roots corresponding to subclusters of roots. A similar idea is underlying Varchenko’s algorithm for the construction of adapted coordinates. Then we shall again distinguish between the corresponding three cases arising in the new coordinates and, if, necessary, iterate this procedure until no further branching of roots will take place. The details of this algorithm, which will

FIRST RESULTS WHEN hlin (φ) ≥ 2

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allow us to deal with the problems related to Case 3, will be outlined in the next subsection. Finally, the problems related to Case 2 will be handled in this chapter in an easy way by means of Drury’s Fourier restriction theorem for curves with nonvanishing torsion, as indicated in Remark 6.6. This will work when h lin ≥ 5, but in order to handle the general case where h lin ≥ 2, substantially more refined methods will be required. These will be developed in the next chapters. The study of the domain Dpr will thus require finer decompositions into further transition and homogeneous domains (with respect to further weights). These will be devised by means of an iteration scheme, somewhat resembling Varchenko’s algorithm. If φ is analytic, then the effect of any further step of the stopping time algorithm that we shall devise can be interpreted as a localization from a given domain containing one cluster of roots to a finite number of smaller, hornshaped subdomains, each of them containing either exactly one subcluster of the given cluster of roots of φ (“homogeneous domains”) or none (“transition domains; compare Remark 1.8). We also note that Varchenko’s algorithm shows that the principal root jet ψ is actually a polynomial ψ(x1 ) = cx1m + · · · + cpr x1a of degree ≤ a = alpr in Cases (a) and (b) (cf. [IM11a]). Notice that in Case (a), we will have cpr = 0, and the same holds true in Case (b) if a is not an integer. 6.5 REFINED DOMAIN DECOMPOSITION OF Dpr : THE STOPPING-TIME ALGORITHM 6.5.1 First step of the algorithm Let us begin with Case (a), where Dpr ∩ = {(x1 , x2 ) : 0 < x1 < δ, |x2 − ψ(x1 )| ≤ N x1a }, with a possibly large constant N > 0. We then put D(1) := Dpr ∩ , φ (1) := φ a , ψ (1) := ψ and a(1) := a, κ (1) := κ lpr, so that D(1) can be rewritten as a

D(1) = {(x1 , x2 ) : 0 < x1 < δ, |x2 − ψ (1) (x1 )| ≤ N x1 (1) }. As in the discussion of the domains Dl in the previous chapter, we can cover the domain D(1) by finitely many narrow domains of the form a

a

D(1) (c0 ) := {(x1 , x2 ) : 0 < x1 < δ, |x2 − ψ(x1 ) − c0 x1 (1) | ≤ εx1 (1) }, where ε > 0 can be chosen as small as we need and where 0 ≤ |c0 | ≤ N. Fix any of these domains, and again put v := (1, c0 ). We distinguish again between the cases where ∂2 φκ(1) (1) (v)  = 0 (Case 1), (1) (1) (1) ∂2 φκ (1) (v) = 0 and ∂1 φκ (1) (v)  = 0 (Case 2), and ∇φκ (1) (v) = 0 (Case 3). Now, in Case 1, we can argue as in the corresponding case in Section 6.3, since our arguments in that case did not make use of the condition l > lpr . In Case 2, the argument given in Section 6.3 may fail, since it made use of the estimate B < d/2, which here no longer may hold true. However, as explained

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in Remark 6.6, if h lin ≥ 5, we may use the alternative argument based on Drury’s restriction estimate for curves in this case. If Case 3 does not appear for any choice of c0 , then we stop our algorithm and are done. a Otherwise, assume Case 3 applies to c0 , so that c0 x1 (1) is a root of φκ(1) (1) , say of multiplicity M1 ≥ 2. In this case, we define new coordinates y by putting y1 := x1

and

y2 := x2 − ψ (2) (x1 ),

(6.27)

where a

ψ (2) (x1 ) := ψ(x1 ) + c0 x1 (1) . We denote by x = s(2) (y) the corresponding change of coordinates, which in general is a fractional shear only, since the exponent a(1) = a may be noninteger (but rational). In these coordinates (y1 , y2 ), φ is given by φ (2) := φ ◦ s(2) , and the domain D(1) (c0 ) becomes the domain a

a D(1) := {(y1 , y2 ) : 0 < y1 < δ, |y2 | ≤ εy1 (1) },

which is still κ (1) homogeneous. Let us see to which extent the Newton polyhedra of φ (1) and φ (2) will differ. Claim 1. The Newton polyhedra of φ (1) and φ (2) agree in the region above the bisectrix. In particular, the line (m) intersects the boundary of the augmented Newton polyhedron N r (φ (1) ) = N r (φ a ) at the same point as the augmented Newton polyhedron N r (φ (2) ) of φ (2) , so that we can use the modified “adapted” coordinates (6.27) in place of our earlier adapted coordinates to compute the r-height of φ. a To see this, observe that φ (2) (x1 , x2 ) = φ (1) (x1 , x2 + c0 x1 (1) ), where the exponent a(1) is just the reciprocal of the (modulus of the) slope of the line containing the principal face of the Newton polyhedron of φ (1) = φ a . This implies that the edges of N (φ (1) ) and N (φ (2) ) that lie strictly above the bisectrix are the same (compare corresponding discussions in [IM11a] or in Chapter 9). Moreover, if γ(1) = [(A(0) , B(0) ), (A(1) , B(1) )] = [(Alpr −1 , Blpr −1 ), (Alpr , Blpr )] is the principal face of N (φ (1) ), then it is easy to see that the principal face of N (φ (2) ) is given by the   := [(A(0) , B(0) ), (A(1) , B(1) )], where edge γ(1) A(1) := A(1) + a(1) (B(1) − M1 ),

 B(1) = M1 a

(1) (write φκ(1) is a root of multiplicity M1 (1) in the normal form (1.15) and use that c0 x1 a of φκ(1) ). Observe also that M ≤ h, because φ is in adapted coordinates. We thus 1 (1)  still lies on or below the bisectrix. This proves see that the right endpoint of γ(1) the claim. Our considerations show that it suffices to study the contributions of narrow domains of the form

a

 = {(x1 , x2 ) : 0 < x1 < δ, |x2 − ψ (2) (x1 )| ≤ εx1 (1) } D(1)

in place of D(1) (these actually depend on the choice of the real root of φκ(1) (1) —this corresponds to a “fine splitting” of roots of φ in the case where φ is analytic).

FIRST RESULTS WHEN hlin (φ) ≥ 2

127

 Case A. N (φ (2) ) ⊂ {(t1 , t2 ) : t2 ≥ B(1) = M1 }. In this case, we again stop our algorithm.  =M1 . Case B. N (φ (2) ) contains a point below the horizontal line given by t2 =B(1)

Then N (φ (2) ) will contain a further compact edge  γ(2) = [(A(1) , B(1) ), (A(2) , B(2) )],   so that (A(1) , B(1) ) is a vertex at which the edges γ(1) and γ(2) meet. Determine the (2) weight κ by requiring that γ(2) lies on the line

κ1(2) t1 + κ2(2) t2 = 1, and put a(2) := κ2(2) /κ1(2) . Then clearly a(1) < a(2) .  into the domains Next, we decompose the domain D(1) a

a

E(1) := {(x1 , x2 ) : 0 < x1 < δ, Nx1 (2) < |x2 − ψ (2) (x1 )| ≤ εx1 (1) } and a

D(2) := {(x1 , x2 ) : 0 < x1 < δ, |x2 − ψ (2) (x1 )| ≤ N x1 (2) }, where N > 0 will be a sufficiently large constant. The contributions by the transition domain E(1) can be estimated in exactly the same way as we did for the domains El in Section 6.2. Indeed, notice that our arguments for the domains El did apply to any l ≥ l0 as long as Bl ≥ 1, so that this statement is immediate when c0 = 0, where the coordinates y in (6.27) do agree with our original adapted coordinates. When c0  = 0, there are two minor twists in the arguments needed: first, observe that Lemma 6.1 remains valid for = φ (2) and the domain a

a

a E(1) := {(y1 , y2 ) : 0 < y1 < δ, 2M y1 (2) < |y2 | ≤ 2−M y1 (1) }

corresponding to the domain E(1) in the coordinates (6.27) when ε = 2−M and N = 2M . The fact that a(2) may be noninteger, but rational, say a(2) = p/q, with p, q ∈ N, requires only minor changes of the proof: just consider the Taylor expanq sion of the smooth function (y1 , y2 ). Second, if we define, in analogy with hl in  and γ(2) of N (φ (2) ) (1.17), the corresponding quantity associated to the edges γ(1) by h(1) :=

1 + mκ1(1) − κ2(1) κ1(1) + κ2(1)

= hlpr

and

h(2) :=

1 + mκ1(2) − κ2(2) κ1(2) + κ2(2)

,

then Claim 1 shows that max{h(1) , h(2) } ≤ hr (φ), which replaces the condition max{hl , hl+1 } ≤ hr (φ) that was needed in the proof of Proposition 6.3. 6.5.2 Further steps of the algorithm We are thus left with the domains D(2) , which formally look exactly like D(1) , only with ψ (1) replaced by ψ (2) and a(1) replaced by a(2) . This allows us to iterate our

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first step of the algorithm that led from D(1) to D(2) , producing in this way nested sequences of domains, Dpr = D(1) ⊃ D(2) ⊃ · · · ⊃ D(l) ⊃ D(l+1) ⊃ · · · , of the form a

D(l) := {(x1 , x2 ) : 0 < x1 < δ, |x2 − ψ (l) (x1 )| ≤ N x1 (l) }, where N = Nl may be large and where the functions ψ (l) are of the form ψ (l) (x1 ) = ψ(x1 ) +

l−1

a

cj −1 x1 (j ) ,

j =1

with real coefficients cj , and where the exponents a(j ) form a strictly increasing sequence a = a(1) < a(2) < · · · < a(l) < a(l+1) < · · · of rational numbers. Moreover, each of the domains D(l) will be covered by a finite number of narrow domains of the form a

a

D(l) (cl ) := {(x1 , x2 ) : 0 < x1 < δ, |x2 − ψ (l) (x1 ) − cl x1 (l) | ≤ εx1 (l) }, with |cl | ≤ Nl , for every given ε = εl > 0. Notice that these domains can be rewrit of the form ten as domains D(l) a

 = {(x1 , x2 ) : 0 < x1 < δ, |x2 − ψ (l+1) (x1 )| ≤ εx1 (l) }; D(l)

(6.28)

here ε > 0 can be chosen as small as we please. These, in turn, will decompose as  D(l) = E(l) ∪ D(l+1) ,

where E(l) is a transition domain of the form a

a

E(l) := {(x1 , x2 ) : 0 < x1 < δ, Nx1 (l+1) < |x2 − ψ (l+1) (x1 )| ≤ εx1 (l) }. Putting φ (l) (x1 , x2 ) := φ(x1 , x2 + ψ (l) (x1 )), one finds that the Newton polyhedron N (φ (l+1) ) agrees with that one of φ a = φ (1) in the region above the bisectrix, and it will have subsequent “edges”      γ(1) = [(A(0) , B(0) ), (A(1) , B(1) )], γ(2) = [(A(1) , B(1) ), (A(2) , B(2) )], . . . ,     = [(A(l−1) , B(l−1) ), (A(l) , B(l) )], γ(l+1) = [(A(l) , B(l) ), (A(l+1) , B(l+1) )], γ(l)

crossing or lying below the bisectrix, at least (possibly more). In fact, it is possible that some of these “edges” degenerate and become a single point (we then shall still speak of an edge, with a slight abuse of notation). The edge with index l will lie on a line L(l) := {(t1 , t2 ) ∈ R2 : κ1(l) t1 + κ2(l) t2 = 1},

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129 a

where a(l) = κ2(l) /κ1(l) . Moreover, cl−1 x1 (l) is any real root of the κ (l) -homogeneous polynomial φκ(l)(l) , of multiplicity Ml ≥ 2. Notice that when φ is real analytic, then this just means that ψ (l) is a leading term of a root of φ belonging to the cluster of roots defined by ψ. Our algorithm thus follows any possible “fine splitting” of the roots belonging to this cluster, and the domains D(l) , and so on, depend on the branches of these roots that we choose along the way.  , which shows that the sequence of By our construction, we see that Ml = B(l) multiplicities is decreasing, that is, M1 ≥ M2 ≥ · · · ≥ Ml ≥ Ml+1 ≥ · · · .

(6.29)

Observe also that the transition domains E(l) can be handled by the same reasoning that we had applied to E(1) . When will our algorithm stop? Clearly, this will happen at step l, when φκ(l)(l) has no real root, so that only Case 1 and Case 2 will arise at this step. In that case, we do obtain the desired Fourier restriction estimate for the piece of surface corresponding to D(l) , just by the same reasoning that we applied in Section 6.3. Otherwise, we shall also stop our algorithm in step l when  = Ml }. N (φ (l+1) ) ⊂ {(t1 , t2 ) : t2 ≥ B(l)  In this situation, the domain that still needs to be understood is the domain D(l) given by (6.28). Notice that in this case, by Condition (R) there exists a function ψ˜ (l+1) ∼ ψ (l+1) such that φ can be factored as

˜ 1 , x2 ), φ(x1 , x2 ) = (x2 − ψ˜ (l+1) (x1 ))Ml φ(x

(6.30)

˜ This means that Lemma 6.2 (respectively, with a fractionally smooth function φ. its immediate extension to fractionally smooth functions) applies to the function  can be regarded as

(y1 , y2 ) := φ(y1 , y2 + ψ˜ (l+1) (y1 )), and since the domain D(l) a generalized transition domain, like the domains Elpr −1 that appeared when the principal face of N (φ a ) was an unbounded horizontal edge, we can argue in the same way as we did for the domains Elpr −1 in Section 6.2 to derive the required  . restriction estimates for the piece of S corresponding to D(l) There is, finally, the possibility that our algorithm does not terminate. In this case, (6.29) shows that the sequence of integers Ml will eventually become constant. We then choose L minimal so that Ml = ML for all l ≥ L. Note that, by our construction, ML ≥ 2. For every l ≥ L + 1, the point (A, B) := (A(L) , B(L) ) = (AL , ML ) will be a vertex of N (φ (l) ) that is contained in the line L(l) , whose slope 1/a(l) tends to zero as l → ∞, and N (φ (l) ) is contained in the half-plane bounded by L(l) from below. Notice also that there is a fixed rational number 1/q, with q an integer, such that every a(l) is a multiple of 1/q. This can be proven in the same way as the corresponding statement in [IKM10] on page. 240. We can thus apply a classical theorem of E. Borel [Bo95] (see also [H90], Theorem 1.2.6) in a way similar to [IM11a] in order to show that there is a smooth

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function h of x1 whose Taylor series expansion is given by the formal series h(x1 ) ∼

q ψ(x1 )

+

∞

qa(j )

cj −1 x1

.

j =1

If we put ψ (∞) (x1 ) := h(x1 ) and set φ (∞) (y1 , y2 ) := φ(y1 , y2 + ψ (∞) (y1 )), then it is easily seen by means of a straightforward adaption of the proof of Theorem 5.1 in [IM11a] that N (φ (∞) ) ⊂ {(t1 , t2 ) : t2 ≥ B}. Therefore, Condition (R) in Theorem 1.14 guarantess that, possibly after adding a flat function to ψ (∞) , we may ˜ 1 , x2 ), which means that assume that φ factors as φ(x1 , x2 ) = (x2 − ψ (∞) (x1 ))B φ(x the analogue of (6.30) holds true. We can thus argue as before to complete also this case and hence also the discussion of the Case (a), in which the principal face of N (φ a ) is a compact edge. Finally, in Case (b), where the principal face of N (φ a ) is a vertex, we have that  in Dpr = {(x1 , x2 ) : |x2 − ψ(x1 )| ≤ εx1a }, which corresponds to the domain D(1) the discussion of Case (a). This means that we can just drop the initial step of the algorithm described before and from then on may proceed as in Case (a). We have thus established the desired restriction estimates for the piece of the surface S corresponding to the remaining domain Dpr , which completes the proof of Proposition 6.7 and hence also of Theorem 1.14, in the case where h lin (φ) ≥ 5. 1/q

Chapter Seven How to Go beyond the Case hlin (φ) ≥ 5 So far, we have been able to cover the cases where h lin (φ) < 2 or h lin (φ) ≥ 5. In the latter case, we had made use of Drury’s restriction estimate for curves, but this approach is indeed limited to the case where h lin (φ) ≥ 5. In order also to cover the case where 2 ≤ h lin (φ) < 5, quite different and substantially more involved arguments are needed, which will be developed in the sequel. It turns out that these arguments will work whenever h lin (φ) ≥ 2. In Section 7.1 we shall briefly remind the reader of the regions for which we still  , need to prove restriction estimates, namely, the narrow domains of the type D(l) (l+1) which in the modified adapted coordinates y1 := x1 and y2 := x2 − ψ (x1 ) are a a := {(y1 , y2 ) : 0 < y1 < ε, |y2 | ≤ εy1 (l) }. Since we are left with of the form D(l) (1, 0) = 0 and ∂1 φκ(l+1) (1, 0)  = the Case 2 situation, we will assume that ∂2 φκ(l+1) (l) (l) 0. Again, we shall decompose these domains dyadically, and Littlewood-Paley theory allows us to reduce everything to proving uniform restriction estimates for the dyadic constituents µk of the part of the measure µ corresponding to the domain  (cf. Proposition 7.2). D(l)  , which are asIt will turn out that the treatment of the domains of type D(1) sociated with the principal face of the Newton polyhedron of φ a , is by far more  with l ≥ 2, involved than the treatment of the remaining domains of type D(l) which are associated with edges of the Newton polyhedra lying below the bisectrix.  and only briefly We shall therefore mainly concentrate on the domains of type D(1)  with l ≥ 2 indicate the modifications also needed to cover the domains of type D(l) in Section 7.10. In Section 7.2, we shall basically follow our approach from earlier chapters and first reduce our restriction estimates for the measures µk to corresponding uniform estimates for families of normalized measures νδ (compare (7.8)) associated with phase functions of the form B−1
j n φδ (x) = x2B b(x1 , x2 , δ) + x1n α(δ1 x1 ) + δj +2 x2 x1 j αj (δ1 x1 ). j =1

These phase functions will depend on small perturbation parameters δ = (δ0 , . . . , δB+1 ), where each δi is a fractional power of 2−k . In some situations, it is possible to remove the term of order j = B − 1 in the last sum by means of another smooth change of coordinates of nonlinear shear type, as shown in Section 7.3. This will turn out to be quite useful, in particular for the discussion of the case where B = 3 (which will create the most serious difficulties among all), for instance in our discussion of the operators TδI I I1 in Section 7.6, as well as in Chapter 8. Next, in Section 7.4, we shall establish lower bounds for the r-height hr = hr (φ). These will turn out to be very useful in the subsequent sections, since we will

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be able to prove in many situations that for our normalized measures νδ , uniform Lp -L2 Fourier restriction estimates will hold true for wider Lp -ranges of the form p  ≥ 2hrb + 2 than the range p ≥ 2hr + 2, where hrb is one of the lower bounds that we have found for hr . Indeed, these restriction estimates will be established in Section 7.5, where we shall follow to some extent the approach that we had devised in Chapter 4. Again, we perform an additional spectral localization to dyadic frequency domains given by |ξi | ∼ λi , i = 1, 2, 3, and denote by νδλ the corresponding complex measures. These should be compared with the measures νjλ from Section 4.1, where δ0 will now take over the role that 2−j played in Chapter 4. Again, we will distinguish various cases, depending on the relative sizes of the quantities λ1 , λ2 , λ3 , and δ0 , in a way analogous to what we did in Section 4.1. The most difficult situation will arise when λ1 ∼ λ2 ∼ λ3 , and the study of this case will therefore be initiated only later in Section 7.7. In all other situations, we can obtain the desired Fourier restriction estimates for the measures νδλ in a comparatively easy way, by making use of our lower bounds for hr , with the exception of a few cases in which interpolation arguments are needed in order to cover the endpoint p = pc . In some cases, this can be done by means of the real interpolation method due to Bak and Seeger, more precisely by means of Proposition 2.6 (compare Section 7.6), and in others we need again to apply complex interpolation. In all these exceptional situations, we shall have m = 2 and B = 2 or B = 3. In Sections 7.7 and 7.8, where we assume that λ1 ∼ λ2 ∼ λ3 , we will again be able to handle most cases in a relatively easy way, with the exception of a few cases. One of these exceptional situations concerns the case where B = 5, but it turns out that this case can still be dealt with in an easy way by means of a slight improvement of our previous methods (see Section 7.8). As already mentioned, the most difficult situations will arrive when m = 2 and B = 3. An important tool for the study of this situation will be provided by Proposition 7.9, which gives a description of the augmented Newton polyhedron of φ˜ a under the additional assumption that 3 < hr + 1 ≤ 3.5. Only when hr + 1 lies within this range, our lower bounds for hr are insufficient in order to obtain the desired estimates in the chapter, but the precise value for hr provided by Proposition 7.9 will work. What remains in the end are a few exceptional cases, which will be compiled in Section 7.9. In these remaining cases, we have in particular that m = 2 and either B = 3 or B = 4. Their treatment will be rather involved, and it will form the content of Chapter 8.

7.1 THE CASE WHEN hlin (φ)≥2: REMINDER OF THE OPEN CASES Assume from now on that h lin (φ) ≥ 2. We recall the following two cases from Chapter 6: (a) The principal face π(φ a ) of the Newton polyhedron N (φ a ) of φ a is a compact edge, lying on the line La , which we call the principal line of N (φ a ); (b) π(φ a ) is the vertex (h, h).

GOING BEYOND THE CASE hlin (φ) ≥ 5

133

What has remained open in our preceding discussion is the study of the piece of the surface S corresponding to the domain Dpr containing the principal root jet ψ in Cases (a) and (b), that is,  {(x1 , x2 ) : 0 < x1 < ε, |x2 − ψ(x1 )| ≤ N x1a }, in Case (a), Dpr ∩ := in Case (b), {(x1 , x2 ) : 0 < x1 < ε, |x2 − ψ(x1 )| ≤ εx1a }, when 2 ≤ h lin (φ) < 5. Indeed, we shall develop an approach that will work whenever h lin (φ) ≥ 2. Our goal will thus be to prove the following extension of Proposition 6.7 to the case where h lin (φ) ≥ 2. Proposition 7.1. Assume that h lin (φ) ≥ 2, and that we are in Case (a) or (b). Then for any given N > 0 in Case (a), respectively, every sufficiently small ε in Case (b), we have 1/2  2 ρlpr  |f | dµ ≤ Cp f Lp (R3 ) , f ∈ S(R3 ), Dpr

whenever p  ≥ pc . In order to prove this proposition, we follow the domain decomposition algorithm for the domain Dpr developed in Section 6.5 and begin with Case (a). In this case, this algorithm led to a finite family of subdomains E(l) (so-called transition  , l ≥ 1 of the form domains), and domains D(l) a

 D(l) := {(x1 , x2 ) : 0 < x1 < δ, |x2 − ψ (l+1) (x1 )| ≤ εx1 (l) },

where the functions ψ (l) are of the form (l)

ψ (x1 ) = ψ(x1 ) +

l−1


a

cj −1 x1 (j ) ,

j =1

with real coefficients cj , and where the exponents a(j ) form a strictly increasing sequence a = a(1) < a(2) < · · · a(l) < a(l+1) < · · · of rational numbers. Moreover, in the modified adapted coordinates given by y1 := x1 ,

y2 := x2 − ψ (l+1) (x1 ),

the function φ is given by φ (l+1) (y1 , y2 ) := φ(y1 , y2 + ψ (l+1) (y1 )). Notice that we have ψ (1) = ψ and φ (1) = φ a . Recall also that the domains and modified adapted coordinates that we encounter here do depend on the choice of the real numbers cl that we make in every step of our algorithm.    is associated with an “edge” γ(l) = [(A(l−1) , B(l−1) ), Moreover, the domain D(l)   (A(l) , B(l) )] (which is, indeed, an edge or can degenerate to a single point) of the Newton polyhedron of φ (l+1) in the following way.

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The edge with index l will lie on a line L(l) := {(t1 , t2 ) ∈ R2 : κ1(l) t1 + κ2(l) t2 = 1} of “slope” 1/a(l) , where a(l) = κ2(l) /κ1(l) . Introduce corresponding “κ (l) -dilations” (l) (l) (l) δr = δrκ by putting δr (y1 , y2 ) = (r κ1 y1 , r κ2 y2 ), r > 0. Then the domain a

a := {(y1 , y2 ) : 0 < y1 < ε, |y2 | ≤ εy1 (l) }, D(l)  in the coordinates (y1 , y2 ), is invariant under which represents the domain D(l) these dilations for r ≤ 1, and the Newton diagram of the κ (l) -principal part φκ(l+1) (l)  of φ (l+1) consists exactly of the edge γ(l) .  Recall also that the first edge γ(1) agrees with the principal face π(φ (2) ) of φ (2) a and lies on the principal line L of the Newton polyhedron of φ a , and it intersects  will lie within the closed half-space the bisectrix , whereas for l ≥ 2 the edge γ(l) below the bisectrix (this case will turn out to be easier). Moreover, the Newton polyhedra of φ a and of φ (l) do agree in the closed half space above the bisectrix. Now, as we have seen in Section 6.5, what remained to be controlled is the  = D(l) (cl ), that is, the situation Case 2 situation for any given narrow domain D(l) (l) (l) where ∂2 φκ (l) (1, cl ) = 0 and ∂1 φκ (l) (1, cl )  = 0. But, observe that if we define our modified adapted coordinates y1 := x1 , y2 := x2 − ψ (l+1) (x1 ) in such a way that a ψ (l+1) (x1 ) := ψ (l) (x1 ) + cl x1 (l) , with cl as in the definition of the domain D(l) (cl ), then in these coordinates y the point (1, cl ) (given in the coordinates associated to φ (l) ) corresponds to the point v = (1, 0), and we may rewrite the previous Case 2 condition as follows:

(v) = 0 and ∂2 φκ(l+1) (l)

∂1 φκ(l+1) (v)  = 0. (l)

Only in Case 2, we had made use of the assumption h lin (φ) ≥ 5, so that we may concentrate in the sequel on this case. Notice also that our decomposition algorithm worked as well in Case (b), only we had to skip the first step of the algorithm. We shall therefore study the Fourier restriction estimates for the pieces of the surface S corresponding to the domains   and shall begin with the most difficult case l = 1, that is, the domain D(1) in D(l) Case (a). In the last section we shall describe the minor modifications needed to also treat  for l ≥ 2, which will then also cover Case (b) at the same time. the domains D(l)  by means of a cutoff Observe finally that we can localize to the domain D(l) function   x2 − ψ (l+1) (x1 ) , ρ(l) (x1 , x2 ) := χ0 a εx1 (l) where χ0 ∈ D(R). Let us again fix a suitable smooth cutoff function χ ≥ 0 on R2 supported in an annulus A := {x ∈ R2 : 12 ≤ |y| ≤ R} such that the functions

GOING BEYOND THE CASE hlin (φ) ≥ 5

135 (l)

χka := χ ◦ δ2k form a partition of unity. Here, δr = δrκ denote the dilations associated with the weight κ (l) . In the original coordinates x, these correspond to the functions χk (x) := χka (x1 , x2 − ψ (l+1) (x1 )). We then decompose the measure µρ(l) dyadically as
µρ(l) = µk , (7.1) k≥k0

where χk ρ(l) µk := µ(l) . k := µ

Notice that by choosing the support of η sufficiently small, we can choose k0 ∈ N as large as we need. It is also important to observe that this decomposition can essentially be achieved by means of a dyadic decomposition with respect to the variable x1 , which again allows to apply Littlewood-Paley theory. Moreover, changing to modified adapted coordinates in the integral defining µk and scaling by δ2−k , we find that  (l) (l) (l) (l) −k|κ (l) | f (2−κ1 k x1 , 2−κ2 k x2 + 2−mκ1 k x1m ω(2−κ1 k x1 ), 2−k φk (x))

µk , f  = 2 ×η(x) dx, where ω = ω

(l)

(7.2)

is given by ψ (l+1) (x1 ) = x1m ω(x1 ),

so that ω(0)  = 0, η = ηk(l) is a smooth function supported where x1 ∼ 1, |x2 | < ε (for some small ε > 0), whose derivatives are uniformly bounded in k and where (x) + error terms of order O(2−δk ) φk (x) = φk(l+1) (x) := 2k φ (l+1) (δ2−k x) = φκ(l+1) (l) with respect to the C ∞ topology (and δ > 0). In order to prove Proposition 7.1, we still need to prove the following. Proposition 7.2. Assume that h lin ≥ 2, that we are in Case 2, that is, (1, 0) = 0 and ∂1 φκ(l+1) (1, 0)  = 0, and recall that pc = 2hr + 2. When ∂2 φκ(l+1) (l) (l) ε > 0 is sufficiently small and k0 ∈ N is sufficiently large, then for every l ≥ 1, 1/2  2  |f | dµk ≤ Cpc f Lpc (R3 ) , f ∈ S(R3 ), k ≥ k0 , where the constant Cp is independent of k. 7.2 RESTRICTION ESTIMATES FOR THE DOMAINS D (1) : REDUCTION TO NORMALIZED MEASURES ν δ Let us assume that we are in Case (a), where the principal face π(φ a ) is a compact edge. In the enumeration of edges γl of the Newton polyhedron associated with φ a in Chapter 6, this edge corresponds to the index l = lpr , that is, π(φ a ) = γlpr .

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Here, the weight κ (1) is the principal weight κ lpr from Chapter 6, and the line L(1) is the principal line La = Llpr of the Newton polyhedron of φ a . We then put κ˜ := κ (1) ,

so that

a=

κ˜ 2 a , φ a = φpr . κ˜ 1 κ˜

In particular, hlpr + 1 is the second coordinate of the point of intersection of the line (m) = {(t, t + m + 1) : t ∈ R} with the line La , and according to the identity (1.17), we have hlpr + 1 =

1 + (m + 1)κ˜ 1 . |κ| ˜

(7.3)

 The domain D(1) that we have to study is then of the form  = {(x1 , x2 ) : 0 < x1 < ε, |x2 − ψ(x1 ) − c0 x1a | ≤ εx1a }, D(1)

where ψ(x1 ) + c0 x1a = ψ (2) (x1 ). Moreover, φ (2) (x1 , x2 ) = φ a (x1 , x2 + c0 x1a ) =: φ˜a (x1 , x2 ),

(7.4)

˜a

so that φ represents φ in the modified adapted coordinates y1 := x1 ,

y2 := x2 − ψ(x1 ) − c0 x1a ,

(7.5)

compared to the adapted coordinates given by y1 := x1 , y2 := x2 − ψ(x1 ), in which φ is represented by φ a . Notice that the exponent a may be noninteger (but rational), so that ψ (2) is, in general, only fractionally smooth, that is, a smooth function of x2 and some fractional power of x1 only. The same applies to every ψ (l) with l ≥ 2, whereas φ a is still smooth; that is, when we express φ in our adapted coordinates, we still get a smooth function, whereas when we pass to modified adapted coordinates, we may only get fractionally smooth functions. a , that is, We shall write D a for the domain D(1) D a := {(y1 , y2 ) : 0 < y1 < ε, |y2 | < εy1a },  so that D a represents our domain D(1) in our modified adapted coordinates, in a ˜ which φ is represented by φ . We assume that we are in Case 2, so that ∂2 φ˜a κ˜ (1, 0) = 0 and ∂1 φ˜a κ˜ (1, 0)  = 0. ˜ Next we choose B ≥ 2 to be minimal so that ∂2B φ˜ κa˜ (1, 0)  = 0. Since φ˜ κa˜ is κhomogeneous, the principal part of φ˜ a is then of the form (cf. (6.18))

φ˜ κa˜ (y1 , y2 ) = y2B Q(y1 , y2 ) + c1 y1n ,

c1  = 0, Q(1, 0)  = 0,

(7.6)

where Q is a κ-homogeneous ˜ smooth function. Note that n is rational but not necessarily an integer since we are in modified adapted coordinates. Observe also that this implies that we may write φ˜ a (y1 , y2 ) = y2B bB (y1 , y2 ) + y1n α(y1 ) +

B−1
j =1

j

y2 bj (y1 ),

(7.7)

GOING BEYOND THE CASE hlin (φ) ≥ 5

137

with fractionally smooth functions bB , α and bj such that α(0)  = 0 and bB (y1 , y2 ) = Q(y1 , y2 ) + terms of κ-degree ˜ strictly bigger than that of Q. Moreover, the functions b1 , . . . , bB−1 of y1 are either flat or of “finite type,” that n is, bj (y1 ) = y1 j αj (y1 ), with rational exponents nj > 0 and fractionally smooth functions αj such that αj (0)  = 0. n For convenience, we shall also write bj (y1 ) = y1 j αj (y1 ) when bj is flat, keeping in mind that in this case we may choose nj ∈ N as large as we please (but αj (0) = 0). j Notice that then, for j = 1, . . . , B − 1, y2 bj (y1 ) consists of terms of κ-degree ˜ strictly bigger than 1. Recall that the Newton diagram of the κ-principal ˜ part φ˜ κa˜ is the line segment      γ(1) = [(A(0) , B(0) ), (A(1) , B(1) )], which consequently must contain a point of the Newton support of φ˜a , with the second coordinate given by B and none below that point, with the exception of (n, 0). It then follows easily that the following relations hold true: κ˜ 2 < 1,

2≤m
0. Observe that n

2(1−j κ˜2 )k bj (2−κ˜1 k x1 ) = x1 j 2−(j κ˜2 +nj κ˜1 −1)k αj (2−κ˜1 k x1 ), where (j κ˜ 2 + nj κ˜ 1 − 1) > 0.

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CHAPTER 7

In the sequel, we shall rewrite νk as νδ by putting  

νδ , f  := f x1 , δ0 x2 + x1m ω(δ1 x1 ), φδ (x) η(x) dx,

(7.8)

where φδ is of the form φδ (x) := x2B b(x1 , x2 , δ) + x1n α(δ1 x1 ) + r(x1 , x2 , δ),

(7.9)

with r(x1 , x2 , δ) :=

B−1


j n

δj +2 x2 x1 j αj (δ1 x1 ),

(7.10)

j =1

and δ = (δ0 , δ1 , δ2 , δ3 , . . . , δB+1 ) is given by δ := (2−k(κ˜2 −mκ˜1 ) , 2−kκ˜1 , 2−kκ˜2 , 2−(n1 κ˜1 +κ˜2 −1)k , . . . , 2−(nB−1 κ˜1 +(B−1)κ˜2 −1)k) ). (7.11) Recall that α(0)  = 0 and that either αj (0)  = 0, and then nj is fixed (the type of the finite-type function bj ) or αj (0) = 0, and then we may assume that nj is as large as we please. Observe that the components of δ can be viewed as small perturbation parameters, since δ → 0 as k → ∞, and that every δj is a power of δ0 , q

δj = δ0 j ,

j = 1, . . . , B + 1,

with positive exponents qj > 0 that are fixed rational numbers, except for those j ≥ 3 for which αj −2 (0) = 0, for which we may choose the exponents qj as large as we please. Moreover, b(x1 , x2 , δ) is a smooth function of all three arguments, and b(x1 , x2 , 0) = Q(x1 , x2 ).

(7.12)

For δ sufficiently small, this implies in particular that b(x1 , x2 , δ)  = 0 when x1 ∼ 1 and |x2 | < ε. Assume we can prove that for δ sufficiently small, we have 1/2  |fˆ|2 dνδ ≤ Cf Lpc , (7.13) with C independent of δ. Then straightforward rescaling by means of the κ− ˜ dilations leads to the estimate 1/2  2 ˜ (1−2(hlpr +1)/pc )/2 ˆ ≤ C2−k|κ| f Lpc , (7.14) |f | dµk where pc ≥ 2(hlpr + 1) (compare (6.16) and the subsequent identity). So, our goal is to verify (7.13). Observe also that the κ-principal ˜ parts of φ˜ a and φδ do agree. 7.3 REMOVAL OF THE TERM y2B−1 bB−1 (y1 ) IN (7.7) In the case where the left endpoint of the principal face π(φ˜ a ) of the Newton polyhedron of φ˜ a (which agrees with the one for φ a ) is of the form (A, B), it is possible

GOING BEYOND THE CASE hlin (φ) ≥ 5

139

to remove the term y2B−1 bB−1 (y1 ) in our formula (7.7) for φ˜ a by means of an additional smooth change of coordinates (again of nonlinear shear type). The new coordinates turn out to be smooth and can be used as alternative adapted coordinates. This will become important in some situations—in particular, when B = 3. Proposition 7.3. Assume that the left endpoint of the principal face π(φ˜ a ) of N (φ˜ a ) is of the form (A, B), where φ˜ a and B are as in (7.7). Then the numbers a = κ˜ 2 /κ˜ 2 , A, and n are integers, and φ˜ a is smooth. Moreover, there exists an additional, smooth change of coordinates of the form z 1 = x1 , z 2 = x2 − (ψ (2) (x1 ) + ρ(x1 )), which allows us to remove the term of order j = B − 1 in (7.7), that is, after applying this change of coordinates and denoting the corresponding coordinates again by (y1 , y2 ), we may assume that φ˜ a (y1 , y2 ) = y2B bB (y1 , y2 ) + y1n α(y1 ) +

B−2


j

y2 bj (y1 ).

(7.15)

j =1

Proof. In view of our definition of B (cf. (7.6)), we see that our assumptions imply that the κ˜ homogeneous function Q in (7.6) must be of the form Q(y1 , y2 ) = c0 y1A , with c0  = 0. Let us assume without loss of generality that c0 = 1, so that by (7.6) φ˜ κa˜ (y) = y1A y2B + c1 y1n ,

c1  = 0.

(7.16)

But then φκa˜ will be given by φκa˜ (x) = x1A (x2 − c0 x1a )B + c1 x1n (compare (7.4)). This must be a polynomial in (x1 , x2 ), since φ a is smooth. Expanding (x2 − c0 x1a )B , it is clear that A and a must be integers. But then, φ˜ a is also necessarily smooth, which implies that n must be an integer, too. Observe next that (7.16) also implies that ∂1A ∂2B−1 φ˜ κa˜ (y1 , 0) ≡ 0 and ∂1A ∂2B φ˜ κa˜ (y1 , 0) ≡ A!B!  = 0 for |y1 | < ε. Moreover, since (A, B) is a vertex of N (φ˜ a ), we also have ∂1A ∂2B φ˜ a (0, 0) = A!B!  = 0, whereas ∂1A ∂2B−1 φ˜ a (0, 0) = 0. Recall also that a ≥ 2. We can now argue in a similar way as in the proof of Proposition 2.11: if A = 0, then the implicit function theorem shows that, for ε sufficiently small, there is a smooth function ρ(y1 ), −ε < y1 < ε, such that ∂2B−1 φ˜ a (y1 , ρ(y1 )) ≡ 0, and comparing κ-principal ˜ parts, it is easy to see that the κ-principal ˜ part of ρ has κ-degree ˜ strictly bigger than the degree of y1a . Thus, if we perform the further change of coordinates (z 1 , z 2 ) := (y1 , y2 − ρ(y1 )), in which φ(x1 , x2 ) is repre˜ 1 , z 2 ), it is easily seen that the Newton polyhedra of φ˜ a and sented, say, by φ(z ˜ φ as well as their κ-principal ˜ parts are the same (cf. similar arguments in [IM11a]). Replacing ψ (2) (y1 ) by ψ (2) (y1 ) + ρ(y1 ) and modifying ω(y1 ) accordingly, we then find that in the corresponding modified adapted coordinates z 1 = x1 , z 2 = x2 − (ψ (2) (x1 ) + ρ(x1 )), the function φ˜ satisfies ˜ 1 , 0) ≡ 0. ∂2B−1 φ(z

(7.17)

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On the other hand, like φ˜ a , it must be of the form ˜ 1 , z 2 ) = z 2B bB (z 1 , z 2 ) + z 1n α(z 1 ) + φ(z

B−1


j

z 2 bj (z 1 ),

j =1

where bB (z 1 , z 2 ) = 1 + terms of κ-degree ˜ strictly bigger than 0, and then (7.17) implies that bB−1 (z 1 ) ≡ 0. Assume next that A ≥ 1. Then, by (7.16), ∂2B−1 φ˜ a (y1 , y2 ) = c2 y1A y2 + R(y1 , y2 ), ˜ strictly bigger than the with c2 := B! and where R consists of terms of κ-degree degree of y1A y2 . Consider the equation ∂2B−1 φ˜ a (y1 , y2 ) = c2 y1A y2 + R(y1 , y2 ) = 0.

(7.18)

It is easily seen that the line given by t1 + at2 = C, where C := A + a ∈ N, is a supporting line to the Newton polyhedron N (∂2B−1 φ a ), which intersects N (∂2B−1 φ a ) in exactly one point, namely, the point (A, 1). Moreover, R is a smooth j j function consisting of terms cj1 ,j2 y11 y22 for which j1 + aj2 ≥ 1 + C (more precisely, the Newton polyhedron of R is contained in the half space where t1 + at2 ≥ 1 + C). In analogy to the proof of the last part of Proposition 2.11, we write, for y1  = 0, y2 = y1a z. Then (7.18) is equivalent to the equation c2 y1C z + R(y1 , y1a z) = 0.

(7.19)

R(y1 , y1a z)

Clearly, is a smooth function, and in view of our previous observation we may factor it as R(y1 , y1a z) = y1C+1 g(y1 , z), with a smooth function g(y1 , z). Thus, for y1  = 0, equation (7.19) is equivalent to the equation c2 z + y1 g(y1 , z) = 0. Now we can again apply the implicit function theorem to conclude that near the origin this equation has a unique smooth solution z = ψ1 (y1 ),

with ψ1 (0) = 0.

In particular, we find that c2 y1C ψ1 (y1 ) + R(y1 , y1a ψ1 (y1 )) ≡ 0. Thus, putting ρ(y1 ) := y1a ψ1 (y1 ), we see that ρ is smooth, ρ(0) = 0, and ∂2B−1 φ˜ a (y1 , ρ(y1 )) ≡ 0. Notice also that the κ-principal ˜ part of ρ clearly has κ-degree ˜ strictly bigger than the degree of y1a . From now on we may thus argue as in the previous case where we had A = 0 and change coordinates from (y1 , y2 ) to (z 1 , z 2 ) := (y1 , y2 − ρ(y1 )), with the effect that in these new coordinates, we again have (7.17). On the other hand, like φ˜ a , φ˜ must here be of the form ˜ 1 , z 2 ) = z 2B bB (z 1 , z 2 ) + z 1n α(z 1 ) + φ(z

B−1


j

z 2 bj (z 1 ),

j =1

z 1A

˜ strictly bigger than κ˜ 1 A, and thus (7.18) where bB (z 1 , z 2 ) = + terms of κ-degree Q.E.D. implies that bB−1 (z 1 ) ≡ 0, and we obtain (7.15).

GOING BEYOND THE CASE hlin (φ) ≥ 5

141

7.4 LOWER BOUNDS FOR hr (φ) Recall that B ≥ 2 and d ≥ 2. We shall often use the interpolation parameter θc :=

2 1 = r . pc h +1

Since, by definition, hr ≥ d, the second assumption implies that 1 . (7.20) 3 We first derive some useful estimates from the following for pc = 2(hr + 1). We let H := 1/κ˜ 2 , so that θc ≤

n=

1 , κ˜ 1

H =

1 . κ˜ 2

(7.21)

Note that H is rational but not necessarily entire. We next define mH 2 m+1 m+1 , p˜ c := 2(h˜ r + 1), θ˜c :=  = ≤ θ˜B := . h˜ r := m+1 p˜ c mH + m + 1 mB + m + 1 We also let p˜ B := 2/θ˜B ≤ p˜ c and  := pH

12H , 3+H

θH :=

2 1 1 +  = pH 2H 6

and define pB , θB accordingly, with H replaced by B. Lemma 7.4. (a) We have pc > p˜ c , unless hr = h˜ r = d and hr + 1 ≥ H . In the latter case, pc = p˜ c = 2(d + 1). (b) If m ≥ 3 and H ≥ 2 or m = 2 and H ≥ 3, then  ≥ pB , p˜ c ≥ pH  where the inequality p˜ c ≥ pH is even strict unless m = 2 and H = 3.

Proof. (a) The Newton polyhedron N (φ˜ a ) of φ˜ a is contained in the closed half space bounded from below by the principal line La of φ˜ a , which passes through the points (0, H ) and (n, 0). Moreover, it is known that the principal line L of φ is a supporting line to N (φ˜ a ) (this follows from Varchenko’s algorithm), and it has slope 1/m. It is therefore parallel to the line L˜ passing through the points (0, H ) and (mH, 0) and lies “above” L˜ (see Figure 7.1). Thus the second coordinate d + 1 of the point of intersection of L with (m) is greater than or equal to the second coordinate t2 of the point of intersection (t1 , t2 ) of L˜ with (m) , so that hr ≥ d ≥ t2 − 1. But, the point (t1 , t2 ) is determined by the equations t2 = m + 1 + t1 and t2 = H − t1 /m, so that t2 = (mH + m + 1)/(m + 1) = h˜ r + 1. This shows that hr ≥ h˜ r ; hence, pc ≥ p˜ c . ˜ Notice also that d + 1 > t2 ; hence, pc > p˜ c , unless L = L. ˜ So, assume that L = L.

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CHAPTER 7

∆(m)

H d+1 t2 m+1

N (φ˜a )

B ˜ L

L

mH

n Figure 7.1

Then d = 1/(1/H + 1/mH ) = h˜ r , and the principal face π(φ˜ a ) of N (φ˜ a ) must be the edge [(0, H ), (n, 0)] (see Figure 7.1). Thus, if hr + 1 ≥ H , then clearly hr = d = h˜ r and pc = p˜ c . And, if hr + 1 < H , then we see that hr + 1 is the second coordinate of the point of intersection of (m) with π(φ˜ a ), and thus 1 + (m + 1)κ˜ 1 h˜ r + 1 < hr + 1 = hlpr + 1 = |κ| ˜ (cf. (7.3)).  is equivalent to (b) The inequality p˜ c ≥ pH mH 2 − (2m + 5)H + 3m + 3 ≥ 0, so that the remaining statements are elementary to check.

Q.E.D.

The following corollary is a straightforward consequence of the definition of θH and Lemma 7.4.

GOING BEYOND THE CASE hlin (φ) ≥ 5

143

Corollary 7.5. (a) If m ≥ 3 and H ≥ 2 or m = 2 and H ≥ 3, then θc < θB unless m = 2 and H = B = 3 (where θc = θB = 13 ). (b) If hr + 1 ≤ B, then θc < θ˜c unless B = H = hr + 1 = d + 1, where θc = θ˜c . (c) If H ≥ 3, then θc < 13 unless H = 3 and m = 2. 7.5 SPECTRAL LOCALIZATION TO FREQUENCY BOXES WHERE |ξi |∼λi : THE CASE WHERE NOT ALL λi s ARE COMPARABLE Observe next that by (7.8) the Fourier transform νδ of νδ can be written as  νδ (ξ ) = e−i(x,δ,ξ ) η(x) dx, where the complete phase  corresponding to φδ is given by (x, δ, ξ ) := ξ1 x1 + ξ2 (δ0 x2 + x1m ω(δ1 x1 )) + ξ3 φδ (x1 , x2 ), with φδ (x) = x2B b(x1 , x2 , δ) + x1n α(δ1 x1 ) + r(x1 , x2 , δ), so that (x, δ, ξ ) = ξ1 x1 + ξ2 x1m ω(δ1 x1 ) + ξ3 x1n α(δ1 x1 )  +ξ2 δ0 x2 + ξ3 x2B b(x1 , x2 , δ) + r(x1 , x2 , δ) . By Tδ we shall denote the operator of convolution with νδ , which we need to  estimate as an operator from Lp to Lp . In order to estimate the operator Tδ , we follow the approach in Section 4.1 (compare with (4.10), where 2−j plays the same role that δ0 does here) and decompose
νδ = νk = νδλ , λ

where the sum is taken over all triples λ = (λ1 , λ2 , λ3 ) of dyadic numbers λi ≥ 1 and where νkλ is localized to frequencies ξ such that |ξi | ∼ λi if λi > 1 and |ξi |
1 if λi = 1. The cases where λi = 1 for at least one λi can be dealt with in the same way as the corresponding cases where λi = 2, and therefore we shall always assume in the sequel that λi > 1,

i = 1, 2, 3.

The spectrally localized measure νδλ (x) is then given by       ξ1 ξ2 ξ3 χ1 χ1 νδ (ξ ), νδλ (ξ ) := χ1 λ1 λ2 λ3 that is, νδλ (x) = λ1 λ2 λ3



 χˇ 1 (λ1 (x1 − y1 )) χˇ 1 λ2 (x2 − δ0 y2 − y1m ω(δ1 y1 )

×χˇ 1 (λ3 (x3 − φδ (y)) η(y) dy,

(7.22)

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CHAPTER 7

where χˇ 1 again denotes the inverse Fourier transform of χ1 . Recall also that supp η ⊂ {y1 ∼ 1, |y2 | < ε},

(ε  1).

(7.23)

Arguing as in the proof of estimate (4.37) in Section 4.2, by making use of the localizations given by the first and the third factor of the integrand in (7.22) for the integrations in y1 and y2 by means of the van der Corput–type estimates of Lemma 2.1, we obtain −1/B

νδλ ∞
λ1 λ2 λ3 λ−1 1 λ3

.

Similarly, the localizations given by the first and second factor imply −1 νδλ ∞
λ1 λ2 λ3 λ−1 1 (λ2 δ0 ) ,

and consequently (B−1)/B

νδλ ∞
min{λ2 λ3

, λ3 δ0−1 }.

(7.24)

We have to distinguish various cases. Notice first that it is easy to see that the phase function  has no critical point with respect to x1 if one of the components λi of λ is much bigger than the two others, so that integrations by parts yield that νδλ ∞
|λ|−N

for every N ∈ N,

which easily implies estimates for the operator Tδλ : ϕ  → ϕ ∗ νδλ that are better than needed. We may therefore concentrate on the following, remaining cases. Recall the interpolation parameter θc = 2/pc ≤ 13 . Case 1. λ1 ∼ λ3 , λ2  λ1 . First applying the method of stationary phase in x1 and then van der Corput’s lemma in x2 , we find that νδλ ∞
λ1

−1/2−1/B

.

By interpolation, using this estimate and the first one in (7.24), we obtain −1/B−1/2+3θ/2 θ λ2 ,

Tδλ p→p
λ1

where θ = 2/p . Summation over all dyadic λ2 with λ2  λ1 yields
−1/B−1/2+5θ/2 Tδλ p→p
λ1 . λ2 λ1

Notice that for θ := θB we have −

1 5 1 1 − + θ= B 2 2 4



1 1 − B 3

 ≤ 0,

if B ≥ 3,

and that strict inequality holds when θ < θB . But, Corollary 7.5(a) shows that if H ≥ 3, then indeed θc < θB , unless m = 2 and H = B = 3. Consequently, for θ := θc we can sum over all dyadic λ1 (unless H = B = 3 and m = 2) and obtain
TδI pc →pc
Tδλ pc →pc
1, (7.25)

λ1 ∼λ3 , λ2 λ1

where TδI := λ1 ∼λ3 , λ2 λ1 Tδλ denotes the contribution by the operators Tδλ that arise in this case. The constant in this estimate does not depend on δ.

GOING BEYOND THE CASE hlin (φ) ≥ 5

145

If H = B = 3 and m = 2, then we get only a uniform estimate,
Tδλ pc →pc
1. λ2 λ1

Finally, assume that B = 2. Since we assume that θc ≤ 13 , we then again find that 1 5 1 1 5 − − + θc ≤ −1 + · < 0, B 2 2 2 3 so that (7.25) remains valid. Let us return to the case where H = B = 3 and m = 2; hence, θc = 13 , which will require more refined methods. In a first step, we shall take the sum of the νδλ over all dyadic λ2  λ1 . Moreover, since λ1 ∼ λ3 , we may reduce this to the case where λ3 = 2M λ1 , where M ∈ N is fixed and not too large. For the sake of simplicity of notation, we then assume that M = 0. All this then amounts to considering the functions σδλ1 given by       ξ1 ξ2 ξ3 λ1  σδ (ξ ) = χ1 χ0 χ1 νδ (ξ ), λ1 λ1 λ1 where now χ0 is smooth, compactly supported in an interval [−ε, ε], where ε > 0 is sufficiently small, and χ0 ≡ 1 in the interval [−ε/2, ε/2]. In particular, σδλ1 (x) is given again by expression (7.22), only with the second factor χˇ 1 (λ2 (x2 − δ0 y2 − y1m ω(δ1 y1 )) in the integrand replaced by χˇ 0 λ1 (x2 − δ0 y2 − y1m ω(δ1 y1 ) and λ2 replaced by λ1 . Thus we obtain the same type of estimates as in (7.24), that is, λ1 σ δ ∞
λ1

−5/6

σδλ1 ∞
λ1 min{λ1 , δ0−1 λ1 } = λ1 min{λ1 , δ0−1 }. (7.26) λ1 λ1  By Tδ we shall denote the operator of convolution with σδ . In view of (7.26), we shall distinguish between two subcases. 2/3

,

−3/2

1.1. The subcase where λ1 ≤ δ0

λ1 σ δ ∞
λ1

1/3

2/3

. In this case, by (7.26) we have

−5/6

σδλ1 ∞
λ1 , 5/3

,

(7.27)

so that −5/9 5/9 λ1

Tδλ1 pc →pc
λ1

= 1,

and summing these estimates does not lead to the desired uniform estimate. Let us denote by
TδI1 := Tδλ1 −3/2

λ1 ≤δ0

the contribution by the operators Tδλ that arise in this subcase. In order to prove the desired estimate TδI1 pc →pc
1,

(7.28)

we shall therefore have to apply an interpolation argument (see Subsection 7.6.1).

146

CHAPTER 7 −3/2

1.2. The subcase where λ1 > δ0 interpolation yields

. In this case we have σδλ1 ∞
δ0−1 λ1 , and −1/3 −2/9 λ1 .

Tδλ1 pc →pc
δ0

If we denote the contribution by the operators Tδλ that arise in this subcase

by TδI2 := λ1 >δ−3/2 Tδλ1 , we thus obtain 0
−1/3 −2/9 TδI2 pc →pc
δ0 λ1
1. (7.29) −3/2

λ1 >δ0

Case 2. λ2 ∼ λ3 and λ1  λ2 . Here, we can estimate νδλ in the same way as in the −1/2 −1/B −1/2−1/B ∼ λ2 . Moreover, by (7.24), previous case and obtain νδλ ∞
λ2 λ3 (B−1)/B −1 λ , δ0 }. Both these estimates are independent of we have νδ ∞
λ2 min{λ2 , we therefore consider λ1 . Assuming here without loss of generality that λ2 = λ3 the sum over all νδλ such that λ1  λ2 , by putting σδλ2 := λ1 λ2 νδ(λ1 ,λ2 ,λ2 ) . This means that       ξ1 ξ2 ξ3 λ2 (ξ ) = χ χ1 χ1 νδ (ξ ), σ 0 δ λ2 λ2 λ2 where now χ0 is smooth and compactly supported in an interval [−ε, ε], with ε > 0 sufficiently small. In particular, σδλ2 (x) is given again by expression (7.22), only with the first factor χˇ 1 (λ1 (x1 − y1 )) in the integrand replaced by χˇ 0 (λ2 (x1 − y1 )) and λ1 replaced by λ2 . Thus we obtain the same type of estimates λ2 σ δ ∞
λ2

−1/2−1/B

,

(B−1)/B

σδλ2 ∞
λ2 min{λ2

, δ0−1 }.

(7.30)

λ2 Denote by Tδλ2 the operator of convolution with σ δ . Interpolating between the first estimate in (7.30) and the the estimate σδλ2 ∞
(B−1)/B , we get λ2 λ2 −1/B−1/2+5θ/2

Tδλ2 p→p
λ2

.

Arguing as in the first case, we see that this still suffices to sum over all dyadic λ2 for θ = θc = 2/pc to obtain the desired estimate
λ Tδ 2 pc →pc
1, (7.31) TδI I pc →pc
λ2

unless H = B = 3 and m = 2. Here TδI I denotes the contribution by the operators Tδλ that arise in this case. So, assume that H = B = 3 and m = 2, so that θc = 13 . Then we distinguish two subcases. −3/2

2.1. The subcase where λ2 ≤ δ0

. In this case, (7.30) reads

λ2 σ δ ∞
λ2

−5/6

,

σδλ2 ∞
λ2 ,

which implies our previous estimate, Tδλ2 pc →pc
1.

5/3

(7.32)

GOING BEYOND THE CASE hlin (φ) ≥ 5

147

Let us denote by TδI I1 :=




Tδλ2

−3/2

λ2 ≤δ0

the contribution by the operators Tδλ which arise in this subcase. In order to prove the desired estimate TδI I1 pc →pc
1,

(7.33)

we shall thus have to apply an interpolation argument once more (see Subsection 7.6.1). −3/2 2.2. The subcase where λ2 > δ0 . Then (7.30) implies that σδλ2 ∞
λ2 δ0−1 , hence (3/2+1/3)θc −1/2−1/3

Tδλ2 pc →pc
δ0−θc λ2

−1/3 −2/9 λ2 .

= δ0

As in Subcase 1.2, this implies the desired estimate, TδI I2 pc →pc
1,

(7.34)

for the contributions TδI I2 of the operators Tδλ , with λ satisfying the assumptions of this subcase, to the operator to Tδ . Case 3. λ1 ∼ λ2 and λ3  λ1 . If λ3  λ2 δ0 , then the phase function has no critical point in x2 , and so integrations by parts in x2 and the stationary phase method in x1 yield νδλ ∞
λ1

−1/2

−1/2

(λ2 δ0 )−N
λ2

λ−N 3

for every N ∈ N, and the second estimate in (7.24) implies that νδλ ∞
λ3 δ0−1 . Interpolating these estimates, we obtain 

−(1−θ)/2 −θ δ0 ,

Tδλ p→p
λ−N 3 λ2

where N  can be chosen arbitrarily large if θ < 1. But, if θ = θc , then θ ≤ 13 , and since λ2 δ0 ≥ 1 if λ3  λ2 δ0 , we see that
(1−3θ)/2 Tδλ pc →pc
δ0
1. λ1 ∼λ2 , λ3 λ2 δ0

Let us therefore assume from now on that in addition λ3  λ2 δ0 . Then we can first apply the method of stationary phase to the integration in x1 and, subsequently, van der Corput’s estimate to the x2 -integration and obtain the estimate νδλ ∞
λ2

−1/2 −1/B λ3 .

In view of (7.24), we distinguish two subcases.

(7.35)

148

CHAPTER 7 1/B

3.1. The subcase where λ3 > λ2 δ0 . Then interpolation of the first estimate in (7.24) with (7.35) yields 3θ−1/2 θ−1/B λ3 .

Tδλ p→p
λ2

Since θc ≤ 13 , we have 3θc − 1 ≤ 0 (even with strict inequality, unless B = 2 or H = B = 3 and m = 2, because of Corollary 7.5(c)). If 3θc − 1 < 0, we can sum over all dyadic λ2  λ3 and obtain
−1/B−1/2+5θ/2 Tδλ pc →pc
λ3 . 1

{λ2 :λ3 λ2 0 6B

(7.37)

GOING BEYOND THE CASE hlin (φ) ≥ 5

149

for B ≥ 2, and since θ = θc ≤ θB = 1/(2B) + 16 by Corollary 7.5, we see that we can sum the estimates in (7.37) over all dyadic λ3  λ2 and get
[(3B+2)θ−B−2]/2B −θ Tδλ pc →pc
λ2 δ0 . λ3 λ2

Again using that θ ≤ θB , we find that for B ≥ 2,   1 1 + −B −2 (3B + 2)θ − B − 2 ≤ (3B + 2) 6 2B =

−3B 2 − B + 6 < 0, 6B −B/(B−1)

so we can also sum in λ2 ≥ δ0 and find that TδI I I2a pc →pc


−[5Bθ−B−2]/2(B−1) δ0 , where TδI I I2a := λ1 ∼λ2 ≥δ−B/(B−1) , λ3 λ2 Tδλ . But, 0   1 1 3−B 5Bθ − B − 2 ≤ 5BθB − B − 2 = 5B + −B −2= ≤0 2B 6 6 (7.38) if B ≥ 3, and thus for B ≥ 3 we get TδI I I2a pc →pc
1.

(7.39)

There remains the case B = 2. Here, for θ = θc , we have (B + 1)θ − 1 = 3θ − 1 ≤ 0, and −(1−θ)/2 (3θ−1)/2 −θ λ3 δ0 .

Tδλ pc →pc
λ2

Assume first that θc < 13 . Then we can first sum in λ3 ≥ λ2 δ0 (notice that λ2 δ0 > 1) and obtain
(θ−1)/2 Tδλ pc →pc
λ22θ−1 δ0 . λ3 ≥λ2 δ0 (1−3θ)/2

Then we sum over λ2 ≥ δ0−1 and get an estimate by Cδ0
1, so that (7.39) remains true also in this case. −1/3 −1/3 Assume finally that θc = 13 . Then Tδλ pc →pc
λ2 δ0 . Summing −1/3 −1/3 first in λ3  λ2 , we get an estimate by C(log λ2 ) λ2 δ0 , and summa−B/(B−1) −2 = δ0 leads to an estimate of order tion over all λ2 ≥ δ0 1/3 log(1/δ0 )δ0
1. Thus, (7.39) again holds true. −B/(B−1) 1/B . Then we use λ3 ≤ (λ2 δ0 )B , that is, λ2 ≥ λ3 δ0−1 , (b) λ2 < δ0 and summation of the estimate (7.37) over these λ2 yields
[(3+2B)θ−3]/2B (1−3θ)/2 Tδλ pc →pc
λ3 δ0 . {λ2 :λ2 ≥λ3 δ0−1 } 1/B

If the exponent of λ3 on the right-hand side of this estimate is strictly negative, then we see that TδI I I2b pc →pc
1, where TδI I I2b :=


−B/(B−1) λ1 ∼λ2 |D|/4c. Here we may estimate |χˇ1 (y1 )|
|D|−N (1 + |y1 |)−N for every N ∈ N, and combining this with (7.59) clearly yields (7.60). Denote next by µ2λ1 ,λ3 (x) the contribution by the region where |y1 | ≤ |D|/4c. Then |ri (y1 )| ≤ |D|/2. But notice also that 1/2

|Ey2 |  |E|λ3

≤ λ1 δ0  1,

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CHAPTER 7

which shows that |D − Ey2 + r1 (y1 )| ≥ |D|/4. Thus, the second factor in Fδ can be estimated by CN |D|−N , and we again arrive at (7.60). λ3 λ3 2. The part where |D| < 4c. Denote by σ1+it,2 (x) the contribution to σ1+it (x) by the terms for which |D(x, λ1 , δ)| < 4c. As in the discussion in the previous subsection (part 2.) we see that the oscilla−1/2 λ3 (x) can essentially be written in the form (2.15), with tory sum defining λ3 σ1+it,2 3 α := − 2 , l = k1 and −1/2

β l β l u1 = 2β l a1 := λ−1 1 λ3 , u2 = 2 a2 := λ1 QD (x), u3 = 2 a3 := λ1 (λ3 1

2

3

δ0 ),

and where the function H = Hλ3 ,x,δ of u := (u1 , u2 , u3 ) is now given by  −1/2 y2 ) χˇ 1 (u2 − u3 y2 + r1 (y1 ; λ−1 H (u) := χˇ 1 (y1 )η(x1 − u1 λ−1 3 y1 , λ3 3 u1 , x1 , δ)) −1/2

× χˇ 1(A − By2 − y22 b(x1 − λ−1 3 u1 y1 , λ3 −1/2

+(λ3

y2 , δ) + u1 r2 (y1 ; λ−1 3 u1 , x1 , δ)

y2 )δ3 r3 (y1 ; λ−1 3 u1 , x1 , δ)) dy1 dy2 .

Moreover, the cuboid Q in Lemma 2.7 is here defined by the conditions |u1 | ≤ 2−M ,

|u2 | < 4c,

|u3 | ≤ 2−3M/2 .

Let us estimate the C 1 -norm of H on Q. Because of (7.59), we clearly have H C(Q)
1. We next consider partial derivatives of H . From our integral formula for H (u), it is obvious that the partial derivative of H with respect to u1 will −1/2 −1 2 y2 , essentially produce only additional factors of the form λ−1 3 y1 , λ3 y1 y2 , λ3 −1 and λ3 y1 under the double integral. However, powers of y1 can be absorbed by 2 the rapidly decaying factor χˇ 1 (y1 ), and |λ−1 3 y2 | ≤ ε  1, so that |∂u1 H (u)|
1 too, and the same applies to |∂u2 H (u)|. The main problem is again caused by the partial derivative with respect to u3 , which produces an additional factor y2 . However, arguing as in the preceding subsection, we find that for  ∈]0, 1] and s ∈ [0, 1],  (1 + |y1 |)−N |∂u3 H (su)|
|u3 |−1 s −1 1/2

|y2 |≤λ3

  −1/2 ×χˇ 1 (A − By2 − y22 b(x1 − λ−1 y2 , δ) 3 su1 y1 , λ3 −1/2

+ su1 r2 (y1 ; λ−1 3 su1 , x1 , δ) + (λ3

  y2 )δ3 r3 (y1 ; λ−1 su , x , δ)  1 1 3

×|y2 | dy1 dy2 . /2

Estimating |y2 | in a trivial way by |y2 | ≤ λ3 , we see by means of (7.59) that /2

|∂u3 H (su)|
|u3 |−1 s −1 λ3 ,

for all u ∈ Q.

By means of Lemma 2.7 (and our choice of γ (ζ )), this implies that −1/2

|λ3

/2

λ3 σ1+it,2 (x)|
λ3 ,

which completes the proof of (7.57) and hence also of Proposition 7.7.

Q.E.D.

GOING BEYOND THE CASE hlin (φ) ≥ 5

163

7.6.4 Estimation of TδI I I2 : Complex interpolation The discussion of this operator will somewhat resemble the one of the operator VI Tδ,j in Section 5.3, which arose from the same Subcase 3.2(b) (in the latter case in Section 4.2), with δ0 playing here the role that 2−j played there, and where we did have B = 2 in place of B = 3 here. Assuming again without loss of generality that λ1 = λ2 , we see that here we have to prove the following. Proposition 7.8. Let m = 2 and B = 3, and consider the measure

νδ(λ1 ,λ1 ,λ3 ) , νδI I I2 := −3/2

2M δ0−1 ≤λ1 3|D|/2 is 1 (x) are again of the order |D|−N for every N ∈ N, and their contributions to ρt,λ 0 again admissible. What remains is the region where |y1 |  |D|ε and |D|/2 ≤ |y2 | ≤ 3|D|/2. In addition, we may assume that y2 and D have the same sign, since otherwise we can estimate as before. Let us therefore assume, for example, that D > 0 and that D/2 ≤ y2 ≤ 3D/2. The change of variables y2  → Dy2 then allows us to rewrite the corresponding contribution to fλ0 ,x (λ4 ) as   ˜ fλ0 ,x (λ4 ) := D χˇ 1 (y1 )F˜δ (λ0 , λ4 , x, y1 , y2 ) dy2 dy1 , |y1 ||D|ε

1/2≤y2 ≤3/2

where here

 F˜δ (λ0 , λ4 , x, y1 , y2 ) := η x1 − δ0 (λ0 λ4 )−1 y1 , (λ0 λ4 )−1 Dy2 , δ × χˇ 1 (D − Dy2 + r1 (y1 )) χ1 (y2 )  × χˇ 1 A − BDy2 − ED 3 y23 b (x1 − δ0 × (λ0 λ4 )−1 y1 , (λ0 λ4 )−1 Dy2 , δ

2 −1 + δ3 δ0 λ4 λ−2 0 D r3 (y1 ) y2 + δ0 λ4 λ0 r2 (y1 )

and where χ1 is supported where y2 ∼ 1. In the subsequent discussion, we may and shall assume that fλ0 ,x (λ4 ) is replaced by f˜λ0 ,x (λ4 ). Recall also from (7.43) that b(x1 , x2 , δ) = b3 (δ1 x1 , δ2 x2 ), and that δ2  δ0 . The last estimate implies that |δ2 (λ0 λ4 )−1 D| =

δ2 |QD (x)|  1, δ0

which shows that the second derivative of the argument of the last factor of our function F˜δ (λ0 , λ4 , x, y1 , y2 ) with respect to y2 is comparable to |ED 3 |. We may

GOING BEYOND THE CASE hlin (φ) ≥ 5

167

therefore apply van der Corput’s estimate to the integration in y2 (case (i) in Lemma 2.1(b)) and obtain 3/2 |f˜λ0 ,x (λ4 )|
|D||ED 3 |−1/2 = λ0 |D|−1/2 .

Interpolation with the trivial estimate |f˜λ0 ,x (λ4 )|
1 then leads to |f˜λ0 ,x (λ4 )|
1/2 λ0 |D|−1/6 . The second factor allows us to sum in λ4 , since we are assuming that 1/2 1 |D|  1, and we obtain |ρt,λ (x)| ≤ Cλ0 , in agreement with (7.66). 0 We may thus in the sequel assume that |D|
1. Here we go back to (7.67) and observe that χˇ 1 (y1 )χˇ 1 (D − y2 + r1 (y1 )) can be estimated by CN (1 + |y1 |)−N (1 + |y2 |)−N . This shows, in particular, that any power of y1 or y2 can be “absorbed” by these two factors. We shall still have to distinguish between the cases where |B| ≥ 1 and where |B| < 1. 2 (x) the contribution to 2. The part where |D|
1 and |B| ≥ 1. Denote by ρt,λ 0 ρt,λ0 (x) by the terms for which |D|
1 and |B| ≥ 1. If |y2 |  (|B|/|E|)1/2 , then we see that we can estimate the contribution to −3/2 fλ0 ,x (λ4 ) by a constant times (|B|/|E|)−1/2 = λ0 |B|−1/2 . Summing over all λ4 such that |B| ≥ 1 then leads to a uniform estimate for the contributions of these 2 (x). regions to ρt,λ 0 So, assume that |y2 |  (|B|/|E|)1/2 =: H . Applying the change of variables y2  → Hy2 , we then see that we may replace fλ0 ,x (λ4 ) by  ˜ fλ0 ,x (λ4 ) = H χˇ 1 (y1 )F˜δ (λ0 , λ4 , x, y1 , y2 ) dy1 dy2 ,

where  F˜δ (λ0 , λ4 , x, y1 , y2 ) := η x1 − δ0 (λ0 λ4 )−1 y1 , (λ0 λ4 )−1 Hy2 , δ × χˇ 1 (D − Hy2 + r1 (y1 )) χ0 (y2 )   × χˇ 1 A − H B y2 + y23 sgn (B)  × b x1 − δ0 (λ0 λ4 )−1 y1 , (λ0 λ4 )−1 Hy2 , δ 2 −1 + δ3 δ0 λ4 λ−2 0 H y2 r3 (y1 ) + δ0 λ4 λ0 r2 (y1 ) . We claim that 3/4 |f˜λ0 ,x (λ4 )| ≤ CH |H B|−1/2 = λ0 |B|−1/4 .

(7.70)

Since we are here assuming that |B| ≥ 1, this estimate would imply the estimate 3/4 2 |ρt,λ (x)| ≤ Cλ0 , again in agreement with (7.66). 0 In order to prove (7.70), observe first that the contribution to f˜λ0 ,x (λ4 ) by the region where |y1 | > |H B| clearly can be estimated by the right-hand side of (7.70) because of the rapidly decaying factor χˇ 1 (y1 ) in the integrand. And, on the

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remaining region where |y1 | ≤ |H B|, we have −2 |δ3 δ0 λ4 λ−2 0 H r3 (y1 )| ≤ δ3 δ0 λ4 λ0 |H | |H B| = δ3

δ0 λ24 |QB (x)|1/2 | |H B| λ0


λ−3 0 δ3 |H B|  |H B|. 3/2

Observe also that (λ0 λ4 )−1 H = |QB (x)|1/2
δ3  1. This shows that if γ denotes the argument of the last factor of F˜δ (λ0 , λ4 , x, y1 , y2 ), then there are constants 0 < C1 < C2 , such that  ∂    ∂ 2      C1 |H B| ≤  γ (y2 ) +  γ (y2 ) ≤ C2 |H B|, |y2 |
1, ∂y2 ∂y2 1/2

uniformly in x, y1 , and δ. We may thus apply a van der Corput–type estimate (see case (ii) in Lemma 2.1(b)) to the integration in y2 and again arrive at an estimate by the right-hand side of (7.70) and also for the contribution by the region where |y1 | ≤ |H B|. 3 3. The part where |D|
1, |B| < 1, and |A|  1. Denote by ρt,λ (x) the contri0 bution to ρt,λ0 (x) by the terms for which these conditions are satisfied. We claim that here we get

|fλ0 ,x (λ4 )| ≤ C|A|−N ,

(7.71) 3/4

3 for every N ∈ N. This estimate will imply the estimate |ρt,λ (x)| ≤ Cλ0 , again in 0 agreement with (7.66). In order to prove (7.71), observe that the contributions to fλ0 ,x (λ4 ) by the regions where |y1 |  |A|1/3 or |y2 |  |A|1/3 can be estimated by a constant times (|A|1/3 )−N , because of the rapid decay in y1 and y2 of Fδ . So, assume that |y1 | + |y2 |  |A|1/3 . Then we see that

|By2 |  |B||A|1/3  |A|

and

|Ey23 |  |EA|  |A|,

as well as 2 −1 2/3 |δ3 δ0 λ4 λ−2  |A|, 0 y2 r3 (y1 ) + δ0 λ4 λ0 r2 (y1 )|  |A|

and thus the last factor of Fδ is of order |A|−N . Consequently, the contribution to fλ0 ,x (λ4 ) by the region where |y1 | + |y2 |  |A|1/3 can also be estimated as in (7.71). 4 (x) the contribution 4. The part where max{|A|, |B|, |D|}
1. Denote by ρt,λ 0 to ρt,λ0 (x) by the terms for which max{|A|, |B|, |D|}
1. Then we can easily 4 (x) by means of Lemma 2.7 in a very similar way as we did in the last estimate ρt,λ 0 4 (x)| ≤ C. part of Section 5.3 and obtain that |ρt,λ 0 This completes the proof of estimate (7.66) (with  := 34 ) and, hence, also the proof of Proposition 7.8. Q.E.D.

GOING BEYOND THE CASE hlin (φ) ≥ 5

169

7.7 THE CASE WHERE λ1 ∼ λ2 ∼ λ3 We shall assume for the sake of simplicity that λ1 = λ2 = λ3  1. The more general case where λ1 ∼ λ2 ∼ λ3  1 can be treated in a very similar way. By changing notation slightly, we shall denote in this section by λ the common value of the λj . We change coordinates from ξ = (ξ1 , ξ2 , ξ3 ) to s1 , s2 and s3 := ξ3 /λ, that is, ξ1 = s1 ξ3 = λs1 s3 ,

ξ2 = λs2 ξ3 = λs2 s3 ,

ξ3 = λs3 ,

and write in the sequel s := (s1 , s2 , s3 ) s  := (s1 , s2 ). Then we may rewrite ˜ (x, δ, ξ ) = λs3 (x, s  ), where ˜ (x, s  ) := s1 x1 + s2 x1m ω(δ1 x1 ) + x1n α(δ1 x1 )  +s2 δ0 x2 + x2B b(x1 , x2 , δ) + r(x1 , x2 , δ) ,

(7.72)

where ω(0)  = 0, α(0)  = 0, and b(x1 , 0, δ)  = 0, if x1 ∼ 1, and where δ and r(x1 , x2 , δ) are given by (7.11) and (7.10), respectively. ˜ has at worst an Airy-type singularity with respect to x1 , Now, the first part of  and the derivative of order B with respect to x2 does not vanish, so that we obtain νδλ ∞
λ−1/3−1/B

(7.73)

(indeed, by localizing near a given point x and looking at the corresponding ˜ at this point, this follows more precisely from the main Newton polyhedron of  result in [IM11b]). On the other hand, standard van der Corput–type arguments (compare Lemma 2.1) show that here 0

νδλ ∞
min{λ3 λ−1 λ−1/B , λ3 λ−1 (λδ0 )−1 } = λ min{λ(B−1)/B , δ0−1 }.

(7.74)

We therefore distinguish the following cases: −B/(B−1)

Case A. λ ≤ δ0

. Then νδλ ∞
λ2−1/B , and by interpolation we get Tδλ pc →pc
λ−1/3−1/B+7θc /3 ,

again with θc =

−B/(B−1)

Case B: λ > δ0

(7.75)

2/pc . . Then νδλ ∞
λδ0−1 , and by interpolation we get

Tδλ pc →pc
λ−1/3−1/B+(4/3+1/B)θc δ0−θc .

(7.76)

Observe that in both cases, the exponents of λ in these estimates become strictly smaller and the one of δ increases if we replace θ by a strictly smaller number.

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7.7.1 The case where hr + 1 > B We observe that then pc > 2B, and thus θc
0 if B ≥ 3, we see that (7.76) implies in Case B that
(B+3−7Bθc )/3(B−1) (B−4)/3(B−1) Tδλ pc →pc
δ0 < δ0 ; −B/(B−1)

λ>δ0

hence, for B ≥ 4,




Tδλ pc →pc
1,

(B ≥ 4).

(7.78)

−B/(B−1) λ>δ0

The case where B = 3 requires more refined estimates, whereas we shall see that the case B = 2 is rather easy to handle because of the estimate θc ≤ 13 in (7.20). Assume first that B = 2. Then, by (7.75), if λ ≤ δ0−2 , Tδλ pc →pc
λ−1/3−1/2+7θc /3 = λ(14θc −5)/6 , and since θc ≤ 13 , the exponent of λ in this estimate is strictly negative, so that we can sum over λ and again obtain (7.77). Similarly, by (7.76), if λ > δ0−2 , Tδλ pc →pc
λ−1/3−1/2+(4/3+1/2)θc δ0−θc = λ(11θc −5)/6 δ0−θc . But, 11θc − 5 ≤

11 3

− 5 < 0, and so we get
(5−14θc )/3 Tδλ pc →pc
δ0 ≤ 1, λ>δ0−2

so that (7.78) also holds true in this case. −3/2

Assume next that B = 3. Then in Case A, where λ ≤ δ0 Tδλ pc →pc


λ

(7θc −2)/3

,

, we have by (7.75)

GOING BEYOND THE CASE hlin (φ) ≥ 5

171 −3/2

and thus, if 7θc − 2 < 0, then we can sum these estimates in λ ≤ δ0 and obtain (7.78). Let us therefore assume henceforth that θc ≥ 27 . Observe that by Lemma 7.4 we have θc < θ˜c , unless h˜ r = d and hr + 1 ≥ H , in which case we have θc = θ˜c and p˜ c = pc . Thus ˜

Tδλ pc →pc
λ(7θc −2)/3 , with θ˜c > 27 , unless θc = θ˜c = 27 , h˜ r = d and hr + 1 ≥ H . Note that in the latter case, H = B = 3, and since θ˜c = 27 , we find that m = 5 and d = 52 . In this particular case, we get only a uniform estimate Tδλ pc →pc
1. However, in this case have νδλ ∞
λ5/3 , since we are in Case A, and νδλ ∞
λ−2/3 , whereas θc = 27 , and thus −(1 − θc )2/3 + θc 5/3 = 0. Moreover, νδλ = νδ ∗ φλ ,    where the Fourier transform of φλ is given by φλ (ξ ) = χ1 ξλ1 χ1 ξλ2 χ1 ξλ3 . This implies a uniform estimate of the L1 -norms of the φλ for all dyadic λ. We may thus estimate the operator T I V1 of convolution with the Fourier transform of the complex measure
νδλ νδI V1 := −3/2

λ≤δ0

by means of the real interpolation Proposition 2.6, in the same way as we estimated the operators T I1 and T I I1 in Section 7.6.1, by adding the measure νδI V1 to the family of measures µi , i ∈ I from the second class in (7.41). So, assume that θ˜c > 27 . Then we find that


(2−7θ˜c )/2

Tδλ pc →pc
δ0

.

−3/2 λ≤δ0

−3/2

Let us next turn to Case B, where λ > δ0

. Then, by (7.76),

Tδλ pc →pc
λ(5θc −2)/3 δ0−θc . Since θc ≤ 13 , we can sum in λ and obtain


(2−7θc )/2

Tδλ pc →pc
δ0

(2−7θ˜c )/2

≤ δ0

.

−3/2

λ>δ0

Combining these estimates, we obtain
(2−7θ˜c )/2 Tδλ pc →pc
δ0 . λ1

Observe next that 2h˜ r − 5 2 − 7θ˜c 7 , =1−  = 2 p˜ c p˜ c

(7.79)

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and recall that δ0 = 2−(κ˜2 −mκ˜1 )k . In combination with the rescaling estimate (6.16), the estimate in (7.79) thus leads to 1/2  ˜ κ˜ 1 (1+m)+1]/pc +(κ˜ 2 −mκ˜ 1 )(2h˜ r −5)/2p˜ c ) |fˆ|2 dµ1,k
2−k(|κ|/2−[ f Lpc ˜ κ˜ 1 (1+m)+1]/p˜ c +(κ˜ 2 −mκ˜ 1 )[2h −5]/2p˜ c ) f Lpc . ≤ C2−k(|[κ|/2− 

˜r



(7.80) where µ1,k denotes the measure corresponding to the frequency domains that we are here considering; that is, µ1,k corresponds to the rescaled measure
(λ,λ,λ) ν1,δ := νδ . λ1

But

 E

:= 2p˜ c

2h˜ r − 5 κ˜ 1 (1 + m) + 1 |κ| ˜ + ( κ ˜ − m κ ˜ ) − 2 1 2 p˜ c 2p˜ c



= |κ|(2 ˜ h˜ r + 2) − 2(κ˜ 1 (1 + m) + 1) + (κ˜ 2 − mκ˜ 1 )(2h˜ r − 5) = κ˜ 2 (4h˜ r − 3) + κ˜ 1 (3m − 2h˜ r (m − 1)) − 2, where 4m − 3κ˜ 2 , m+1   κ˜ 1 3 m−1 −2 . κ˜ 1 (3m − 2h˜ r (m − 1)) = m κ˜ 2 H m+1 κ˜ 2 (4h˜ r − 3) =

Since H ≥ B = 3, we see that 3/H − 2(m − 1)/(m + 1) ≤ (3 − m)/(m + 1) ≤ 0 if m ≥ 3. Thus, if m ≥ 3, then since κ˜ 1 /κ˜ 2 = 1/a < 1/m, we see that m−1 κ˜ 1 (3m − 2h˜ r (m − 1)) ≥ 3κ˜ 2 − 2 , m+1 and altogether we find that E ≥ 0 (even with strict inequality, if H > 3). We thus have proved 1/2  |fˆ|2 dµ1,k ≤ Cf Lpc , (7.81) with a constant C not depending on k, provided that m ≥ 3. Assume finally that B = 3 and m = 2. Recall also that we are still assuming that hr + 1 > B = 3 and θc ≥ 27 , so that 3 < hr + 1 ≤ 72 . We shall prove that the Newton polyhedron of φ˜ a (defined in Section 7.1), respectively, of φ, will have a particular structure. Indeed, if φ is analytic, then one can show that φ is of type Z, E, J, Q, and so on, in the sense of Arnol’d’s classification of singularities (compare [AGV88]). We shall, however, content ourselves with a little less precise information, which will nevertheless be sufficient for our purposes.

GOING BEYOND THE CASE hlin (φ) ≥ 5

173

Recall from Chapter 1 the notion of augmented Newton polyhedron N r (φ˜ a ) of a ˜ φ . If L denotes the principal line of N (φ), then it is a supporting line to N (φ˜ a ) too, and if (A+ , B + ) is the right endpoint of the line segment L ∩ N (φ˜ a ), then let us denote by L+ the half line L+ ⊂ L contained in the principal line of N (φ) whose right endpoint is given by (A+ , B + ). Then N r (φ˜ a ) is the convex hull of the union of N (φ˜ a ) with the half line L+ . Recall also that N r (φ˜ a ) and N r (φ a ) do agree within the closed half space bounded from below by the bisectrix , so that hr + 1 is the second coordinate of the point at which the line (m) intersects the boundary of N r (φ˜ a ). Proposition 7.9. If B = 3, m = 2, and 3 < hr + 1 ≤ 3.5, then (A+ , B + ) = (1, 3), and N r (φ˜ a ) has exactly two edges, L+ and the line segment [(1, 3), (0, n)], which is contained in the principal line La of N (φ˜ a ). In particular,     7 1 n−1 1 2 , , hr = d = , and κ˜ = , , (7.82) κ= 7 7 3 n 3n where n > 7.  Proof. Denote by (A , B  ) := (A(0) , B(0) ) ∈ La the left endpoint of the principal face π(φ˜a ) of the Newton polyhedron of φ˜a . Then B  ≥ B = 3. In a first step, we prove that B  = 3. Assume, to the contrary, that we had B  ≥ 4 (observe that B  is an integer). Since the line La has slope strictly less 1/m = 12 , then it easily seen that the line La would intersect the line (2) at some point with second coordinate z 2 strictly bigger than 3.5, so that hr + 1 ≥ z 2 > 3.5, which would contradict our assumption (Figure 7.2). Thus, B  = B = 3. In a second step, we show that A = 1. To this end, let us here work with φ a in place of φ˜a . Note the point (A , B  ) is also the left endpoint of the principal face of N (φ a ), and that the principal faces of the Newton polyhedra of φ a and φ˜a both lie on the same line La , since the last step in the change to modified adapted coordinates (7.5) preserves the homogeneity κ. ˜ This shows that also A ∈ N. Moreover, A ≥ 1, for otherwise we had A = 0 and thus hr + 1 = 3. Assume that A ≥ 2. We have to distinguish two cases. (a) If the line L, which has slope 12 , contains the point (A , B  ), then the assumption A ≥ 2 would imply that hr + 1 > 3.5 (see Figure 7.3). (b) If not, then π(φ a ) will have an edge γ = [(A , B  ), (A , B  )] with right endpoint (A , B  ), and L must touch N (φ a ) in a point contained in an edge strictly to the left of γ . But then the line L containing γ must have slope strictly less than the slope 12 of L, and necessarily B  > B  = 3; hence B  ≥ 4. It is then again easily seen that the line L would intersect the line (2) at some point with second coordinate z 2 strictly bigger than 3.5, so that again hr + 1 ≥ z 2 > 3.5, which would contradict our assumption (Figure 7.3). We have thus found that (A , B  ) = (1, 3). Assume finally that N (φ) had a vertex (A , B  ) to the left of (1, 3). Then, necessarily, A = 0 and B  ≥ 4, so that the line passing through the points (A , B  ) and (1, 3) would have slope at least 1, a contradiction. We have seen that N (φ) is contained in the half plane where

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CHAPTER 7

L

∆(2) N r (φ˜a )

hr + 1

(A
, B
)

4 3, 5 B=3 La 1

n Figure 7.2

L



L (A

, B

)

∆(2)

4 3, 5

γ 3

L L

1

A
≥ 2 Figure 7.3

n

GOING BEYOND THE CASE hlin (φ) ≥ 5

175

t1 ≥ 1, and thus the line L must pass through the point (1, 3), and the claim on the structure of N r (φ˜a ) is now obvious. But then clearly (2) will intersect the boundary of N r (φ˜a ) in a point of L+ , so that hr = d. The remaining statements in (7.82) are now easily verified. Q.E.D. With this structural result, we can now conclude the discussion of this case. In3 > 27 , and, arguing as before, only with θc deed, by Proposition 7.9 we have θc = 10 in place of θ˜c , we obtain
(2−7θc )/2 Tδλ pc →pc
δ0 . λ1

Following the rescaling arguments from the previous discussion, we see that this implies that here the right-hand side of the estimate (7.80) can be written as 2−kE f Lpc , with exponent E satisfying   2hr − 5 κ˜ 1 (1 + 2) + 1 |κ| ˜ 2pc E := 2pc + ( κ ˜ − 2 κ ˜ ) − 2 1 2 pc 2pc = κ˜ 2 (4hr − 3) + κ˜ 1 (6 − 2hr ) − 2.

By means of (7.82) one then computes that 19 n − 1 4 1 n−7 · + · −2= > 0, 9 n 3 n 9n so that the uniform estimate (7.81) remains valid also in this case. 2pc E =

7.7.2 The case where hr + 1 ≤ B In this case, since d < h ≤ hr + 1, we have d < B, and since we are assuming that d > 2, we see that we may assume that B ≥ 3. Moreover, it is obvious from the structure of the Newton polyhedron of φ a that necessarily m + 1 ≤ B, and hr + 1 = hlpr + 1 =

1 + (m + 1)κ˜ 1 . |κ| ˜

(7.83)

Indeed, this follows from the geometric interpretation of the notion of r-height given directly after Remarks 1.9, since the line (m) intersects the principal face π(φ a ) (cf. Figure 1.4). Therefore, in passing from the measure νδ to µk , no further gain is possible in this situation in (7.14). First consider Case A. Corollary 7.5(b) implies that θc ≤ θ˜B . Thus, by (7.75) we have Tδλ pc →pc
λ−1/3−1/B+7(m+1)/3(mB+m+1) = λ−M(m,B)/3B(mB+m+1) , where M(m, B) := mB 2 − (3m + 6)B + 3(m + 1)

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is increasing in B if B ≥ 3 and m ≥ 2. Since B ≥ m + 1, we thus have M(m, B) ≥ m3 − m2 − 5m − 3, and the right-hand side of this inequality is increasing in m if m ≥ 2 and assumes the value 0 if m = 3. Therefore, M(m, B) ≥ 0 if m ≥ 3 and even with strict inequality if B > m + 1. We thus see that our estimates for Tδλ pc →pc do sum for m ≥ 3 over all dyadic −B/(B−1) λ ≤ δ0 when B > m + 1, and they are at least uniform, if B = m + 1. Finally, when m = 2, then M(2, B) = 2B 2 − 12B + 9 = 2[(B − 3)2 − 9/2] > 0, iff B ≥ 6. We thus find that
Tδλ pc →pc
1, −B/(B−1)

λ≤δ0

except possibly when m = 2, B = 3, 4, 5, or m = 3, B = 4.

(7.84)

Moreover, we have Tδλ pc →pc
1

if m = 3, B = 4.

But, recall that here θc < θ˜c ≤ θ˜B , unless 4 = B = H = hr + 1 = d + 1, so that θc = 14 . Therefore, in those cases where m = 3 and B = 4 and in which we can obtain only a uniform estimate for the norms Tδλ pc →pc , we will have θc = 14 . But, by (7.73) and (7.74) we have νδλ ∞
λ7/4 , since we are in Case A, and νδλ ∞
λ−7/12 , where −(1 − θc )7/12 + θc 7/4 = 0. We may thus again use real interpolation in order to estimate

the operator of convolution with the Fourier transform of the complex measure λ≤δ−4/3 νδλ , namely, by means of Proposition 2.6, in 0 the same way as we did just before in the corresponding case where m = 5 and B = 3. We are thus left with the cases where m = 2, hr + 1 ≤ B, and B = 3, 4, 5. So assume in the sequel that that m = 2 and hr + 1 ≤ B. If m = 2 and B = 3, then B = m + 1, and since we are assuming hr + 1 ≤ B, a look at the Newton polyhedron of φ˜a shows that necessarily H = B = 3 = hr + 1. Lemma 7.10. Assume that m = 2 and B = 4, 5. Then we have 1 1 7 − − + θ˜c < 0, 3 B 3 provided 9 , if B = 4, 2 H > H (B) := 81 , if B = 5. 16 Proof. For m = 2 we have 1 1 7 1 7 1

− , 2 B +3 2

(7.85)

(7.86)

GOING BEYOND THE CASE hlin (φ) ≥ 5

177

and it is easily checked that for B = 4, 5, this holds true if and only if H > H (B). Q.E.D. Since by Corollary 7.5 (b) θc ≤ θ˜c , the previous lemma shows that for m = 2, (7.84) can be sharpened as follows:
Tδλ pc →pc
1, (7.87) −B/(B−1)

λ≤δ0

with the possible exceptions of the cases where B = H = 3 = hr + 1, or B = 4, 5, hr + 1 ≤ B and H ≤ H (B). Finally, consider Case B. Then, since θc ≤ θ˜B , ˜

Tδλ pc →pc
λ−1/3−1/B+(4/3+1/B)(m+1)/(mB+m+1) δ0−θB ˜

= λ−B(mB−3)/3B(mB+m+1) δ0−θB . The exponent of λ is negative, so we can sum these estimates in λ and obtain
[1/3+1/B−(4/3+1/B)θ˜B ]B/(B−1)−θ˜B Tδλ pc →pc
δ0 −B/(B−1)

λ>δ0

(B+3−7B θ˜B )/3(B−1)

= δ0

M(m,B)/3(B−1)(mB+m+1)

= δ0

.

But, our previous discussion of M(m, B) shows that M(m, B) ≥ 0, unless m = 2 and B = 3, 4, 5 (notice that the case m = 3, B = 4 still works here ). In the latter cases, we can improve our estimates again by using θ˜c in place of θ˜B . Indeed, notice that the condition B + 3 − 7B θ˜c > 0 is equivalent to (7.86), which by Lemma 7.10 does hold true if H ≤ H (B). We thus find that
Tδλ pc →pc
1, (7.88) −B/(B−1)

λ>δ0

unless m = 2 and B = H = 3 = hr + 1, or B = 4, 5 hr + 1 ≤ B and H ≤ H (B). Finally, we observe that the following consequence of Lemma 7.10. Corollary 7.11. Assume that m = 2 and that B = H = 3, or B = 4, 5, hr + 1 ≤ B, and H ≤ H (B). Then the left endpoint of the principal face of the Newton polyhedron of φ˜ a is of the form (A, B), where A ∈ {0, 1, . . . , B − 3}, and we have   n+3 1 n−A , and hr + 1 = B, κ˜ = n Bn n+B −A where n must satisfy n > 2B and (Bn)/(n − A) = H ≤ H (B). Moreover, φ˜ a is smooth here, so that in particular a and n are integers. Proof. Adopting the notation of the proof of Proposition 7.9, let (A, B  ) denote the left endpoint of π(φ˜ a ), which, as we recall, is also the left endpoint of π(φ a ). If B = 4, 5, then B ≤ B  ≤ H ≤ H (B) < B + 1, so that B  = B. For B = 3 = H ,

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the same conclusion applies. Next, since we assume that hr + 1 ≤ B, and since the line (2) intersects the line where t2 = B in the point (B − 3, B), we must have 0 ≤ A ≤ B − 3. This implies the first statement of the corollary, because A is an integer. Moreover, Proposition 7.3 now shows that φ˜ a is smooth and that a and n are integers. The remaining statements follow easily (for the identity for hr + 1, recall (7.83)). Q.E.D.

7.8 THE CASE WHERE B = 5 The case where B = 5 can now be treated quite easily. We begin by recalling that by (7.75) we have the estimate Tδλ pc →pc
λ−1/3−1/5+7θc /3 . This estimate is valid in Case A as well as in Case B (in the latter case, an even stronger estimate holds true, but we won’t need it). And, according to Corollary 7.11, if B = 5, then we have the precise formula, θc =

n+5−A , 5(n + 3)

for θc , where A ∈ {0, 1, 2} and n > 2B = 10. Since here n is an integer, we find that n ≥ 11, and thus θc ≤ (11 + 5)/70 = 8/35, with strict inequality, unless A = 0 and n = 11. This shows that the exponent of λ in the estimate for Tδλ pc →pc is strictly negative, so that we can sum these estimates over all dyadic λ ≥ 1, unless A = 0 and n = 11, where we only get a uniform estimate Tδλ pc →pc
1. 8 However, if A = 0, B = 5, and n = 11, then θc = 35 , and moreover, by (7.73)  λ λ 9/5 −8/15 , where −(1 − θc )8/15 + and (7.74), we have νδ ∞
λ and νδ ∞
λ the operator of convolution with the Fourier θc 9/5 = 0. We may thus again estimate

transform of the complex measure λ1 νδλ by means of the real interpolation method of Proposition 2.6 by adding this measure to the list of measures µi , i ∈ I , of the second class in (7.41).

7.9 COLLECTING THE REMAINING CASES Recall that the integer B ≥ 2 is chosen so that (7.6) holds true, that is, that φ˜ κa˜ (y1 , y2 ) = y2B Q(y1 , y2 ) + c1 y1n ,

c1  = 0, Q(1, 0)  = 0,

and that according to (7.21) H = 1/κ˜ 2 is the second coordinate of the point of intersection of the principal line La with the t2 -axis. The quantity H (B) has been defined in (7.85).

GOING BEYOND THE CASE hlin (φ) ≥ 5

179

The estimates that remain to be established when m = 2 and B = 3 have essentially already been addressed in the previous sections. Their treatment will again require Airy-type analysis of a similar form as in Chapter 5, which had been devoted to the case where m = 2 and B = 2, but the discussion in the next chapter will be even more involved. Also in the case m = 2 and B = 4, we shall need some Airy-type analysis, but of a simpler form. Combining all the previous estimates and applying (7.14), we see that we have proved the following. Corollary 7.12. Assume that m and H, B are not such that m = 2 and B = H = 3 = hr + 1 or m = 2, B = 4, hr + 1 ≤ 4, and H ≤ H (4). Then the estimates in Proposition 7.2 hold true for l = 1. More precisely, in view of Corollary 7.11, what remains open are the following situations: m = 2, B = 3, 4, hr + 1 ≤ B,

and λ1 ∼ λ2 ∼ λ3 ,

(7.89)

where the left endpoint of the principal face of the Newton polyhedron of φ˜ a is of the form (A, B), where A ∈ {0, . . . , B − 3}. In these situations, we have   n+3 1 n−A , and hr + 1 = B, (7.90) κ˜ = n Bn n+B −A where n must be an integer satisfying n > 2B, and (Bn)/(n − A) = H ≤ H (B) if B = 4. Moreover, according to Corollary 7.11, in all these cases the exponent a = κ˜ 2 /κ˜ 1 is an integer, and the modified adapted coordinates given by (7.5) are in fact smooth, so that we could use them as well as adapted coordinates for φ. In these adapted coordinates, φ is represented by the smooth function φ˜a . The discussion of these cases will occupy the major part of remainder of this monograph. Before we come to this, let us first study also the contributions by the  , l ≥ 2, which will turn out to be easier to handle. remaining domains D(l) 7.10 RESTRICTION ESTIMATES FOR THE DOMAINS D  (l) , l ≥ 2  For the domans D(l) , l ≥ 2, we can essentially argue as in the preceding sections by letting

φ˜ a := φ (l+1) ,

κ˜ := κ (l) ,

a D a := D(l) ,

La := L(l) ,

H and n are defined correspondingly. We then have the following analogue of Lemma 7.4. Lemma 7.13. (a) If l ≥ 2, then pc > p˜ c . (b) If m ≥ 3 and H ≥ 2 or m = 2 and H ≥ 3, then  ≥ pB . p˜ c ≥ pH

and so on.

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Proof. (a) We can follow the proof of Lemma 7.4(a) and see again that pc > p˜ c , unless the principal face π(φ˜ a ) of φ˜ a is the edge [(0, H ), (n, 0)]. However, for  , which lies below the l ≥ 2, we know from Section 7.1 that π(φ˜ a ) is the edge γ(l)   bisectrix , so that we cannot have pc = p˜ c . (b) follows as before. Q.E.D. This implies the following, stronger analogue of Corollary 7.5. Corollary 7.14. Assume that l ≥ 2. (a) If m ≥ 3 and H ≥ 2 or m = 2 and H ≥ 3, then θc < θB . (b) If hr + 1 ≤ B, then θc < θ˜c , unless B = H = hr + 1 = d + 1, where θc = θ˜c . (c) If H ≥ 3, then θc < 13 unless H = 3 and m = 2. Finally, in place of Proposition 7.9, we have the following. Proposition 7.15. Assume that l ≥ 2 and that B = 3, m ≥ 2. Then hr + 1 > 3.5.  )∈ Proof. Assume we had hr + 1 ≤ 3.5, and denote by (A , B  ) := (A(l−1) , B(l−1)  a a ˜ L the left endpoint of the principal face π(φ ) = γ(l) of the Newton polyhedron of φ˜a . Then B  ≥ B = 3. Arguing in the same way as in the first step of the proof of Proposition 7.5, we see that B  = B = 3.  will have a left endpoint (A , B  ) with B  ≥ 4. Then the preceding edge γ(l−1) Moreover, since the line L(1) has slope 1/a < 1/m ≤ 12 , the line L(l−1) containing  γ(l−1) has slope strictly less than 12 . But then it will intersect the line (2) at some point with second coordinate z 2 strictly bigger than 3.5, so that we would again Q.E.D. arrive at hr + 1 ≥ z 2 > 3.5, which contradicts our assumption.

These results allow us to proceed exactly as in Sections 7.5 and 7.7, even with some simplifications. Indeed, a careful inspection of our arguments in these sections reveals that here all the series that appear do sum, and no further interpolation arguments are required. This is because of the stronger estimate pc > p˜ c of Lemma 7.13 and the stronger statement of Proposition 7.15, which implies in particular that θc < 13 when B = 3. Moreover, in Subsection 7.7.1, the more delicate case where B = 3 and θc ≥ 27 does not appear anymore, since by Proposition 7.14 we have θc < 27 . Observe finally that if m = 2, B = 3, 4, 5, hr + 1 ≤ B, and H ≤ H (B), then Corollary 7.11, whose proof applies equally well when l ≥ 2, shows that left endpoint of the principal face of the Newton polyhedron of φ˜ a is of the form (A, B), where A ≤ B − 3. However, if l ≥ 2, this endpoint must lie on or below the bisectrix, which leads to a contradiction. These cases therefore cannot arise when l ≥ 2. We therefore obtain the following. Corollary 7.16. The estimates in Proposition 7.2 hold true for every l ≥ 2.

Chapter Eight The Remaining Cases Where m = 2 and B = 3 or B=4 We finally turn to the discussion of the remaining cases for l = 1, which have not been covered yet by Corollary 7.12 and which are described by (7.89) and (7.90). In these cases, the exponent B in our expression (7.9) for the phase φδ is either 3 or 4. Our basic approach will be similar to the one that we have developed for the case h lin (φ) < 2 in Chapter 5 (in which the exponent B would be given by B = 2). We shall again have to perform an additional dyadic frequency domain decomposition related to the distance to a certain Airy cone. This is needed in order to be able to control the integration with respect to the variable x1 in the Fourier integral defining the Fourier transform of the complex measures νδλ . Recall that these are already localized to frequency domains on which |ξi | ∼ λ = 2j for i = 1, 2, 3. We shall see by means of Proposition 7.3 that ν
δλ (ξ ) is an oscillatory integral with a phase function −(ξ3 φδ (x) + ξ1 x1 + ξ2 x2 ), where the function φδ , which depends on a finite number of small parameters δi , can be assumed to be of the form φδ (x) = x2B b(x1 , x2 , δ) + x1n α(δ1 x1 ) +

B−2 

j n

x2 x1 j δj +2 αj (δ1 x1 ).

j =1

If B = 2, then the preceding formula shows that the integration with respect to the variable x2 can easily be dealt with by means of the method of stationary phase; this is exactly what we did in Section 5.1. However, if B = 3 or B = 4, this is no longer possible, and a new complication arises, which will necessitate a considerable amount of additional and refined analysis in order to carry out the basic approach of Chapter 5. Indeed, after applying a suitable translation in the x1 -coordinate in a similar way as in Section 5.1, basically by some critical point x1c (δ1 , s2 ) of ∂x1 φδ , in combination with a change of variables, it turns out that ν
δλ (ξ ) can be rewritten as an oscillatory integral with phase λ (x, δ, s), where  (x, δ, s) = x13 B3 (s2 , δ1 , x1 ) − x1 B1 (s, δ1 ) + B0 (s, δ1 ) + φ (x, δ, s2 ), with φ (x, δ, s2 ) = x2B b(x, δ, s2 ) + 

B−2 



j

δj +2 x2 α˜ j (x1 , δ, s2 )

j =2

+x2 s2 δ0 + δ3 x1c (δ1 , s2 ) + x1

n1



α˜ 1 (x1 , δ1 , s2 )

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CHAPTER 8

and with ξ = λ(s1 s3 , s2 s3 , s3 ). Here the si , i = 1, 2, 3, are integration variables of size |si | ∼ 1. A major problem will then be caused by the term δ3 (x1c (δ1 , s2 ) + x1 )n1 × α˜ 1 (x1 , δ1 , s2 ), which does not yet appear when B = 2. Indeed, without this term, the coefficient of the linear term in x2 would be given by s2 δ0 and hence would be of a well-defined size δ0 . However, in the presence of the additional term depending on the small parameter δ3 , the situation becomes less clear. We therefore have to perform a more refined analysis of the phase  in Section 8.2, which leads us to distinguish between a nondegenerate Case ND and a degenerate Case D in Lemma 8.3. In Case ND, we do have good control of the size of the coefficient of x2 , and the same is true of Case D, in which, however, an additional term, roughly of the form δ0 x1 x2 , does appear. The new form of the phase arising in this way will depend on a slightly modified vector δ˜ of perturbation parameters δ˜i in place of δ. Next, following the proof of Proposition 4.2 in Chapter 5, we decompose  λ λ νδ, νδλ = νδ, Ai + l, M0 ≤ 2l ≤ λ/M1 λ λ 2/3 −l 2/3 B1 (s, δ1 ))ν
δλ (ξ ) and ν B1 (s, δ1 )) where ν δ, Ai (ξ ) := χ0 (λ δ, l (ξ ) := χ1 ((2 λ)
λ × νδ (ξ ), and estimate the corresponding terms separately. This decomposition is motivated by the proof of Lemma 2.2 and it allows for a precise understanding of the integration with respect to the variable x1 in the phase  (x, δ, s). A major problem that remains is to also control the integration in x2 . We cannot simply view the phase  as a perturbation of the phase given when δ˜ = 0 but need more precise estimates in terms of the (relative) sizes of λ and the components δ˜i ˜ Here we pick up some ideas going back to Duistermaat [Dui74] and define of δ. ˜ in such a way that under the natural dilations a kind of homogeneous gauge ρ(δ) 1/3 1/B σr (x1 , x2 ) := (r x1 , r x2 ), the rescaled phase

   r (x1 , x2 , δ, s) := r σ1/r (x1 , x2 ), δ, s looks essentially like  , only with the coefficients δ˜i replaced by coefficients δ˜ir ˜ In particular, if we choose r = 1/ρ(δ), ˜ then in such a way that ρ(δ˜r ) = rρ(δ). r ˜ the rescaled coefficients will satisfy ρ(δ ) = 1, and thus at least one component of δ˜r will be of size one. The effect is that the rescaled phase will have at worst a singularity of order B − 1 with respect to x2 , in contrast to the original phase, which had a singularity of order B, so that we can obtain a better decay of the corresponding oscillatory integral. Notice, however, that the price to be paid here ˜ and that is that we have to replace the parameter λ effectively by λ/r = λρ(δ), moreover, we pick up the Jacobian of this change of coordinates. ˜
1 and where This suggest that we distinguish between the cases where λρ(δ) ˜ λρ(δ)  1. Moreover, matters are even more complicated, since we cannot really treat the variables x1 and x2 separately. ˜
1. Here, the change Anyway, in Section 8.3 we first treat the case where λρ(δ) −1/3 −1/B u1 , λ u2 ) turns out to be convenient for the of variables x = σ1/λ u = (λ λ λ treatment of the νδ, Ai (and related scalings will work for the treatment of the νδ, l ). In fact, this approach is analogous to the proof of part (a) in Lemma 2.2. Under

183

THE REMAINING CASES

˜
1 one finds that the rescaled perturbation parameters δ˜λ the assumption λρ(δ) are still small, so that we do get a rather precise estimate for the oscillatory integrals arising in this situation. This will eventually lead to the desired restriction estimates, at least when B = 4 (compare Proposition 8.6), and when B = 3 we miss only the endpoint estimate for p = pc . That case will therefore again require a complex interpolation argument in order to capture the endpoint as well. ˜  1 and B = 4. Next, in Sections 8.4 and 8.5 we deal with the case where λρ(δ) ˜ + Here, we apply, in fact, the change of variables x = σρ˜ u, where ρ˜ := ρ(δ) 3/2 |B1 (s, δ1 )| captures not only the behavior of our phase with respect to the variable x2 , but also with respect to x1 (compare Lemma 8.8). The basic underlying idea is, however, similar to the one explained before, only that we effectively use ˜ It turns out that by means of these methρ˜ as a scaling parameter in place of ρ(δ). ods, we eventually obtain sufficiently strong estimates, which can even be summed ˜  1, and we arrive at the required absolutely over the dyadic λs for which λρ(δ) restriction estimates. Finally, we turn to the case where B = 3 in Section 8.6. Here, we can basically proceed in a similar way as for B = 4, but it turns out that by these methods we do miss the endpoint estimates for p = pc in a few cases. These remaining cases are listed in Proposition 8.12. The treatment of the remaining four cases in Sections 8.7 and 8.8 will again be based on complex interpolation arguments, similar in spirit to the ones that we had employed in the proof of Proposition 4.2 in Chapter 4. Regretfully, the technical problems to be overcome here will be substantially more involved than in Chapter 4.

8.1 PRELIMINARIES Recall from Corollary 7.11 that in the cases under consideration in this chapter, the function φ˜ a is smooth, so that in particular a = κ˜ 2 /κ˜ 1 and n are integers. Moreover, in the proof of Proposition 7.3, whose assumptions are satisfied here, we have seen (cf. (7.16)) that we may assume that φ˜ κa˜ (y) = y1A y2B + c1 y1n ,

c1  = 0,

so that the function Q in (7.6) is given by Q(y1 , y2 ) = y1A . Then, by (7.9) and (7.12), we may write φδ (x) := x2B b(x1 , x2 , δ) + x1n α(δ1 x1 ) + r(x1 , x2 , δ),

(x1 ∼ 1, |x2 | < ε), (8.1)

where b(x1 , x2 , 0) = x1A ∼ 1 and α(0)  = 0, and where r(x1 , x2 , δ) is given by (7.10). Notice, however, that Proposition 7.3 shows that, by passing to alternative smooth adapted coordinates, we may assume without loss of generality that the term of order j = B − 1 in (7.10) vanishes. In the sequel, we shall therefore assume that r(x1 , x2 , δ) is of the form r(x1 , x2 , δ) =

B−2  j =1

j n

x2 x1 j δj + 2 αj (δ1 x1 ).

(8.2)

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CHAPTER 8

Moreover, either αj (0)  = 0, and then nj is fixed (nj is then the type of the finitetype function bj ), or αj (0) = 0, and then we may assume that nj is as large as we please. In particular, in view of (7.11), we may then assume that δ = (δ0 , δ1 , δ2 , δ3 , . . . , δB ) is given by δ := (2−k(κ˜2 −2κ˜1 ) , 2−kκ˜1 , 2−kκ˜2 , 2−k(n1 κ˜1 +κ˜2 −1) , . . . , 2−k(nB−2 κ˜1 +(B−2)κ˜2 −1) ). (8.3) Observe also that if A = 0 (this is necessarily so if B = 3 ), then we have Q(x) ≡ 1, so that κ˜ 2 = 1/B and, consequently, b(x1 , x2 , δ) = bB (δ1 x1 , δ2 x2 )

(8.4) k

˜a

−κ˜ 1 k

in (8.1). This indeed follows from our definition φk (x1 , x2 ) := 2 φ (2 x1 , 2−κ˜2 k x2 ) of the rescaled phase function φk and the corresponding function φδ in Section 7.2, in combination with (7.15). We finally recall that the complete phase corresponding to φδ is given by (x, δ, ξ ) := ξ1 x1 + ξ2 (δ0 x2 + x1m ω(δ1 x1 )) + ξ3 φδ (x1 , x2 ). Recall also that we are interested in the frequency domains where |ξj | ∼ λj , j = 1, 2, 3, assuming that λ1 ∼ λ2 ∼ λ3 . For the sake of simplicity, we shall assume that λ1 = λ2 = λ3  1. Changing notation in the same way as we did in Section 7.7, in the sequel we shall denote the common value of the λj by λ and accordingly write        ξ1 ξ2 ξ3 e−i(y,δ,ξ ) η(y) dy, χ1 χ1 ν
δλ (ξ ) := χ1 λ λ λ that is, νδλ (x) = λ3



  χˇ 1 (λ(x1 − y1 )) × χˇ 1 λ(x2 − δ0 y2 − y12 ω(δ1 y1 )

× χˇ 1 (λ(x3 − φδ (y)) η(y) dy. Recall also that supp η ⊂ {y1 ∼ 1, |y2 | < ε}, (ε  1). As before, we change coordinates from ξ = (ξ1 , ξ2 , ξ3 ) to s1 , s2 , and s3 := ξ3 /λ by writing ξ = ξ(s, λ), with ξ1 = s1 ξ3 = λs1 s3 ,

ξ2 = s2 ξ3 = λs2 s3 ,

ξ3 = λs3 .

Accordingly, we write   (y, δ, ξ ) = λs3 1 (y1 , δ1 , s) + s2 δ0 y2 + y2B b(y1 , y2 , δ) + r(y1 , y2 , δ) , (8.5) where 1 (y1 , δ1 , s) := s1 y1 + s2 y12 ω(δ1 y1 ) + y1n α(δ1 y1 ). Notice that here |sj | ∼ 1, j = 1, 2, 3, and that from now on we shall denote by s the pair s := (s1 , s2 ) (in order to simplify the notation, we deviate slightly here

185

THE REMAINING CASES

from our previous notation in Section 7.7, where this pair had been denoted by s  ). By passing from sj to −sj , if necessary, we shall in the sequel always assume that sj ∼ 1,

j = 1, 2, 3

(notice that these changes of signs may cause a change of sign of the xj and ω, respectively, of ). Let us fix s 0 = (s10 , s20 ) such that s10 ∼ 1 ∼ s20 , and consider the phase function 1 (x1 , 0, s 0 ) when δ1 = 0. Assume first that this phase has at worst nondegenerate critical points x1c ∼ 1. Then the same is true for sufficiently small δ1 , and the estimate (7.73) for ν
δλ in Section 7.7 can be improved to ν
λ
λ−1/2−1/B , δ ∞

and thus in Case A of Section 7.7 we obtain the better estimate Tδλ pc →pc
λ−1/2−1/B+5θc /2 compared to (7.75). Moreover, by means of (7.89) and (7.90), one checks easily that the exponent of λ in this estimate is strictly negative if B = 4 and zero if B = 3. Thus we can sum these estimates over all dyadic λ  1 if B = 4 and obtain at least a uniform estimate when B = 3. It is easy to see that this case can then still be treated by means of the real interpolation Proposition 2.6, since the relevant frequencies will here be restricted essentially to cuboids in the ξ -space. −B/(B−1) , we obtain the better estimate In Case B, where λ > δ0 Tδλ pc →pc
λ−1/2−1/B+(3/2+1/B)θc δ0−θc , compared to (7.76). The exponent of λ is strictly negative (compare the discussion −B/(B−1) leading to (7.88)), so summing over all λ > δ0 , we find that  (B+2−5Bθ c )/(2B−2) Tδλ pc →pc
δ0 . −B/(B−1)

λ>δ0

And, since θc ≤ θ˜B , one easily checks that the exponent of δ0 in this estimate is nonnegative if B ≥ 3. 8.2 REFINED AIRY-TYPE ANALYSIS We are thus left with the more subtle situation where the phase function 1 (x1 , 0, s 0 ) has a degenerate critical point x1c of Airy type. Denoting by 1 , 1  , . . . , derivatives with respect to x1 and arguing as in Section 5.1, we see by the implicit function theorem that for s sufficiently close to s 0 and δ sufficiently small, there is a unique, nondegenerate critical point x1c = x1c (δ1 , s2 ) ∼ 1 of 1 , that is, 

1 (x1c (δ1 , s2 ), δ1 , s) = 0,

|s − s 0 | < ε, |δ| < ε,

if ε is sufficiently small. We then shift this critical point to the origin, by putting 1  (x, δ, ξ ) := (8.6) (x1c (δ1 , s2 ) + x1 , x2 , δ, ξ ), |(x1 , x2 )|  1 s3 λ

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CHAPTER 8

(notice that we may indeed assume that |(x1 , x2 )|  1, since away from x1c , we have at worst nondegenerate critical points, and the previous argument applies). From Lemma 5.2 (with β := α, σ = 1, δ3 = 0 and b0 = 0), we then immediately get the following result, after scaling in x1 so that we may assume that −

2ω(0) = 1. n(n − 1)α(0)

Lemma 8.1.  is of the form  (x, δ, ξ ) = x13 B3 (s2 , δ1 , x1 ) − x1 B1 (s, δ1 ) + B0 (s, δ1 ) + φ (x, δ, s2 ),

(8.7)

with

φ (x, δ,

s2 ) := x2B

b(x, δ, s2 ) +

B−2 

j

δj +2 x2 α˜ j (x1 , δ, s2 )

j =2

+x2 (s2 δ0 + δ3 (x1c (δ1 , s2 ) + x1 )n1 α1 (δ1 (x1c (δ1 , s2 ) + x1 ))), (8.8) and where the following hold true: The functions b (which will be different from b in (8.5)) and α˜ j are smooth, b(x, δ, s2 ) ∼ 1, and also |α˜ j | ∼ 1, unless αj is a flat function. Moreover, B0 , B1 , and B3 are smooth functions, and (n−3)/(n−2)

B3 (s2 , δ1 , 0) = s2

1/(n−2)

G4 (δ1 s2

),

where G4 (0) =

n(n − 1)(n − 2) α(0). 6

Furthermore, we may write  1/(n−2) 1/(n−2) c  G1 (δ1 s2 ), x1 (δ1 , s2 ) = s2   1/(n−2) 1/(n−2) n/(n−2) 1/(n−2) G1 (δ1 s2 ) − s2 G2 (δ1 s2 ), B0 (s, δ1 ) = s1 s2     (n−1)/(n−2) 1/(n−2) G3 (δ1 s2 ), B1 (s, δ1 ) = −s1 + s2 where

  G1 (0)     G2 (0)     G (0) 3

(8.9)

= 1, n2 − n − 2 α(0), 2 = n(n − 2)α(0). =

(8.10)

Notice that all the numbers in (8.10) are nonzero, since we assume n > 2B > 5. Finally, if we put G5 := G1 G3 − G2 , then we have G3 (0)  = 0,

G5 (0)  = 0.

(8.11)

Observe that we here obtain a more specific dependency of x1c , B0 , B1 , and B3 on δ1 and s2 than in Section 5.1, due to the fact that the equation for the critical point depends only on the parameter δ1n−2 s2 in the coordinate y1 := δ1 x1 . Nevertheless, with a slight abuse of notation, we shall frequently also use the 1/(n−2) ), j = 1, . . . , 4. shorthand notation Gj (s2 , δ) in place of Gj (δ1 s2

187

THE REMAINING CASES

We also remark that the part of the measure νδ corresponding to the small neighborhood of the critical point x1c defined in (8.6) and which by some slight abuse of notation we shall again denote by νδ , is given by an expression for its Fourier transform of the form  (with ξ = ξ(s, λ)), (8.12) ν
δ (ξ ) = e−is3 λ (x,δ,ξ ) a(x, δ, s) dx where a is a smooth function with compact support in x such that |x| ≤ ε on supp a. Remark 8.2. It will be important in the sequel to observe that every single δj is a fractional power of 2−k , hence, also of δ0 ; that is, there is some positive rational q r number r > 0 such that δj = δ0 j , with positive integers qj (cf. (8.3)). In the sequel, we shall need more precise information on the structure of the last term of φ in (8.8). Lemma 8.3. Let a1 (x1 , δ, s2 ) := s2 δ0 + δ3 (x1c (δ1 , s2 ) + x1 )n1 α1 (δ1 (x1c (δ1 , s2 ) + x1 )) be the coefficient of x2 in the last term of φ in (8.8). Then a1 can be rewritten in the form a1 (x1 , δ, s2 ) = δ3,0 α˜ 1 (x1 , δ0r , s2 ) + δ0 x1 α1,1 (x1 , δ0r , s2 ), q

r

with smooth functions α˜ 1 and α1,1 , where δ3,0 is of the form δ3,0 = δ0 3,0 , with some positive integer q3,0 . Moreover, two possible cases may occur. Case ND: The max{δ0 , δ3 } ≥ δ0 .

nondegenerate

case. α1,1 ≡0,

|α˜ 1 | ∼ 1

and

δ3,0 =

Case D: The degenerate case. |α1,1 | ∼ 1, α˜ 1 = α˜ 1 (δ0r , s2 ) is independent of x1 , δ0 = δ3 , and δ3,0  δ0 . Moreover, either |α˜ 1 | ∼ 1, or we can choose q3,0 ∈ N as large as we wish. In particular, we may write φ (x, δ, s) = x2B b(x, δ0r , s2 ) +

B−2 

j

δj +2 x2 α˜ j (x1 , δ0r , s2 )

j =2

+δ3,0 x2 α˜ 1 (x1 , δ0r , s2 ) + δ0 x1 x2 α1,1 (x1 , δ0r , s2 ),

(8.13)

with smooth functions α˜ j and b, where |b| ∼ 1 and α˜ 1 and α1,1 are as in Case D (respectively, ND). Moreover, in both cases we have max{δ0 , δ3 } = max{δ0 , δ3,0 }. Proof. Recall that δ0 = 2−k(κ˜2 −2κ˜1 ) and δ3 = 2−k(n1 κ˜1 +κ˜2 −1) . We therefore distinguish two cases. Case 1. n1
= n − 2. Then κ˜ 2 − 2κ˜ 1  = n1 κ˜ 1 + κ˜ 2 − 1, since κ˜ 1 = 1/n, and thus either δ0  δ3 or δ0  δ3 , for k sufficiently large. Notice that we may assume this

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CHAPTER 8

to be true in particular if the function α1 is flat, since we may then choose n1 as large as we want. By putting δ3,0 := max{δ0 , δ3 }, we then clearly may write a1 as in Case ND. Case 2. n1 = n − 2. Then δ0 = δ3 , and we may assume that |α1 | ∼ 1. Thus, expanding around x1 = 0 and applying (8.9), we see that we may write a1 (x1 , δ, s2 ) = δ0 (s2 + x1c (δ1 , s2 )n1 α1 (δ1 x1c (δ1 , s2 )) + x1 α1,1 (x1 , δ0r , s2 )) 1/(n−2)

= δ0 s2 (1 + G1 (δ1 s2

1/(n−2)

) α1 (δ1 s2

1/(n−2)

G1 (δ1 s2

)))

+ δ0 x1 α1,1 (x1 , δ0r , s2 ) 1/(n−2)

= δ0 s2 (1 + g(δ1 s2

)) + δ0 x1 α1,1 (x1 , δ0r , s2 ),

with smooth functions 1 + g and α1,1 , where |α1,1 | ∼ 1. By means of a Taylor expansion of g around the origin, we thus find that 1/(n−2) N

a1 (x1 , δ, s2 ) = δ0 s2 (δ1 s2

1/(n−2)

) gN (δ1 s2

) + δ0 x1 α1,1 (x1 , δ0r , s2 )

= (δ0 δ1N )α˜ 1 (δ0r , s2 ) + δ0 x1 α1,1 (x1 , δ0r , s2 ), with N ∈ N and gN smooth. Moreover, we may either assume that |α˜ 1 | ∼ 1 (if 1 + g is a finite type N at the origin) or that we may choose N as large as we wish (if 1 + g is flat). Notice that if N = 0, then we can include the second term into the first term and arrive again at Case ND. In all other cases we arrive at the situation described by Case D, where the second term cannot be included into the first term. Notice that then δ3,0 := δ0 δ1N  δ0 . Q.E.D. Let us next introduce the quantity 

B/(B−1) B/(B−j )  + B−2 δ3,0 j =2 δj +2 ρ :=  δ 3B/(2B−3) + δ B/(B−1) + B−2 δ B/(B−j ) j =2 j +2 0 3,0

in Case ND, (8.14) in Case D,

˜ of the coefficients which we shall view as a function ρ(δ)  (δ3,0 , δ4 , . . . , δB ) in Case ND, δ˜ := (δ , δ , δ , . . . , δ ) in Case D. 0 3,0 4 B Remark 8.4. Observe that if we scale the complete phase  from (8.7) in x1 by the factor r −1/3 and in x2 by r −1/B , r > 0, and multiply by r, that is, if we look at  r (u1 , u2 , δ, s) := r (r −1/3 u1 , r −1/B u2 , δ, s),

189

THE REMAINING CASES

then the effect is essentially that δ˜ in (8.8) is replaced by  (r (B−1)/B δ3,0 , . . . , r (B−j )/B δj +2 , . . . , r 2/B δB ) in Case ND, δ˜r := (r (2B−3)/3B δ , r (B−1)/B δ , . . . , r (B−j )/B δ , . . . , r 2/B δ ) in Case D, 0 3,0 j +2 B (8.15) whereas B1 (s, δ1 ) is replaced by r B1 (s2 , δ1 ) and B0 (s, δ1 ) by rB0 (s, δ1 ). More precisely, if denote by σr the dilations σr (x1 , x2 ) := (r 1/3 x1 , r 1/B x2 ), so that r (u, δ, s) = r (σ1/r u, δ, s), then we have 2/3

 r (u1 , u2 , δ, s) = u31 B3 (s2 , δ1 , r −1/3 u1 ) − u1 r 2/3 B1 (s, δ1 ) + rB0 (s, δ1 ) + φr (u1 , u2 , δ˜r , s2 ), where φr (u, δ˜r , s) := uB2 b(σr −1 u, δ0r , s2 ) +

B−2 

j δ˜jr +2 u2 α˜ j (r −1/3 u1 , δ0r , s2 )

j =2 r + δ˜3,0

u2 α˜ 1 (r

−1/3

u1 , δ0r , s2 ) + δ˜0r u1 u2 α1,1 (r −1/3 u1 , δ0r , s2 ). (8.16)

And, under these dilations, ρ is homogeneous of degree 1, that is, ˜ ρ(δ˜r ) = rρ(δ),

r > 0.

˜ we see that we have In particular, after scaling  in this way by r := 1/ρ(δ), ˜ = 1. normalized the coefficients of φ in such a way that ρ(δ)

This observation, which is based on ideas by Duistermaat [Dui74], will become important in the sequel. ˜
1 8.3 THE CASE WHERE λρ(δ) Assume now that B ∈ {3, 4}. Following the proof of Proposition 4.2 in Chapter 5, λ λ we define the functions νδ, Ai and νδ, l by 2/3 λ B1 (s, δ1 ))ν
δλ (ξ ), ν δ, Ai (ξ ) := χ0 (λ −l 2/3 λ B1 (s, δ1 ))ν
δλ (ξ ), ν δ, l (ξ ) := χ1 ((2 λ)

so that λ νδλ = νδ, Ai +



M0 ≤ 2l ≤

(8.17) λ , M1

(8.18)

λ νδ, l.

M0 ≤2l ≤λ/M1

Here, χ0 , χ1 ∈ C0∞ (R), and χ1 (t) is supported where 2−4/3 ≤ |t| ≤ 24/3 , whereas 2/3 χ0 (t) ≡ 1 for |t| ≤ M0 . Thus, by choosing M0 sufficiently large, we may assume that 2−l ≤ 1/M0  1. Denote by Tδ,λ Ai and Tδ,λ l the corresponding operators of convolution with the Fourier transforms of these functions.

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CHAPTER 8

Our goal will be to adjust the proofs of the estimates in Lemma 5.3 and Lemma 5.4 to our present situation in order to derive the following estimates, which are analogous to the corresponding ones in those lemmas (formally, we have only to replace a factor λ−1/2 by the factor λ−1/B ): −1/B−1/3 λ ν , δ, Ai ∞ ≤ C1 λ

(8.19)

λ 5/3−1/B , νδ, Ai ∞ ≤ C2 λ

(8.20)

−l/6 −1/B−1/3 λ λ , ν δ, l ∞ ≤ C1 2

(8.21)

λ l/3 5/3−1/B . νδ, l ∞ ≤ C2 2 λ

(8.22)

as well as

λ 8.3.1 Estimates for νδ, Ai

Changing coordinates from x to u by putting x = σ1/λ u = (λ−1/3 u1 , λ−1/B u2 ) in the integral (8.12) and making use of Remark 8.4 (with r := λ), we find that −1/B−1/3 λ χ1 (s, s3 ) χ0 (λ2/3 B1 (s, δ1 )) e−is3 λB0 (s,δ1 ) ν δ,Ai (ξ ) = λ  3 −1/3 2/3 ˜λ × e−is3 (u1 B3 (s2 ,δ1 ,λ u1 )−u1 λ B1 (s,δ1 )+φλ (u1 ,u2 , δ ,s2 ))

× a(σλ−1 u, δ, s) du1 du2 ,

(8.23)

where χ1 (s, s3 ) := χ1 (s1 s3 )χ1 (s2 s3 )χ1 (s3 ) localizes to the region where sj ∼ 1, j = 1, 2, 3. Observe that here we are integrating over the possibly large domain where |u1 | ≤ ελ1/3 and |u2 | ≤ ελ1/B . Recall also that φλ is given by (8.16) and that λ ˜
1, and so we have ρ(δ˜ ) = λρ(δ) |δ˜λ |
1

and

λ2/3 |B1 (s, δ1 )|
1.

λ By means of this integral formula for ν δ,Ai (ξ ), we easily obtain the following.

˜
1, then we may write Lemma 8.5. If λρ(δ)   −1/B−1/3 λ ν χ1 (s, s3 ) χ0 λ2/3 B1 (s, δ1 ) e−is3 λB0 (s,δ1 ) δ,Ai (ξ ) = λ   × a λ2/3 B1 (s, δ1 ), δ˜λ , s, s3 , δ0r , λ−1/3B ,

(8.24)

where a is again a smooth function of all its (bounded) variables. Proof. Given L  1, we decompose the integral in (8.23) by means of suitable smooth cutoff functions into the integral I1 over the region where |(u1 , u2 )| ≤ L, the integral I2 over the region where |(u1 , u2 )| > L and |u2 |B−1  |u1 |, and the integral I3 over the region where |(u1 , u2 )| > L and |u2 |B−1
|u1 |. For each of these contributions Ij , we then show that it is of the form aj (λ2/3 B1 (s, δ1 ), δ˜λ , s, s3 , δ0r , λ−1/3B ), with a suitable smooth function aj , provided L is sufficiently large. For I1 , this claim is obvious.

191

THE REMAINING CASES

On the remaining region where |(u1 , u2 )| ≥ L, we may use iterated integrations by parts with respect to u1 or u2 in order to convert the integral into an absolutely convergent integral, to which we then may apply the standard rules for differentiation with respect to parameters (such as sj , etc.). Denote to this end the complete phase function appearing in this integral by c . It is then easily seen that we may estimate |∂u2 c |  |u2 |B−1 − c|u1 |,

(8.25)

|∂u1 c |  u21 − cλ−1/3 |u2 |B − c|u2 |,

(8.26)

with a fixed constant c > 0. The last terms appear only in the degenerate case D, due to the presence of the term δ˜0λ u1 u2 α1,1 (λ−1/3 u1 , δ0r , s2 ) in φλ . Denote by I2 the contribution by the subregion on which |u2 |B−1  |u1 |. On this region, we may gain factors of order |u2 |−2N(B−1) (for any N ∈ N) in the amplitude by means of iterated integrations by parts in u2 . And, since |u2 |−2N(B−1)
|u1 |−N |u2 |−N(B−1) , we see that we arrive at an absolutely convergent integral. Similarly, denote by I3 the contribution by the subregion on which |u2 |B−1
|u1 |. Observe that |u2 |B
|u1 |B/(B−1)  u21 , since B ≥ 3, and since we may assume that |u1 |  1. This shows that in this region, we have |∂u1 c |  u21 , and thus we may gain factors of order |u1 |−2N (for any N ∈ N) in the amplitude, by means of iterated integrations by parts in u1 . And, since |u1 |−2N
|u1 |−N |u2 |−N(B−1) , we arrive again at an absolutely convergent integral. Q.E.D. Lemma 8.5 implies, in particular, estimate (8.19). As for the more involved estimate (8.20), with Lemma 8.5 at hand, we can basically follow the arguments from Section 5.2.2, only with the factor λ−1/2 appearing there replaced by the factor λ−1/B here and with the amplitude g(λ2/3 B1 (s  , δ, σ ), λ, δ, σ, s) replaced by the amplitude a(λ2/3 B1 (s, δ1 ), δ˜λ , s, s3 , δ0r , λ−1/3B ) here (compare with (5.17)). λ 8.3.2 Estimates for νδ,l

Changing coordinates from x to u by letting x = σ2l /λ u in the integral (8.12), and making use of Remark 8.4 (with r := λ/2l ), we find that   −l −1/B−1/3 λ χ1 (s, s3 ) χ1 (2−l λ)2/3 B1 (s, δ1 ) e−is3 λB0 (s,δ1 ) ν δ,l (ξ ) = (2 λ)  l × e−is3 2 (u1 ,u2 ,s,δ,λ,l) a(σ2l λ−1 u, δ, s) du1 du2 , (8.27) with phase function (u1 , u2 , s, δ, λ, l) −l := u31 B3 (s2 , δ1 , (2−l λ)−1/3 u1 ) − u1 (2−l λ)2/3 B1 (s, δ1 ) + φ2 −l λ (u1 , u2 , δ˜2 λ , s2 ).

(8.28) Observe that here we are integrating over the possibly large domain where |u1 | ≤ (ε2−l λ)1/3 and |u2 | ≤ ε(2−l λ)1/B . Recall also that φ2−l λ is given by (8.16) and that −l ˜
2−l , so that we have ρ(δ˜2 λ ) = 2−l λρ(δ) |δ˜2

−l

λ

|1

and (2−l λ)2/3 |B1 (s, δ1 )| ∼ 1.

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CHAPTER 8

Notice that this implies that   −l φ2 −l λ (u1 , u2 , δ˜2 λ , s2 ) = uB2 b σ(2−l λ)−1 u, δ0r , s2 + small error.

(8.29)

Arguing in a somewhat similar way as in Section 5.2, we first decompose λ λ λ = νl,0 + νl,∞ , νδ,l

where   −l −1/B−1/3 λ χ1 (s, s3 ) χ1 (2−l λ)2/3 B1 (s, δ1 ) e−is3 λB0 (s,δ1 ) ν l,0 (ξ ) := (2 λ)  l × e−is3 2 (u1 ,u2 ,s,δ,λ,l) a(σ2l λ−1 u, δ, s) χ0 (u) du1 du2 ,   −l −1/B−1/3 λ χ1 (s, s3 ) χ1 (2−l λ)2/3 B1 (s, δ1 ) e−is3 λB0 (s,δ1 ) ν l,∞ (ξ ) := (2 λ)  l × e−is3 2 (u1 ,u2 ,s,δ,λ,l) a(σ2l λ−1 u, δ, s) (1 − χ0 (u)) du1 du2 . Here, we choose χ0 ∈ C0∞ (R2 ) such that χ0 (u) ≡ 1 for |u| ≤ L, where L is supposed to be a sufficiently large positive constant. λ : Arguing as in the proof of Let us first consider the contribution given by the νl,∞ Lemma 8.5, we can easily see by means of integrations by parts that, given k ∈ N, then for every N ∈ N we may write −lN λ (2−l λ)−1/B−1/3 χ1 (s, s3 ) χ1 ((2−l λ)2/3 B1 (s, δ1 )) e−is3 λB0 (s,δ1 ) ν l,∞ (ξ ) = 2 −l × aN,l ((2−l λ)2/3 B1 (s, δ1 ), s, s3 , δ˜2 λ , δ0r , (2−l λ)−1/3 , λ−1/3B ),

(8.30) where aN,l is a smooth function of all its (bounded) variables such that aN,l C k is uniformly bounded in l. In particular, we see that −lN −1/B−1/3 λ λ ν l, ∞ ∞
2

∀N ∈ N.

(8.31)

Next, applying the Fourier inversion formula and changing coordinates from s1 to z := (2−l λ)2/3 B1 (s, δ1 ),

(n−1)/(n−2)

that is, s1 = s2

G3 (s2 , δ1 ) − (2−l λ)−2/3 z, (8.32)

as in Section 5.2, we find that

 (n−1)/(n−2) λ (x) = λ3 2−lN (2−l λ)−1/B−1 e−is3 λ(z,s2 ,δ) χ1 (s2 G3 (s2 , δ1 ) νl,∞ −(2−l λ)−2/3 z, s2 , s3 ) χ1 (z) −l ×aN,l (z, s2 , s3 , δ0r , δ˜2 λ , (2−l λ)−1/3 , λ−1/3B ) dz ds2 ds3 ,

(8.33)

where the phase function  is given by n/(n−2)

(z, s2 , δ) := s2

(n−1)/(n−2)

G5 (s2 , δ) − x1 s2

+ (2l λ−1 )2/3 z (x1 −

G3 (s2 , δ) − s2 x2 − x3

1/(n−2) s2 G1 (s2 , δ)).

(8.34)

193

THE REMAINING CASES

Applying the van der Corput–type Lemma 2.1 (of order M = 3) to the integration in s2 , which allows for the gain of a factor λ−1/3 , this easily implies that νl,λ ∞ ∞
2−lN λ5/3−1/B

∀N ∈ N.

(8.35)

Notice that estimates (8.33) and (8.35) are stronger than the desired estimates, (8.21) and (8.22). λ : observe first that on a Let us next look at the contribution given by the νl,0 region where |u1 | is sufficiently small, iterated integrations by parts with respect λ , for every N ∈ N. to u1 allow to gain factors 2−lN in the integral defining νl,0 Afterwords, freezing u1 , we can reduce this to the integration in u2 alone. A similar argument applies whenever we are allowed to integrate by parts in u1 (this is also the case when B1 and B3 have opposite signs). We shall therefore assume from now on that B1 and B3 have the same sign. Moreover, let us assume without loss of generality that u1 > 0. Then there is a unique nondegenerate critical point uc1 = −l uc1 ((2−l λ)−1/B u2 , δ˜2 λ , ...) of the phase  in (8.28), of size |uc1 | ∼ 1, and we may restrict the integration in u1 to a small neighborhood of uc1 . That is, we may replace the cutoff function χ0 (u1 ) by a cutoff function χ1 (u1 ) supported in a sufficiently small neighborhood of a point u01 containing uc1 . Then the method of stationary phase shows that the corresponding integral in u1 will be of order 2−l/2 , so that this term will give the main contribution. For the sake of simplicity, we shall therefore restrict ourselves in the sequel to λ , given by the discussion of this main term νl,1   −l −1/B−1/3 λ χ1 (s, s3 ) χ1 (2−l λ)2/3 B1 (s, δ1 ) e−is3 λB0 (s,δ1 ) ν l,1 (ξ ) := (2 λ)  l × e−is3 2 (u1 ,u2 ,s,δ,λ,l) a(σ2l λ−1 u, δ, s) χ0 (u)χ1 (u1 ) du1 du2 , (8.36) where |u1 | ∼ 1 on the support of χ1 (u1 ). First applying the method of stationary phase to the integration in u1 and, subsequently, van der Corput’s estimate of order B to the integration in u2 , we easily arrive at the estimate −l −1/B−1/3 −l/2−l/B λ 2 , ν l,1 ∞
(2 λ)

which is exactly what we need to verify (8.21) (also recall here estimate (8.31)). λ (x), Fourier inversion allows to write As for the more involved estimation of νl,1 λ (x) = λ3 (2−l λ)−1/B−1/3 νl,1



e−is3 (u,s,x,δ,λ,l) χ1 (s, s3 ) χ1 ((2−l λ)2/3 B1 (s, δ1 ))

× a(σ2l λ−1 u, δ, s) χ0 (u)χ1 (u1 ) du1 du2 ds,

(8.37)

where the complete phase  is given by (u, s, x, δ, λ, l) := 2l (u1 , u2 , s, δ, λ, l) + λ (B0 (s, δ1 ) − s1 x1 − s2 x2 − x3 ) ,

194

CHAPTER 8

with  given by (8.28). Changing again from the coordinate s1 to z as in (8.32), we may also write  ˜ λ (x) = λ3 (2−l λ)−1/B−1 e−is3 (u,z,s2 ,x,δ,λ,l) χ1 (s, s3 ) χ1 (z) νl,1   × a˜ σ2l λ−1 u, (2l λ−1 )2/3 z, s2 , δ χ0 (u)χ1 (u1 ) du1 du2 dz ds2 ds3 , (8.38) with phase 1/(n−2) ˜ (u, z, s2 , x, δ, λ, l) : = λ(2l λ−1 )2/3 z (x1 − s2 G1 (s2 , δ)) n/(n−2)

+ λ(s2

(n−1)/(n−2)

G5 (s2 , δ) − x1 s2

× G3 (s2 , δ) − s2 x2 − x3 ) + 2l (u31 B3 (s2 , δ1 , (2−l λ)−1/3 u1 ) − zu1 −l + φ2−l λ (u1 , u2 , δ˜2 λ , s2 )).

(8.39)

We shall prove the following estimate: λ |νl,1 (x)| ≤ C2l/3 λ5/3−1/B ,

(8.40)

with a constant C that is independent of λ, l, x, and δ. As in Section 5.2.2, this estimate is easily verified if |x|  1, simply by means of integrations by parts in the variables s2 , s3 and z, in combination with the method of stationary phase in u1 and van der Corput’s estimate of order B in u2 . Similarly, if |x|
1 and |x1 |  1, we can arrive at the same conclusion by first integrating by parts in z. Indeed, in these situations we may gain factors 2−2Nl/3 λ−N/3 through integrations by parts, so that the corresponding estimates can be summed in a trivial way. Let us thus assume that |x|
1 and |x1 | ∼ 1. Following Section 5.2, we then decompose λ λ λ λ = νl,I + νl,I νl,1 I + νl,I I I , λ λ λ where νl,I , νl,I I and νl,I I I correspond to the contributions to the integral in (8.38) 1/(n−2) given by the regions where λ(2l λ−1 )2/3 |x1 − s2 G1 (s2 , δ)|  2l , λ(2l λ−1 )2/3 1/(n−2) 1/(n−2) l l −1 2/3 × |x1 − s2 G1 (s2 , δ)| ∼ 2 and λ(2 λ ) |x1 − s2 G1 (s2 , δ)|  2l , respectively. The first and the last term can easily be handled as in Section 5.2 by means of integrations by parts in z, with subsequent exploitation of the oscillations with respect to u1 and u2 , which leads to the following estimate: λ λ −l/3 5/3−1/B |νl,I (x)| + |νl,I λ , I I (x)| ≤ C2

which is better than (8.40) by a factor 2−2l/3 , so that summation in l is no problem for these terms. Nevertheless, summation in λ still will require an interpolation argument if B = 3. λ Let us next concentrate on νl,I I (x), which is of the form  ˜ λ 3 −l −1/B−1 e−is3 (u,z,s2 ,x,δ,λ,l) a(σ νl,I ˜ 2l λ−1 u, (2l λ−1 )2/3 z, s2 , δ) I (x) = λ (2 λ) × χ1 (s2 , s3 ) χ1 ((2l λ−1 )−1/3 (x1 − s2

1/(n−2)

× χ1 (u1 ) χ1 (z) du1 du2 dz ds2 ds3 .

G1 (s2 , δ))) χ0 (u) (8.41)

195

THE REMAINING CASES

Writing n/(n−2) (n−1)/(n−2) ˜ (u, z, s2 , x, δ, λ, l) = λ(s2 G5 (s2 , δ) − x1 s2 G3 (s2 , δ) − s2 x2 − x3 )  1/(n−2) l −l 1/3 + 2 z (2 λ) (x1 − s2 G1 (s2 , δ)) − zu1  −l + u31 B3 (s2 , δ1 , (2−l λ)−1/3 u1 ) + φ2−l λ (u1 , u2 , δ˜2 λ , s2 ) ,

G1 (s2 , δ))| ∼ 1 and |u1 | ∼ 1, we and observing that here |(2−l λ)1/3 (x1 − s2 ˜ may have a critical point (uc1 , z c ) within the support of the see that the phase  amplitude as a function of u1 and z. Moreover, in a similar way as in Section 5.2, we see that this critical point will be nondegenerate. Of course, if there is no critical point, we may obtain even better estimates by means of integrations by parts. Thus, let us assume in the sequel that there is such a critical point. Applying the method of stationary phase to the integration in (u1 , z), we see that we essentially may write  λ 2 −l −1/B e−is3 2 (u2 ,s2 ,x,δ,λ,l) a2 ((2l λ−1 )1/3 u2 , s2 , x, (2l λ−1 )1/3 , δ) νl,I I (x) = λ (2 λ) 1/(n−2)

× χ1 (s2 , s3 ) χ1 ((2l λ−1 )−1/3 (x1 − s2

1/(n−2)

G1 (s2 , δ))) χ0 (u2 ) du2 ds2 ds3 ,

˜ by replacing (u1 , z) by the critical point (uc1 , z c ) where the phase 2 arises from  (which, of course, also depends on the other variables). In order to compute 2 more explicitly, we go back to our original coordinates, in which our complete phase is given by (compare (8.5)) λ(s1 y1 + s2 y12 ω(δ1 y1 ) + y1n α(δ1 y1 ) + s2 δ0 y2 + y2B b(y1 , y2 , δ) +r(y1 , y2 , δ) − s1 x1 − s2 x2 − x3 ). Recall also that we had passed from the coordinates (y1 , s1 ) to the coordinates (u1 , z) by means of a smooth change of coordinates (depending on the remaining variables (y2 , s2 )). Since the value of a function at a critical point does not depend on the chosen coordinates, arguing by means of Lemma 5.6, we find that in the coordinates (y2 , s2 ) the phase 2 is given by λ(s2 x12 ω(δ1 x1 ) + x1n α(δ1 x1 ) + s2 δ0 y2 + y2B b(x1 , y2 , δ) + r(x1 , y2 , δ) − s2 x2 − x3 ). (8.42) And, since y2 = (2−l λ)−1/B u2 , this means that 2 (u2 , s2 , x, δ, λ, l) = λ(s2 x12 ω(δ1 x1 ) + x1n α(δ1 x1 ) − s2 x2 − x3 ) + 2l (uB2 b(x1 , (2−l λ)−1/B u2 , δ) +

B−2 

−l n j u2 δ˜j2+2λ x1 j αj (δ1 x1 )

j =2

+ (2−l λ)(B−1)/B u2 (δ0 s2 + δ3 x1n1 α1 (δ1 x1 ))) 1/(n−2)

G1 (s2 , δ)) ∼ 1 because s2 ∼ 1 and (compare (8.2)). Note that ∂s2 (s2 1/(n−2) G1 (s2 , 0) = 1. Therefore, the relation |x1 − s2 G1 (s2 , δ)| ∼ (2l λ−1 )1/3 can be

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CHAPTER 8

˜ 1 (x1 , δ)| ∼ (2l λ−1 )1/3 , where G ˜ 1 is again a smooth function rewritten as |s2 − G ˜ such that |G1 | ∼ 1. If we write ˜ 1 (x1 , δ), s2 = (2l λ−1 )1/3 v + G then this means that |v| ∼ 1. We shall therefore change variables from s2 to v, which leads to  λ 2 −l −1/B−1/3 e−is3 3 (u2 ,v,x,δ,λ,l) a3 ((2l λ−1 )1/3 u2 , v, x, (x) = λ (2 λ) νl,I I (2l λ−1 )1/3 , δ)χ1 (s3 ) χ1 (v) χ0 (u2 ) du2 dv ds3 ,

(8.43)

with a smooth amplitude a3 and the new phase function 3 (u2 , v, x, δ, λ, l) = λ(v (2−l λ)−1/3 (x12 ω(δ1 x1 ) − x2 ) +(2−l λ)−1/B−1/3 δ0 vu2 + QA (x, δ)) +2l (uB2 b(x1 , (2−l λ)−1/B u2 , δ) +

B−2 

−l n j u2 δ˜j2+2λ x1 j αj (δ1 x1 )

j =2

˜ 1 (x1 , δ) + δ3 x n1 α1 (δ1 x1 ))), (8.44) +u2 (2−l λ)(B−1)/B (δ0 G 1 where ˜ 1 (x1 , δ)(x12 ω(δ1 x1 ) − x2 ) + x1n α(δ1 x1 ) − x3 . QA (x, δ) := G Applying van der Corput’s estimate of order B to the integration in u2 in (8.43), we find that λ 2 −l −1/B−1/3 −l/B 2 = 2l/3 λ5/3−1/B . |νl,I I (x)| ≤ Cλ (2 λ)

This proves (8.40), which completes the proof of estimate (8.22). 8.3.3 Consequences of the estimates (8.19)–(8.22) By interpolation, these estimates imply Tδ,λ Ai pc →pc
λ−1/3−1/B+2θc , Tδ,λ l pc →pc

(8.45)

−l(1−3θc )/6 −1/3−1/B+2θc


2

λ

.

(8.46)

But, by Lemma 7.4, we have θc ≤ θ˜B = 3/(2B + 3), and this easily implies that the exponents of λ and 2l in the preceding estimates are strictly negative if B = 4 and zero if B = 3 (where θc = 13 ). We can therefore sum these estimates over all l, as well as λ  1 if B = 4, and the desired estimate follow, whereas if B = 3, then we get only uniform estimates Tδ,λ Ai pc →pc ≤ C,

Tδ,λ l pc →pc ≤ C,

˜
1), (λρ(δ)

with a constant C not depending on δ. The case where B = 3 will therefore again require a complex interpolation argument in order to capture the endpoint, as in the proof of Proposition 4.2 (c). In particular, we have proven the following.

197

THE REMAINING CASES

Proposition 8.6. If B = 4, then under the assumptions in this section,  Tδλ pc →pc
1, ˜ {λ≥1: λρ(δ)
1}

where the constant in this estimate will not depend on δ. In combination with the following proposition, which will be proved in the next two sections, this will complete the discussion of the remaining case where B = 4 in (7.89) and, hence, also the proof of Proposition 7.1 for this case. Proposition 8.7. If B = 4, then under the assumptions in this section,  Tδλ pc →pc
1, ˜ {λ≥1: λρ(δ)1}

where the constant in this estimate will not depend on δ. 8.4 THE CASE WHERE m = 2, B = 4, AND A = 1 According to Proposition 8.6, we are left with controlling the operators Tδλ with ˜ If A = 1, then according λρ  1, where we have used the abbreviation ρ = ρ(δ). 1 r  to (7.90), we have h + 1 = 4; hence, θc = 4 and pc = 8. We shall then use the first estimate in (7.74), that is, νδλ ∞
λ7/4 .

(8.47)

The crucial observation is that under the assumption λρ  1, we can here improve on estimate (7.73) for ν
δλ . Indeed, we shall prove that ν
δλ ∞
ρ −1/12 λ−2/3 .

(8.48)

Under the assumption that this estimate is valid, we obtain by interpolation from (8.47) and (8.48) that Tδλ pc →pc
(λρ)−1/16 , hence the desired remaining estimate  Tδλ pc →pc
1. λρ1

In order to prove (8.48), recall from (8.12) and Lemma 8.3 that  ν
δ (ξ ) = e−iλs3  (x,δ,s) a(x, δ, s) dx2 dx1 , where the complete phase  is given by  (x, δ, s) = x13 B3 (s2 , δ1 , x1 ) − x1 B1 (s, δ1 ) + B0 (s, δ1 ) + φ (x, δ, s2 ), with φ (x, δ, s2 ) = x24 b(x, δ0r , s2 ) + δ4 x22 α˜ 2 (x1 , δ0r , s2 ) + δ3,0 x2 α˜ 1 (x1 , δ0r , s2 ) + δ0 x1 x2 α1,1 (x1 , δ0r , s2 ),

(8.49)

198

CHAPTER 8

and where a is a smooth amplitude supported in a small neighborhood of the origin in x. Recall also from Lemma 8.3 that |α1,1 | ≡ 0 and |α˜ 1 | ∼ 1 in Case ND, whereas |α1,1 | ∼ 1 and α˜ 1 is independent of x1 in Case D. Recall also that sj ∼ 1 for j = 1, 2, 3. Moreover, in Case ND we have 4/3

ρ = δ3,0 + δ42 , whereas in Case D 12/5

ρ = δ0

4/3

+ δ3,0 + δ42 ,

where δ3,0  δ0 . We shall treat both cases ND and D at the same time, assuming implicitly that δ0 = 0 in Case ND. Estimate (8.48) will thus be a direct consequence of the following lemma, which can be derived from more general results by Duistermaat (cf. Proposition 4.3.1 in [Dui74]). For the convenience of the reader, we shall give a more elementary, direct proof for our situation, which requires only C 2 -smoothness of the amplitude. Our approach will be based on arguments similar to the ones used on pages 196–205 in [IKM10]. Lemma 8.8. Denote by J (λ, δ, s) the oscillatory integral  e−iλ(x,δ,s) a(x, δ, s) dx1 dx2 , J (λ, δ, s) = χ1 (s1 , s2 ) with phase (x, δ, s) = x13 B3 (s2 , δ1 , x1 ) − x1 B1 (s, δ1 ) + φ (x1 , x2 , δ, s2 ), where φ is given by (8.49) and where χ1 (s1 , s2 ) localizes to the region where sj ∼ 1, j = 1, 2. Let us also put ρ˜ := ρ + |B1 (s, δ1 )|3/2 . Then the estimate |J (λ, δ, s)| ≤

C ρ˜ 1/12 λ2/3

(8.50)

holds true, provided the amplitude a is supported in a sufficiently small neighborhood of the origin. The constant C in this estimate is independent of δ and s. Proof. Note that ρ˜
1. We may even assume that ρ˜  1. For, if |B1 (s, δ1 )| ∼ 1 and if we choose the support of a sufficiently small in x, then it is easily seen that the phase  has no critical point with respect to x1 on the support of the amplitude, and thus an integration by parts in x1 allows to estimate |J (λ, δ, s)| ≤ Cλ−1 , which is better than what is needed for (8.50). And, if |B1 (s, δ1 )|  1, then we also have ρ˜  1. We begin with the case where ρλ ˜
1. Here we can argue as in the proof of Lemma 8.5: changing coordinates from x to u by putting x = σ1/λ u = (λ−1/3 u1 ,

199

THE REMAINING CASES

λ−1/4 u2 ) and making use of Remark 8.4 (with r := λ), we find that J (λ, δ, s) = λ−1/4−1/3 χ1 (s1 , s2 )    −is3 u31 B3 (s2 ,δ1 ,λ−1/3 u1 )−u1 λ2/3 B1 (s,δ1 )+φλ (u1 ,u2 , δ˜λ ,s2 ) × e × a(σλ−1 u, δ, s) du1 du2 . Observe that here we are integrating over the possibly large domain where |u1 | ≤ ελ1/3 and |u2 | ≤ ελ1/4 . Recall also that φλ is given by (8.16) and that λ ˜
1, and so we have ρ(δ˜ ) = λρ(δ) |δ˜λ |
1 and

λ2/3 |B1 (s, δ1 )|
1.

It is then easily seen by means of integrations by parts in u1 , respectively, u2 (whenever these quantities are large), that the double integral in this expression is uniformly bounded in δ and s, and thus we arrive at the uniform estimate C . λ7/12 This estimate is stronger than estimate (8.50) when ρλ ˜
1. From now on, we may thus assume that  := ρλ ˜  1. We then apply the change of coordinates x = σρ˜ u = (ρ˜ 1/3 u1 , ρ˜ 1/4 u2 ) and find that |J (λ, δ, s)| ≤

˜ δ, s), J (λ, δ, s) = ρ˜ 7/12 I (λρ, where we have put



I (, δ, s) = χ1 (s1 , s2 )

e−i1 (u,δ,s) a(σρ˜ u, δ, s) du1 du2 ,

(  1), (8.51)

with ˜ δ, s2 ) 1 (u, δ, s) := u31 B3 (s2 , δ1 , ρ˜ 1/3 u1 ) − u1 B1 (s, δ1 ) + φ(u, ρ, and φ(u, ρ, ˜ δ, s2 ) := u42 b(σρ˜ u, δ0r , s2 ) + δ4 u22 α˜ 2 (ρ˜ 1/3 u1 , δ0r , s2 )  u2 α˜ 1 (ρ˜ 1/3 u1 , δ0r , s2 ) + δ0 u1 u2 α1,1 (ρ˜ 1/3 u1 , δ0r , s2 ). (8.52) + δ3,0

Here, B1 (s, δ1 ) :=

B1 (s, δ1 )  δ0 δ3,0 δ4  , δ0 := 5/12 , δ3,0 := 3/4 , δ4 := 1/2 , 2/3 ρ˜ ρ˜ ρ˜ ρ˜

so that, in analogy with Remark 8.4, we have  )4/3 + (δ4 )2 + |B1 (s, δ1 )|3/2 = 1. (δ0 )12/5 + (δ3,0

(8.53)

Note that, in particular,  + δ4 + |B1 (s, δ1 )| ∼ 1. δ0 + δ3,0

In order to prove (8.50), we have thus to verify the following estimate: |I (, δ, s)| ≤ C−2/3 .

(8.54)

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CHAPTER 8

Again, take a smooth cutoff function χ0 ∈ C0∞ (R2 ) such that χ0 (u) = 1 for |u| ≤ L, where L is a sufficiently large, fixed positive number, and decompose I (, δ, s) = I0 (, δ, s) + I∞ (, δ, s), with

 I0 (, δ, s) := χ1 (s1 , s2 )

and



I∞ (, δ, s) := χ1 (s1 , s2 )

e−i1 (u,δ,s) a(σρ˜ u, δ, s) χ0 (u) du1 du2 ,

e−i1 (u,δ,s) a(σρ˜ u, δ, s) (1 − χ0 (u)) du1 du2 .

Note that on the support of 1 − χ0 we have |u1 |  L or |u2 |  L. Thus, by choosing L sufficiently large, we see by (8.53) that the phase 1 has no critical point on the support of 1 − χ0 , and in fact we may use integrations by parts in u1 , respectively, u2 , in order to prove that the double integral in the expression for I∞ (, δ, s) is of order O(−1 ), uniformly in δ and s. This is stronger than what is required for (8.54). There remains the integral I0 (, δ, s). Here we use arguments from [IKM10]  , δ4 ) lies on the (compare pp. 203–5). Recall from (8.53) that (B1 (s, δ1 ), δ0 , δ3,0 “unit sphere”   , δ4 ) ∈ R4 : |B1 |3/2 + (δ0 )12/5 + (δ3,0 )4/3 + (δ4 )2 = 1}.  := {(B1 , δ0 , δ3,0  Following [IKM10] let us fix a point ((B1 )0 , (δ0 )0 , (δ3,0 )0 , (δ4 )0 ) ∈ , a point s 0 in the support of χ1 (s1 , s2 ), and a point u0 = (u01 , u02 ) ∈ supp χ0 and denote by η η a smooth cutoff function supported near u0 . By I0 we denote the corresponding oscillatory integral containing η as a factor in the amplitude:      η ˜ e−i2 (u,B1 ,δ0 ,δ3,0 ,δ4 ,ρ,s) a(σρ˜ u, δ, s) χ0 (u) I0 (, δ, s) := χ1 (s1 , s2 )

× η(u) du1 du2 , where  , δ4 , ρ, ˜ s) := u31 B3 (s2 , δ1 , ρ˜ 1/3 u1 ) − u1 B1 + φ(u, ρ, ˜ δ, s2 ), 2 (u, B1 , δ0 , δ3,0 (8.55) with φ as before. η We shall prove that I0 satisfies the estimate η

|I0 (, δ, s)| ≤ C a(·, δ, s) C 2 −2/3 ,

(8.56)

provided η is supported in a sufficiently small neighborhood of U of u , s lies in  , δ4 ) is in a sufficiently a sufficiently small neighborhood S of s 0 , and (B1 , δ0 , δ3,0  0  0  0 small neighborhood V of the point ((B1 ) , (δ0 ) , (δ3,0 ) , (δ4 )0 ) in . The constant C in these estimates may depend on the “base points” u0 , s 0 , and ((B1 )0 ,  )0 , (δ4 )0 ), as well as on the chosen neighborhoods, but not on , δ (δ0 )0 , (δ3,0 and s. By means of a partition-of-unity argument, this will imply the same type of estimate for I0 and, hence, for I , which will conclude the proof of Lemma 8.8. 0

201

THE REMAINING CASES

 Now, if ∇u 2 (u0 , (B1 )0 , (δ0 )0 , (δ3,0 )0 , (δ4 )0 , ρ, ˜ s 0 )  = 0, then by using an integration-by-parts argument in a similar way as for I∞ , we arrive at the same η type of estimate for I0 as for I∞ , which is better than what is required.  )0 , We may therefore assume from now on that ∇u 2 (u0 , (B1 )0 , (δ0 )0 , (δ3,0  0 0 ˜ s ) = 0, and shall distinguish two cases. (δ4 ) , ρ,

Case 1: u01
= 0. In this case, it is easy to see from (8.55) and (8.52) that  ∂u21 2 (u0 , (B1 )0 , (δ0 )0 , (δ3,0 )0 , (δ4 )0 , ρ, ˜ s 0 )  = 0,

provided ρ˜ is sufficiently small. Then, by the implicit function theorem, the phase  , δ4 , ρ, ˜ s) with respect to u1 , which 2 has a unique critical point uc1 (u2 , B1 , δ0 , δ3,0 is a smooth function of its variables, provided we choose the neighborhoods U , and so on, sufficiently small. Indeed, when ρ˜ = 0, then by (8.55) and (8.52),    B1 − δ0 u2 α11 (0, δ0r , s2 ) 1/2 c     . (8.57) u1 (u2 , B1 , δ0 , δ3,0 , δ4 , 0, s) = 3B3 (s2 , δ1 , 0) We may thus apply the method of stationary phase to the integration with respect η to the variable u1 in the integral defining I0 . Let us denote by  the phase function   (u2 , B1 , δ0 , δ3,0 , δ4 , ρ, ˜ s) := 2 (uc1 (u2 , B1 , δ0 , δ3,0 , δ4 , ρ, ˜ s),  , δ4 , ρ, ˜ s), B1 , δ0 , δ3,0

which arises through this application of the method of stationary phase. We claim that then  , δ4 , ρ, ˜ s)|  = 0. max |∂uj2 (u02 , B1 , δ0 , δ3,0

j =4,5

(8.58)

Notice that it suffices to prove this for ρ˜ = 0, since then the result also follows for ρ˜ sufficiently small. In order to prove (8.58) when ρ˜ = 0, we make use of (8.57). Since |B3 | ∼ 1, (8.57) shows that we may assume that |B1 − δ0 u2 α11 (0, δ0r , s2 )| ∼ |uc1 | ∼ |u01 |.

(8.59)

Note also that by (8.57) we have  (u2 , B1 , δ0 , δ3,0 , δ4 , 0, s) = (u2 ) + u42 b(0, δ0r , s2 ) + δ4 u22 α˜ 2 (0, δ0r , s2 )  u2 α˜ 1 (0, δ0r , s2 ), + δ3,0

where we have put  3/2 . (u2 ) := −2 · 3−3/2 B3 (s2 , δ1 , 0)−1/2 B1 − δ0 u2 α11 (0, δ0r , s2 )  In Case ND we have α11 ≡ 0, and thus |∂u42 (u02 , B1 , δ0 , δ3,0 , δ4 , ρ, ˜ s)|  = 0. Next, if we are in Case D, then |α11 | ∼ 1, and (8.59) implies that | (j ) (u2 )| ∼  , |u01 |3/2 |δ0 /u01 |j . Therefore, if δ0  |u01 |, then we find that |∂u42 (u02 , B1 , δ0 , δ3,0 0   ˜ s)|  = 0, and if δ0  |u1 |, then δ4 , ρ,  , δ4 , ρ, ˜ s)|  |u01 |3/2  = 0. |∂u52 (u02 , B1 , δ0 , δ3,0

202

CHAPTER 8

This verifies (8.58) in this case too. But, (8.58) allows us to apply the van der Corput–type Lemma 2.1 to the integration in u2 (after the application of the method of stationary phase in u1 ), and altogether we obtain the estimate η

|I0 (, δ, s)| ≤ C−1/2−1/5 , which is, again, even stronger than what is required by (8.56). Case 2: u01 = 0. Assume first that (δ0 )0  = 0 and |α11 | ∼ 1 (this situation can occur only in Case D). Then  ∂u1 ∂u2 2 (u0 , (B1 )0 , (δ0 )0 , (δ3,0 )0 , (δ4 )0 , 0, s 0 ) = δ0 α11 (0, 0, s20 )  = 0,  ∂u21 2 (u0 , (B1 )0 , (δ0 )0 , (δ3,0 )0 , (δ4 )0 , 0, s 0 ) = 0.

Therefore, we can apply the method of stationary phase to the integration in both variables (u1 , u2 ) and again obtain an estimate of order O(−1 ), which is again stronger than what we need. From now on, we may thus assume that (δ0 )0 = 0 (recall that in Case ND, we have α11 ≡ 0 and are assuming that δ0 = 0 and, hence, also δ0 = 0, so that this assumption is automatically satisfied). Then, necessarily, (B1 )0 = 0, for otherwise, in view of (8.57) we would have |uc1 | ∼ |(B1 )0 |  = 0 when ρ˜ = 0, which would contradict our assumption   )0 , (δ4 )0 ) ∈ , we thus see that ((δ3,0 )0 )4/3 + that u01 = 0. Since ((B1 )0 , (δ0 )0 , (δ3,0 ((δ4 )0 )2 = 1.  )0 , (δ4 )0 ), the function φ satisTherefore, at the “base point” ((B1 )0 , (δ0 )0 , (δ3,0 fies, for ρ˜ = 0, the inequality 3 

|∂uj2 φ(0, u02 , 0, δ, s20 )|  = 0,

j =2

and this inequality will persist for parameters sufficiently close to this base point. Assume first that we have ∂u22 φ(0, u02 , 0, δ, s20 )  = 0. Then we can first apply the method of stationary phase to the u2 integration and, subsequently, van der η Corput’s estimate in u1 (with M = 3), which results in the estimate |I0 (, δ, s)| ≤ −1/2−1/3 . This is again stronger than what we need. C There remains the case where ∂u22 φ(0, u02 , 0, δ, s20 ) = 0 and ∂u32 φ(0, u02 , 0, δ, s20 )  = 0. In this case the phase function 2 is a small smooth perturbation of a function 02 of the form 02 (u1 , u2 ) = c3 u31 + (u2 − u02 )3 b3 (u2 ) + c0 , where c3 := B3 (s20 , δ1 , 0)  = 0 and where b3 (u2 ) is a smooth function such that b3 (u02 )  = 0. This means that 2 has a so-called D4+ -type singularity in the sense of [AGV88] and the distance between the associated Newton polyhedron and the origin is 32 . Estimate (8.56) therefore follows in this situation from the particular case of D4+ -type singularities in Proposition 4.3.1 of [Dui74]. Alternatively, one could also first treat the integration with respect to u1 by means of Lemma 2.2, with B = 3, and subsequently estimate the integration in u2 by means of van der Corput’s lemma (we leave the details to the interested reader). Q.E.D. This concludes the proof of Lemma 8.8.

203

THE REMAINING CASES

Remark 8.9. Notice that our phase  in Lemma 8.8 is a small perturbation of a phase of the form c1 x13 + c2 x24 , with c1  = 0  = c2 , at least if we assume that |B1 (s, δ1 )|  1 (this has been the interesting case in the preceding proof). It is, however, not true that for arbitrary small perturbations of such a phase function, depending, say, on a small perturbation parameter δ > 0, an estimate analogous  to (8.50) of order O c(δ)λ−2/3 as λ → ∞ holds true. A counterexample is given by the function √ 3 (x, δ) := x13 + (x2 − δ)4 + 4δ(x2 − δ)3 − 3 4δ 2 x1 (x2 − δ)2 + C(δ), where C(δ) is chosen such that (0, δ) ≡ 0. Note that (x, 0) = x13 + x24 .  To see this, consider an oscillatory integral J (λ, δ) := eiλ(x,δ) a(x) dx with phase function , whose amplitude is supported in a sufficiently small neighborhood of the origin and such that a(0) = 1. When δ > 0 is sufficiently small, then  has exactly two critical points, namely, the degenerate critical point xd := (0, δ) and the nondegenerate critical point √ xnd := (6 3 2δ 4/3 , −6δ). Let us consider the contribution of the degenerate critical point xd to the oscillatory integral. The linear change of variables √ √ 3 3 z 1 = x1 − 2δ(x2 − δ), z 2 = x1 + 2 2δ(x2 − δ) ˜ + C(δ), where transforms xd into z d = (0, 0) and the phase function into (z) ˜ (z) := z 12 z 2 +



z2 − z1 √ 3 3 2δ

4 .

˜ reveals that the principal face of N () ˜ A look at the Newton polyhedron of  is given by the compact edge [(0, 4), (2, 1)] that lies on the line given by κ1 t1 + κ2 t2 = 1, with associated weight κ = (κ1 , κ2 ) := ( 38 , 14 ), and the principal part of ˜ is given by  ˜ pr (z) = z 12 z 2 + 

z 24 . √ 3 81 16δ 4

Moreover, the Newton distance is given by d = 85 , whereas the nontrivial roots of ˜ pr have multiplicity 1. Therefore, by Theorem 3.3 in [IM11a], the coordinates  ˜ in a sufficiently small neighborhood of the origin, so (z 1 , z 2 ) are adapted to  ˜ in the sense of Varchenko is also given by h = d = 8 . This that the height h of  5 implies that for every sufficiently small, fixed δ > 0, we have that   as λ → ∞, J (λ, δ) = C(δ)λ−5/8 + O λ−7/8 with a nontrivial constant C(δ), because the contribution of the nondegenerate critical point xnd is of order O(λ−1 ) (compare, for instance, [IM11b]). This shows that an estimate of the type |J (λ, δ)| ≤ C(δ)λ−2/3 cannot hold in this example.

204

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8.5 THE CASE WHERE m= 2, B= 4 AND A = 0 ˜ is given by (8.14). Observe Again, we are assuming that λρ  1, where ρ = ρ(δ) r that if A = 0, then according to (7.90) we have h + 1 = 4(n + 3)/(n + 4), where n ≥ 9, so that θc ≤

13 . 48

Here we shall again perform a frequency decomposition near the Airy cone by λ λ defining functions νδ, Ai and νδ, l as follows:   −2/3 λ B1 (s, δ1 ) ν
δλ (ξ ), ν δ, Ai (ξ ) := χ0 ρ  l −2/3  λ B1 (s, δ1 ) ν
δλ (ξ ), ν δ, l (ξ ) := χ1 (2 ρ) so that



λ νδλ = νδ, Ai +

M0 ≤ 2l ≤

ρ −1 , M1

λ νδ, l.

(8.60)

{l:M0 ≤2l ≤ρ −1 /M1 }

Denote by Tδ,λ Ai and Tδ,λ l the corresponding operators of convolution with the Fourier transforms of these functions. 8.5.1 Estimation of Tδ,λ Ai Here we have |ρ −2/3 B1 (s, δ1 )|
1. In this case, we use the change of variables x =: σρ u := (ρ 1/3 u1 , ρ 1/4 u2 ) in the integral (8.12) defining ν
δ and obtain  7/12 −iλs3 B0 (s,δ1 ) e−iλρ s3 1 (u,s,δ) a(σρ u, δ, s) du, (8.61) ν
δ (ξ ) = ρ e |σρ u| 0 is sufficiently small since this will become important soon. We shall proceed in a somewhat similar way as in Section 5.2, by choosing a cutoff function χ0 ∈ C0∞ (R2 ) such that χ0 (u) = 1 for |u| ≤ R, where R will be chosen sufficiently large, and further decomposing ν
δ (ξ ) = ν δ,0 (ξ ) + ν δ,∞ (ξ ),

205

THE REMAINING CASES

where ν δ,0 (ξ ) := ρ

7/12 −iλs3 B0 (s,δ1 )



e

and 7/12 −iλs3 B0 (s,δ1 ) ν e δ,∞ (ξ ) := ρ



e−iλρ s3 1 (u,s,δ) a(σρ u, δ, s)χ0 (u) du,

e−iλρ s3 1 (u,s,δ,s) a(σρ u, δ, s)(1 − χ0 (u)) du.

Accordingly, we decompose λ λ λ νδ, Ai = νδ,0 + νδ,∞ ,

where we have let

 −2/3  λ B1 (s, δ1 ) χ1 (s, s3 ) ν ν δ,0 (ξ ), δ,0 (ξ ) := χ0 ρ

 −2/3  λ B1 (s, δ1 ) χ1 (s, s3 ) ν ν δ,∞ (ξ ). δ,∞ (ξ ) := χ0 ρ Recall from (8.24) that χ1 (s, s3 ) = χ1 (s1 , s2 , s3 ) localizes to the region where  λ λ sj ∼ 1, j = 1, 2, 3. The corresponding operators of convolution with ν δ,0 and νδ,∞ λ λ will be denoted by Tδ,0 and Tδ,∞ , respectively. λ : By means of integrations by parts, we Let us first consider the operators Tδ,∞ easily see that if R is chosen sufficiently large, then the phase will have no critical point, and thus for every N ∈ N, we have 7/12 λ (λρ)−N . ν δ,∞ ∞
ρ

(8.63)

Moreover, by Fourier inversion we find that  λ λ νδ,∞ (x) = λ3 eiλs3 (s1 x1 +s2 x2 +x3 ) ν δ,∞ (ξ ) ds

(8.64)

R3

(with ξ = λs3 (s1 , s2 , 1)). We then use the change of variables from s = (s1 , s2 ) to (z, s2 ), where z := ρ −2/3 B1 (s, δ1 ), and find that (compare (8.9)) (n−1)/(n−2)

s1 = s2

G3 (s2 , δ) − ρ 2/3 z

and, in particular, 1/(n−2)

B0 (s, δ, σ ) = −ρ 2/3 z s2

n/(n−2)

G1 (s2 , δ) + s2

G5 (s2 , δ).

λ And, if we plug the previous formula for ν δ,∞ into (8.64), we see that we may write λ νδ,∞ (x) as an oscillatory integral  λ (x) = ρ 7/12+2/3 λ3 e−iλs3 2 (u,z,s2 ,δ) χ0 (z)(1 − χ0 (u))a(σρ u, ρ 2/3 z, s, δ) νδ,∞

× χ˜ 1 (s2 , s3 ) du dz ds2 ds3

(8.65)

206

CHAPTER 8

with respect to the variables u1 , u2 , z, s2 , s3 , where the complete phase is given by n/(n−2)

2 (u, z, s2 , δ) := s2

(n−1)/(n−2)

G5 (s2 , δ) − x1 s2 G3 (s2 , δ)   1/(n−2) −s2 x2 − x3 + ρ 2/3 z x1 − s2 G1 (s2 , δ) + ρ 1 (u, z, s2 , δ), (8.66)

where, according to (8.62), the phase 1 is given in the new coordinates by 1 (u, z, s2 , δ) = u31 B3 (s2 , δ1 , ρ 1/3 u1 ) − u1 z + u42 b(σρ u, δ0r , s2 ) + δ4 u22 α˜ 2 (ρ 1/3 u1 , δ0r , s2 )  u2 α˜ 1 (δ0r , s2 ) + δ0 u1 u2 α1,1 (ρ 1/3 u1 , δ0r , s2 ). + δ3,0

(8.67)

Recall also from (8.11) that |G5 | ∼ 1 ∼ |G3 | (where G5 := G1 G3 − G2 ). The new amplitude a is again a smooth function of its arguments, and χ˜ 1 (s2 , s3 ) localizes to the region where |s2 | ∼ 1 ∼ |s3 |. Observe also that here |z|
1 and |ρ 1/3 u1 | ≤ ε, |ρ 1/4 u2 | ≤ ε, so that the sum of the last two terms in 2 can be viewed as a small error term of order O(ρ 2/3 + ε), provided |x|
1. First applying N integrations by parts with respect to the variables u1 , u2 and then van der Corput’s lemma to the integration in s2 , we find that λ 7/12+2/3 3 λ (λρ)−N λ−1/3 |νδ, ∞ (x)|
ρ

if |x|
1. However, if |x|  1, then we may argue as in Section 5.2: if |x1 |  |(x2 , x3 )|, then we easily see by means a further integration by parts with respect to the variλ 7/12+2/3 3 λ (λρ)−N λ−1 , and if |x1 |  |(x2 , x3 )|, then ables s2 or s3 that |νδ, ∞ (x)|
ρ λ an integration by parts in z leads to |νδ, ∞ (x)|
ρ 7/12+2/3 λ3 (λρ)−N (λρ 2/3 )−1 . Both these estimates are stronger than the previous one, and so altogether we have shown that λ 7/12+2/3 8/3 λ (λρ)−N . νδ, ∞ ∞
ρ

(8.68)

Interpolating between this estimate and (8.63), we obtain Tδ,λ ∞ pc →pc
ρ 7/12 (λρ)−N ρ 2θc /3 λ8θc /3 , which implies the desired estimate  Tδ,λ ∞ pc →pc
ρ 7/12−2θc ≤ 1, λρ1

. since θc ≤ 13 48 λ λ and the corresponding operators Tδ,0 . First, We next turn to the main terms νδ,0 we claim that 7/12 λ (λρ)−2/3 . ν δ,0 ∞
ρ

(8.69)

This is an immediate consequence of Lemma 8.8. Indeed our phase 1 in (8.62) is of the form as required in this lemma, if we choose the ρ in the lemma of size 1 and

207

THE REMAINING CASES

replace λ in the lemma by λρ. Therefore, the oscillatory integral in the definition λ 1/12 of ν (λρ)2/3 ). δ,0 can be estimated by C/(1 λ ∞ . In analogy with (8.65), we may write Finally, we want to estimate νδ,0 λ νδ,0 (x) as an oscillatory integral of the form  λ 7/12+2/3 3 e−iλs3 2 (u,z,s2 ,δ) λ νδ,0 (x) = ρ × χ0 (z)χ0 (u)a(σ ˜ ρ u, ρ 2/3 z, s, δ)χ˜ 1 (s2 , s3 ) du dz ds2 ds3 , with 2 given by (8.66). We can reduce this to the following situation, in which the amplitude is independent of z, that is, where  λ 7/12+2/3 3 e−iλs3 2 (u,z,s2 ,δ) νδ,0 (x) = ρ λ × χ0 (z)χ0 (u)a(σρ u, s, ρ 1/3 , δ)χ˜ 1 (s2 , s3 ) du dz ds2 ds3 .

(8.70)

In fact, we may develop the amplitude a˜ into a convergent series of smooth functions, each of which is a tensor product of a smooth function of the variable z with a smooth function depending on the remaining variables only. Thus, by considering each of the corresponding terms separately, we can reduce to the situation (8.70) (the function χ0 (z) will of course have to be different from the previous one). We claim that λ νδ,0 ∞
ρ 7/12 λ2 (λρ)−1/4 .

(8.71)

λ νδ,∞ (x),

λ |νδ,0 (x)|

Indeed, if |x|  1, arguing in a similar way as for we see that
ρ 7/12+2/3 λ3 (λρ 2/3 )−N for every N ∈ N, which is stronger than what is needed for (8.71). Therefore, from now on we shall assume that |x|
1. For such x fixed, we can argue in a similar way as in in Section 5.2 (compare also with the discussion in Subsection 8.7.2): we decompose λ λ λ = ν0,I + ν0,I νδ,0 I, λ ν0,I

where and given by

λ ν0,I I

(8.72)

denote the contributions to the integral (8.70) by the region LI 1/(n−2)

|x1 − s2

G1 (s2 , δ)|  ρ 1/3 ,

and the region LI I where 1/(n−2)

|x1 − s2

G1 (s2 , δ)|
ρ 1/3 ,

respectively. Recall from (8.66) and (8.67) that n/(n−2)

2 (u, z, s2 , δ) = s2

(n−1)/(n−2)

G5 (s2 , δ) − x1 s2

+ zρ(ρ −1/3 (x1 − s2

G3 (s2 , δ) − s2 x2 − x3

1/(n−2)

+ ρu31

B3 (s2 , δ1 , ρ

1/3

G1 (s2 , δ)) − u1 )  u1 ) + ρ u42 b(σρ u, δ0r , s2 )

+ δ4 u22

 α˜ 2 (ρ 1/3 u1 , δ0r , s2 ) + δ3,0 u2 α˜ 1 (δ0r , s2 )  + δ0 u1 u2 α1,1 (ρ 1/3 u1 , δ0r , s2 ) .

(8.73)

208

CHAPTER 8 1/(n−2)

Let us change variables from s2 first to v := x1 − s2 G1 (s2 , δ) and then to 1/(n−2) w := ρ −1/3 v = ρ −1/3 (x1 − s2 G1 (s2 , δ)). In these new coordinates, 2 can be written as 2 = zρ(w − u1 ) + 3 , with 3 of the form 3 (u, w, x, δ) = 3 (ρ 1/3 w, x, δ) + ρu31 B3 (ρ 1/3 w, ρ 1/3 u1 , x, δ)  + ρ u42 b(σρ u, ρ 1/3 w, x, δ0r ) + δ4 u22 α˜ 2 (ρ 1/3 u1 , ρ 1/3 w, x, δ0r )

  + δ3,0 u2 α˜ 1 (ρ 1/3 w, x, δ0r ) + δ0 u1 u2 α1,1 (ρ 1/3 u1 , ρ 1/3 w, x, δ0r ) , (8.74)

where 3 is a smooth, real-valued function. With a slight abuse of notation, we have here used the same symbols B3 , b, . . . , α1,1 as before, since these functions will have the same basic properties here as the corresponding ones in (8.73). A similar remark will apply to the amplitudes, which we shall always denote by the letter a, even though they may change from line to line. Moreover, we may write  λ (x) = (ρ 7/12+2/3 λ3 ) ρ 1/3 e−iλs3 3 (u,w,x,δ) χ
0(λρs3 (w − u1 )) (1 − χ0 (w)) χ0 (u) ν0,I (8.75) × a(σρ u, ρ 1/3 w, s1 , ρ 1/3 , x, δ)χ˜ 1 (s3 ) dw du ds3 .  λ 7/12+2/3 3 λ ) ρ 1/3 e−iλs3 3 (u,w,x,δ) χ
0 (λρs3 (w − u1 )) χ0 (w) χ0 (u) ν0,I I (x) = (ρ × a(σρ u, ρ 1/3 w, s1 , ρ 1/3 , x, δ)χ˜ 1 (s3 ) dw du ds3 .

(8.76)

Here, χ0 (w) will again denote a smooth function with compact support, which is identically 1 on a sufficiently large neighborhood of the origin. Observe that in (8.75) we have |w|  1  |u1 |, so that |χ
0 (λρs3 (w − u1 )) | ≤ CN (λρ|w|)−(N+1) for every N ∈ N, and we immediately obtain the estimate λ ν0,I (x) ∞ ≤ CN (ρ 7/12+2/3 λ3 ) ρ 1/3 (λρ)−(N+1) = CN ρ 7/12 λ2 (λρ)−N ,

(8.77)

which is even stronger than (8.71). In order to estimate the second term, we perform yet another change of variables from u1 to y1 so that u1 = w − (λρ)−1 y1 , that is, y1 = λρ(w − u1 ). This leads to λ the following expression for ν0,I I (x) : λ 7/12+2/3 3 λ ) ρ 1/3 (λρ)−1 ν0,I I (x) = (ρ



e−iλs3 4 (y1 ,u2 ,w,x,δ) χ
0 (s3 y1 )

× χ0 (w) χ0 (w − (λρ)−1 y1 )   × χ0 (u2 ) a4 (λρ 2/3 )−1 y1 , ρ 1/4 u2 , w, s1 , ρ 1/3 , x, δ × χ˜ 1 (s3 ) dy1 du2 dw ds3 ,

(8.78)

209

THE REMAINING CASES

with phase 4 of the form

 3   4 (y1 , u2 , w, x, δ) = 3 ρ 1/3 w, x, δ + ρ w − (λρ)−1 y1   ×B˜ 3 ρ 1/3 w, (λρ 2/3 )−1 y1 , x, δ    + ρ u42 b ρ 1/3 w, (λρ 2/3 )−1 y1 , ρ 1/4 u2 , x, δ0r   + δ4 u22 α˜ 2 ρ 1/3 w, (λρ 2/3 )−1 y1 , ρ 1/4 u2 , x, δ0r      × δ3,0 u2 α˜ 1 ρ 1/3 w, x, δ0r + δ0 u2 w − (λρ)−1 y1   × α1,1 ρ 1/3 w, (λρ 2/3 )−1 y1 , x, δ0r . (8.79)

Observe that in this integral, |u2 | + |w|
1 and |y1 |
λρ. Moreover, the factor χ
0 (s3 y1 ) guarantees the absolute convergence of this integral with respect to the variable y1 . We can thus first apply van der Corput’s estimate of order M = 4 for the integration in u2 , which leads to an additional factor of order (λρ)−1/4 , and then perform the remaining integrations in w, y1 , and s3 . Altogether, this leads to the estimate λ (x) ∞ ≤ C(ρ 7/12+2/3 λ3 ) ρ 1/3 (λρ)−1 (λρ)−1/4 = ρ 7/12 λ2 (λρ)−1/4 . ν0,I

In combination with (8.77), this proves (8.71). Finally, interpolating between the estimates (8.69) and (8.71), we obtain λ pc →pc
ρ 7/12−2θc (λρ)29θc /12−2/3 . Tδ,0 7 , we have 29 θ − 23 < 0 and 12 − 2θc > 0, which implies the But, since θc ≤ 13 48 12 c desired estimate,  λ Tδ,0 pc →pc
ρ 7/12−2θc ≤ 1. λρ1

Altogether, we have thus proved that  λ Tδ,Ai pc →pc
1.

(8.80)

λρ1

8.5.2 Estimation of Tδ,λ l Here we have |(2l ρ)−2/3 B1 (s, δ1 )| ∼ 1. Recall also that 2l ρ ≤ 1/M1  1. In this case, we use the change of variables x =: σ2l ρ u := ((2l ρ)1/3 u1 , (2l ρ)1/4 u2 ) in the integral (8.12) defining ν
δ , and obtain  l e−iλ2 ρ s3 1 (u,s,δ) a(σ2l ρ u, δ, s) du, ν
δ (ξ ) = (2l ρ)7/12 e−iλs3 B0 (s,δ1 ) |σ2l ρ u| 0 is sufficiently small. Observe that the second and third rows in (8.81) are a small perturbation of the leading term, given by the first row. Again, we choose a cutoff function χ0 ∈ C0∞ (R2 ) such that χ0 (u) = 1 for |u| ≤ R, where R will be chosen sufficiently large, and further decompose ν
δ (ξ ) = ν δ,0 (ξ ) + ν δ,∞ (ξ ), where now l

ν δ,0 (ξ ) := (2 ρ)

7/12 −iλs3 B0 (s,δ1 )

e



l

|σ2l ρ u| 4λβk(2) |αk(2) | > · · · > 4n−1 λβk(n) |αk(n) |. This implies that n n−1       λβk αk  ≥ λβk(1) |αk(1) | 1 − 4−l ≥ 23 λβk(1) |αk(1) | ≥ 23 . k=1

l=1

229

THE REMAINING CASES

And, since

βl j −1 j ∈Z:2βl j |αl |≥1 (2 |αl |)

≤ (1 − 2−βl )−1 , we obtain (8.126), with

C2 (β1 , . . . , βn ) :=

1 3 . n! max k 1 − 2−βk 2 Q.E.D.

In order to prove (8.122), let us consider   e−is3 (y,z,w,x,δ,λ,l) χ1 (s3 ) ds3 . F (t, y, z, w, x, δ, l) := λ−2it 2M ≤λ≤2M ρ −1

We shall prove that |F (t, y, z, w, x, δ, λ, l)| ≤ C

2l (1 + |w|3 ) , |2−i2t − 1|

(8.127)

with a constant C not depending on t, y, z, x, δ and l. By choosing N in (8.123) sufficiently large, we see that this estimate will imply (8.122), provided γ (ζ ) contains a factor γ4 (ζ ) :=

22(1−ζ ) − 1 . 3

Let us put β3 := 1, β2 := 23 , β1 := 13 and, given y, w, x, δ and l, α3 := h3 (y, x, δ),     α2 := 2l/3 h2 y, x, δ w, α1 := 22l/3 h1 y, x, δ w 2 . Accordingly, we set  1 = 1 (y, w, x, δ, l) := λ = 2j : 2M ≤ λ ≤ 2M ρ −1 and        max{λ|h3 y, x, δ |, λ2/3 |2l/3 h2 y, x, δ w|, λ1/3 22l/3 |h1 y, x, δ w 2 |} ≥ 1   = λ = 2j : 2M ≤ λ ≤ 2M ρ −1 and max λβk |αk | ≥ 1 k=1,...,3

and

 2 = 2 (y, w, x, δ, l) := λ = 2j : 2M ≤ λ ≤ 2M ρ −1 and        max{λ|h3 y, x, δ |, λ2/3 |2l/3 h2 y, x, δ w|, λ1/3 22l/3 |h1 y, x, δ w 2 |} < 1 .

We also denote by e ⊂  the set of exceptional λs given by Lemma 8.14 for this choice of βk and αk . Correspondingly, we decompose F = Fe + F1 + F2 , where Fe , F1 and F2 are defined as F, only with summation over the dyadic λs restricted to the subsets e , 1 \ e and 2 , respectively. For Fe , we then trivially get the estimate |F (t, y, z, w, x, δ, λ, l)| ≤ C, since the cardinality of e is bounded by a constant not depending on the arguments of Fe . Next, in order to estimate F1 , let us rewrite   β1 β2 β3 e−is3 (y,z,w,x,δ,λ,l) χ1 (s3 ) ds3 = e−is3 (λ α1 +λ α2 +λ α3 )   × e−is3 0 (y,z,w,x,δ,λ,l) χ1 (s3 ) ds3 ,

230

CHAPTER 8

where 0 denotes the phase    −l 0 (y, z, w, x, δ, λ, l) := 2l (w − y1 )z + g1 y, (2l λ−1 )1/3 w, x, (2l λ−1 )1/3 , δ˜2 λ , δ    + h0 y, (2l λ−1 )1/3 w, x, δ w3 . Observe that |0 (y, z, w, x, δ, λ, l)| ≤ C2l (1 + |w|). Integrating by parts in s3 therefore shows that    2l (1 + |w|)  e−is3 (y,z,w,x,δ,λ,l) χ1 (s3 ) ds3  ≤ C . |λβ1 α1 + λβ2 α2 + λβ3 α3 | We may thus control the sum over all λ ∈ 1 by means of Lemma 8.14 and obtain the estimate |F1 (t, y, z, w, x, δ, λ, l)| ≤ C2l (1 + |w|). Finally, F2 can again be estimated by means of Lemma 2.7. Indeed, observe that in the sum defining F2 (t, y, z, w, x, δ, λ, l), the expressions   −l (2l λ−1 )1/3 w, (2l λ−1 )1/3 , δ˜2 λ , λh3 y, x, δ ,     λ2/3 2l/3 h2 y, x, δ w, λ1/3 22l/3 h1 y, x, δ w 2 are all uniformly bounded. Therefore, we may let    l 3 H (u1 , . . . , u6 ) := e−is3 2 [(w−y1 )z+g1 (y,u1 ,x,u2 ,u3 ,δ)+h0 (y,u1 ,x,δ)w ]+u4 +u5 +u6 ×χ1 (s3 ) ds3 , with the ak in Lemma 2.7 given by a1 := 2l/3 2w , a2 := 2l/3 , . . . , a4 := h3 (y, x, δ), a5 := 2l/3 h2 (y, x, δ)w, a6 := 22l/3 h1 (y, x, δ)w2 , and the obvious corresponding cuboid Q. Then clearly H C 1 (Q) ≤ C2l (1 + |w|3 ), and thus Lemma 2.7 yields the estimate |F2 (t, y, z, w, x, δ, λ, l)| ≤ C

2l (1 + |w|3 ) . |2−i2t − 1|

This concludes the proof of estimate (8.127) and, thus, also of (8.122). λ 8.7.5 Contribution by the νl,I II λ The contribution of the terms νl,I I I in (8.106) can be treated in a very similar way as λ the one by the terms νl,I . Indeed, arguing as before, we here arrive at the following

231

THE REMAINING CASES λ expression for µl,λ,I I I := 2−l/3 λ−4/3 νl,I II : (4/3−N)l



µl,λ,I I I (x) = 2

e−is3 (y,z,w,x,δ,λ,l)

  × a σ2l λ−1 y, (2l λ−1 )1/3 , z, (2l λ−1 )1/3 w, δ χ1 (s3 )   w χ0 (y)χ1 (y1 ) × χ0 (2l λ−1 )1/3 w χ0 ε 1 dy1 dy2 dz ds3 dw. × χ1 (z) (w − y1 )N

(8.128)

The phase  is still given by (8.124). Notice that now |w|  1 ∼ u1 . The arguments used in the preceding subsection therefore carry over to this case, with minor modifications (even simplifications). λ 8.7.6 Contribution by the νl,∞

Let us next look at ν∞,1+it := γ (1 + it)





2M ≤λ≤2M ρ −1

{l:M0 ≤2l ≤λ/M1 }

2−itl/2 λ−2it µl,λ,∞ (x),

(8.129)

λ . We want to prove that where we have set µl,λ,∞ := 2−l/3 λ−4/3 νl,∞

|ν∞, 1+it (x)| ≤ C

∀t ∈ R, x ∈ R3 .

(8.130)

λ To this end, recall formulas (8.33) and (8.34) for νl,∞ (x). From these formulas, λ −lN −N it is easy to see that |νl,∞ (x)|
2 λ if |x|  1, and thus summation in l and λ is no problem in this case. So, assume that |x|
1. Then the second term in the phase (z, s2 , δ) in (8.34) can be absorbed into the amplitude aN,l , and we arrive at an expression of the following form for µl,λ,∞ (x):  n/(n−2) (n−1)/(n−2) G5 (s2 ,δ)−x1 s2 G3 (s2 ,δ)−s2 x2 −x3 ) µl,λ,∞ (x) = 2−lN λ1/3 e−is3 λ(s2 −l × aN,l (z, s2 , s3 , δ0r , δ˜2 λ , (2−l λ)−1/3 , λ−1/9 )

× χ1 (z) χ1 (s2 )χ1 (s3 ) dz ds2 ds3 , where aN,l is a smooth function of all its (bounded) variables such that aN,l C k n/(n−2) is uniformly bounded in l. Denote by (s2 ) = (s2 , x, δ) = s2 G5 (s2 , δ) − (n−1)/(n−2) x1 s2 G3 (s2 , δ) − s2 x2 − x3 the phase appearing in this integral. We can now argue in a similar way as in Subsection 8.7.1. Since s2 ∼ 1 in the integral, we see that if |x1 |  1, then |∂s22 (s2 )| ∼ 1, and van der Corput’s estimate implies that |µl,λ,∞ (x)|
2−lN λ1/3−1/2 . We can then sum the series (8.129) absolutely and arrive at (8.130). Let us therefore assume from now on that |x1 | ∼ 1 and that the sign of x1 is such that there is a point s2c (x, δ) ∼ 1 such that ∂s22 (s2c (x, δ), x, δ) = 0. This point is then unique, by the implicit function

232

CHAPTER 8

theorem, since |∂s32 (s2c (x, δ), x, δ)| ∼ 1. Changing coordinates from s2 to v := s2 − s2c (x, δ) and applying a Taylor expansion in v, we see that the phase can be written in the form Q3 (v, x, δ) v 3 − Q1 (x, δ) v + Q0 (x, δ), with smooth functions Qj . Scaling in v by the factor λ−1/3 then leads to an expression of the following form for µl,λ,∞ (x):  −1/3 3 2/3 −lN e−is3 (Q3 (λ w,x,δ) w +λ Q1 (x,δ) w+λQ0 (x,δ)) µl,λ,∞ (x) = 2   −l × aN,l z, λ−1/3 w, s3 , δ0r , δ˜2 λ , (2−l λ)−1/3 , λ−1/9 χ1 (z) χ0 (λ−1/3 w) × χ1 (s3 ) ds3 dw dz.

(8.131)

Performing first the integration in s3 , this easily implies the following estimate:   λ1/3   |µl,λ,∞ (x)|
2−lN 1 + Q3 (λ−1/3 w, x, δ) w 3 −λ1/3

−N + λ2/3 Q1 (x, δ) w + λQ0 (x, δ) χ1 (z) χ0 (λ−1/3 w) dw dz. Putting A := λQ0 (x, δ), B := λ2/3 Q1 (x, δ), and T := λ1/3 in Lemma 2.4, we then find that |µl,λ,∞ (x)|
2−lN max{|A|1/3 , |B|1/2 }−1/2 . This estimate allows us to sum over all λ such that max{λ|Q0 (x, δ)|, λ2/3 |Q1 (x, δ)|} > 1, even absolutely. There remains the summation over all λ such that λ|Q0 (x, δ)| ≤ 1 and λ2/3 |Q1 (x, δ)| ≤ 1. However, in view of (8.131), this sum can easily be controlled by means of Lemma 2.7, as we have done in many similar cases before, and we shall therefore skip the details. Altogether, we arrive at (8.130). λ 8.7.7 Contribution by the νl,00 λ We finally come to the contribution of the terms νl,00 in (8.106). Recall from Subsection 8.3.2 that −l −2/3 λ χ1 (s, s3 ) χ1 ((2−l λ)2/3 B1 (s, δ1 )) e−is3 λB0 (s,δ1 ) ν l,00 (ξ ) := (2 λ)  u  l 1 × e−is3 2 (u1 ,u2 ,s,δ,λ,l) a(σ2l λ−1 u, δ, s) χ0 (u)χ0 du1 du2 , ε

where we assume ε > 0 to be sufficiently small. Moreover, the phase  is given by (8.28) and (8.29), with B = 3. Since |δ˜2

−l

λ

|  1 and

(2−l λ)2/3 |B1 (s, δ1 )| ∼ 1,

we see that we can again integrate by parts in u1 in order to gain factors 2−lN , and λ then the same type of argument that led to expression (8.30) for ν l,∞ (ξ ) can be

233

THE REMAINING CASES

applied in order to see that an analogous expression, −lN −2/3 λ λ χ1 (s, s3 ) χ1 ((2−l λ)2/3 B1 (s, δ1 )) e−is3 λB0 (s,δ1 ) ν l,00 (ξ ) = 2 −l × a˜ N,l ((2−l λ)2/3 B1 (s, δ1 ), s, s3 , δ˜2 λ , δ0r , (2−l λ)−1/3 , λ−1/3B ),

λ can be obtained for ν ˜ N,l is again a smooth function of all its l,00 (ξ ) too, where a (bounded) variables such that aN,l C k is uniformly bounded in l. From here on, λ . we can argue exactly as for νl,∞ This concludes the proof of the second estimate in (8.101) and, thus, the proof of part (a) of Proposition 8.12 as well.

8.8 PROOF OF PROPOSITION 8.12(b): COMPLEX INTERPOLATION In this section, we assume that B = 3 and A = 0 in (7.89) and that λρ  1. III 8.8.1 Estimation of Tδ,Ai III As usual, we embed νδ,Ai into an analytic family of measures   −4/5 5(1−3ζ )/6 λ III νδ, ρ (λρ) νδ,0 , ζ := γ (ζ ) 2M ρ −1 1. We then decompose where F1 (t, w, x, δ) and F (t, w, x, δ) = F1 (t, w, x, δ) + F2 (t, w, x, δ), F2 (t, w, x, δ) are defined like F , only with summation restricted to the subsets (w, x, δ) and  \ (w, x, δ), respectively. Then, by (8.138), we clearly have that  |µλ (w, x)| ≤ C, |F1 (t, w, x, δ)| ≤ λ∈(w,x,δ)

and we are thus left with F2 (t, w, x, δ). In the corresponding sum, we have λ |QA (ρ 1/3 w, x, δ)| ≤ 1 and λ2/3 ×|QB (ρ 1/3 w, x, δ)| ≤ 1, and therefore F2 can again be estimated by means of Lemma 2.7. Indeed, we may here put H (u1 , . . . , u6 )  r 3 1/3 1/3 ˜ 1/3 := e−is3 (u1 +u2 y2 +b(ρ w,u3 y1 ,u4 y2 ,x,δ0 ) y2 +R1 (w,u5 y1 ,ρ ,x,δ) y1 +u6 y2 R2 (w,u5 y1 ,ρ ,x,δ))   × χ
0 (s3 y1 ) χ0 (w − u5 y1 )χ0 (u6 y2 ) a5 u3 y1 , u4 y2 , w, s1 , ρ 1/3 , x, δ × χ˜ 1 (s3 ) ds3 dy1 dy2 , where the variables u1 , . . . , u6 correspond to the bounded expressions λQA (ρ 1/3 w, x, δ), λ2/3 QB (ρ 1/3 w, x, δ), (λρ 2/3 )−1 , λ−1/3 , (λρ)−1 , and (λρ)−1/3 , respectively. By means of integrations by parts in the variable y2 for |y2 |  1 (or, alternatively, in s3 ), it is then easily verified that H C 1 (Q) ≤ C, where Q denotes the obvious cuboid Q appearing in this situation. Thus, estimate (8.139) follows from Lemma 2.7. 8.8.2 Estimation of TδI V The estimation of the operator TδI V will follow similar ideas as the one for TδI I . Nevertheless, for the convenience of the reader, we will give some details. As usually, we embed νδI V into an analytic family of measures    l −4/5 l 5(1−3ζ )/6 λ IV (2 ρ) := γ (ζ ) (λ2 ρ) νl,0 , νδ,ζ {l:M0 ≤2l ≤ρ −1 /M1 } 2M ρ −1 dx = d, which would contradict the maximality of dx . Thus, necessarily, κ2 /κ1 ≥ 2. Conversely, assume without loss of generality that κ2 /κ1 ≥ 2. Consider any ma
a b trix T = c d ∈ GL(2, R) and the corresponding linear coordinates y given by x1 = ay1 + by2 ,

x2 = cy1 + dy2 .

To prove (a), we have to show that dy ≤ dx for all such matrices T Case 1. a
= 0. Then we may factor T = T1 T2 , where    a 0 1 T1 :=  ad − bc  , T2 :=  c 0 a

 b a . 1

We first consider T2 . Since φpr (T2 y) = φκ (y1 + ab y2 , y2 ) and since y2 is κ-homogenous of degree κ2 > κ1 , whereas y1 is κ-homogenous of degree κ1 , we see that the κ-principal part of φ ◦ T2 is given by (φ ◦ T2 )κ = φκ , so that φ ◦ T2 and φ have the same principal face and, in particular, the same Newton distance. This shows that we may assume without loss of generality that b = 0. Then, necessarily, d  = 0. But then our change of coordinates is of the type x1 = ay1 , x2 = cy1 + dy2 considered in Lemma 3.2 of [IM11a], so that this lemma implies that dy ≤ dx . Indeed, one finds more precisely that dy < dx , if c  = 0, and dy = dx otherwise. Case 2. a = 0, d = 0. Since separate scalings of the coordinates have no effect on the Newton polyhedra, T then essentially interchanges the roles of x1 and x2 , that

245

PROOFS OF PROPOSITIONS 1.7 AND 1.17

is, the Newton polyhedron is reflected at the bisectrix under this coordinate change. This shows that here dy = dx .
 0 b Case 3. a = 0, d
= 0. Then we may factor T = c d = T1 T2 , where     0 1 c d T1 := , T2 := . 1 0 0 b We have seen in the previous cases that both T1 and T2 do not change the Newton distance, and thus here dy = dx . This concludes the proof of the first part of Proposition 1.7. Assume finally that x and y are two linearly adapted coordinate systems for φ for which the corresponding principal weights κ and κ  satisfy κ2 /κ1 > 1 and κ2 /κ1 > 1, respectively. Choose T ∈ GL(2, R) such that x = T y. Inspecting the three cases from the previous argument, we see that in Case 1 the mapping T2 does not change the principal face and that necessarily c = 0, since otherwise we had dy < dx . But then T1 also does not change the principal face. Case 2 cannot arise here, since we assume that both κ2 /κ1 > 1 and κ2 /κ1 > 1, and similarly Case 3 cannot apply. This proves also the second statement in the proposition. Q.E.D.

9.2 APPENDIX B: A DIRECT PROOF OF PROPOSITION 1.17 ON AN INVARIANT DESCRIPTION OF THE NOTION OF r-HEIGHT We shall prove both parts (a) and (b) of Proposition 1.17 at the same time. So, let us assume that our coordinates (x1 , x2 ) are linearly adapted to φ, and let f (x1 ) be any nonflat fractionally smooth, real-valued function of x1 , say for x1 > 0, with corresponding fractional shear y1 := x1 ,

y2 := x2 − f (x1 )

+

on the half plane H . In order to prove Proposition 1.17, what still remains to be shown is that d(φ), if the coordinates (x1 , x2 ) are adapted to φ, f h (φ) ≤ (9.1) r h (φ), if the coordinates (x1 , x2 ) are not adapted to φ, without making recourse to the Theorems 1.5 and 1.14, as we did in our “indirect” proof of this proposition in Section 1.5. Indeed, the reverse inequalities had already been shown directly in Section 1.5. Recall from (1.19) and (1.9) that c0 x1m0 denotes the leading term in the Puiseux series expansion of f (x1 ), whereas b1 x1m denotes the leading term of the principal root jet ψ(x1 ); m0 is rational, and m is an integer. In analogy with the definition of the supporting line Lf to the Newton polyhedron N (φ f ), we choose the weight κ 0 = (κ10 , κ20 ) so that κ20 /κ10 = m0 and so that the line L0 := {(t1 , t2 ) ∈ R2 : κ10 t1 + κ20 t2 = 1}

246

CHAPTER 9

is a supporting line to the Newton polyhedron N (φ) of φ. Then N (φ) ∩ L0 is an interval of the form [T1 , T2 ], whose left endpoint we shall denote by T1 = (A, B). Note that if T1 = T2 , then [T1 , T2 ] is a vertex of N (φ) and otherwise is a compact edge. We shall distinguish three cases, which will depend on the relative position of [T1 , T2 ] to the bisectrix  = {t1 = t2 }. The case where [T1 , T2 ] ⊂ {t1 ≤ t2 }. In this case, we will prove that hf (φ) = d f (φ) ≤ d(φ),

(9.2)

where d f (φ) := d f . This clearly will imply (9.1). Indeed, we shall show that the Newton polyhedron N (φ f ) has a compact edge of the form [T1 , T˜2 ] ⊂ L0 , where T1 lies on or above the bisectrix, whereas T˜2 lies on or below the bisectrix. In particular, this will imply that Lf = L0 and that the augmented Newton polyhedron N f (φ f ) will have only one edge within the half space bounded from below by the bisectrix, and this edge lies again on the line Lf . By the definition (1.20) of hf (φ) and its geometric interpretation in terms of the augmented Newton polyhedron N f (φ f ), this will mean that hf (φ) = d f . It will be useful to recall here that the second coordinate of the point of intersection of the line (m0 ) with the line Lf is given by d f + 1. On the other hand, by the definition of d f , (d f , d f ) is the point of intersection of the line Lf with the bisectrix , and thus it obvious from the geometry of the Newton polyhedron N (φ f ) that d f ≤ d, and we arrive at (9.2). We begin with the case where N (φ) ∩ L0 is a vertex, that is, where T1 = T2 = (A, B). Since the point (A, B) lies on or above the bisectrix, we have B ≥ A. Moreover, the κ 0 -principal part of φ is of the form φκ 0 (x1 , x2 ) = cx1A x2B , and it is then easily seen that the κ 0 -principal part of φ f (y1 , y2 ) = φ(y1 , y2 + f (y1 )) is given by the fractionally smooth function cy1A (y2 + c0 y1m0 )B (compare the proof of Lemma 3.2 in [IM11a]). So, the associated edge of the Newton polyhedron N (φ f ) is the closed interval [T1 , T˜2 ] ⊂ L0 , with T1 = (A, B) lying above the bisectrix and T˜2 := (A + m0 B, 0) lying below the bisectrix. Hence, the interval [T1 , T˜2 ] is, in fact, the principal face of the Newton polyhedron N (φ f ), which proves our claim in this case. Let us next consider the case where N (φ) ∩ L0 is a compact edge γ of N (φ). By Proposition 2.2 in [IM11a], we may then assume that κ10 :=

q , k

κ20 :=

p , k

(p, q, k) = 1,

(p, q) = 1,

and that the corresponding polynomial φκ 0 can be factored as φκ 0 := cx1ν1 x2ν2

M

q

p

(x2 − λl x1 )nl ,

l=1

and nl ∈ N \ {0} and with ν1 , ν2 ∈ N \ {0}. We with M ≥ 1, distinct λl ∈ C \ {0}, 0 0 let n := M l=1 nl . Then m0 = κ2 /κ1 = p/q, and the edge γ is given by the interval γ = [(ν1 , ν2 + nq), (ν1 + np, ν2 )]. Since the edge γ is contained in the half space where t2 ≥ t1 , we have ν2 ≥ ν1 + np.

247

PROOFS OF PROPOSITIONS 1.7 AND 1.17 p/q

If c0 x1m0 = c0 x1 we have

does not coincide with any root of the polynomial φκ 0 , then

f

φκ 0 (y1 , y2 ) = ((φκ 0 )f )κ 0 (y1 , y2 ) p/q

= cy1ν1 (y2 + c0 y1 )ν2

M


p/q

p

(y2 + c0 y1 )q − λl y1

nl

.

l=1

So, as before, the principal face of the Newton polyhedron N (φ f ) will be an interval [T1 , T˜2 ] ⊂ L0 , with T1 = (ν1 , ν2 + nq) = (A, B) and T˜2 := (ν1 + ν2 p/q + np, 0) lying on the first coordinate axis, so that again this interval is the principal face of the Newton polyhedron N (φ f ). Assume next that c0 x1m0 does coincide with some real root of the polynomial 1/q function φκ 0 , say, c0 = λl0 . Then we have nl

p/q

((φκ 0 )f )κ 0 (y1 , y2 ) = cy1ν1 (y2 + c0 y1 )ν2 y2 0

q−1




p/q

y2 − c0 (εj − 1)y1

nl0

j =1

×

M


p/q

p

(y2 + c0 y1 )q − λl y1

nl

,

l =l0

where {εj }j =1,...,q−1 denotes the set of qth roots of unity that are different from 1. This shows that the principal edge of the Newton polyhedron N (φ f ) is now given by the interval [T1 , T˜2 ] ⊂ L0 , with T1 = (ν1 , ν2 + nq) = (A, B) as before and now with
 ν2 p (q − 1)pnl0 + + (n − nl0 )p, nl0 . T˜2 := ν1 + q q Observe that T1 lies above the bisectrix, whereas T˜2 lies on or below the bisectrix, which again proves our claim. Indeed, since n ≥ nl0 and ν2 ≥ ν1 + np, we have ν2 p (q − 1)pnl0 (q − 1 + p) + ≥ pnl0 ≥ nl0 . q q q The case where [T1 , T2 ] ⊂ {t1 ≥ t2 }. Then A ≥ B, and so the principal face of N (φ f ) agrees with the principal face of N (φ). Arguing in a similar way as in the preceding case, we see that again L0 = Lf , but this time Lf will touch the Newton polyhedron of φ f only in points lying on or below the bisectrix. This shows that the line (m0 ) will intersect the boundary of the augmented Newton polyhedron N f (φ f ) at some point of the line Lf , so that again hf (φ) = d f (φ). Moreover, from the geometry of the lines L and Lf , it is again clear that d f (φ) ≤ d(φ); that is, (9.1) does hold true also in this case. There remains the case where T1 lies strictly above the bisectrix and T2 strictly below it. Then clearly the compact edge [T1 , T2 ] is the principal face π(φ) ⊂ L of N (φ), and necessarily we have L0 = L and m0 = m. What remains is the following case. The case where π(φ) is a compact edge given by [T1 , T2 ]. Let us first assume that the coordinates (x1 , x2 ) are adapted to φ and denote by κ the principal weight

248

CHAPTER 9

associated to the principal face π(φ) of the Newton polyhedron of φ, so that π(φ) is contained in the principal line L = {(t1 , t2 ) ∈ R2 : κt1 + κ2 t2 = 1}. Since the coordinates of φ are adapted to φ, Proposition 1.2 shows that the κ-principal part φκ has only real roots of multiplicity ≤ d(φ). By a similar reasoning as in the case where [T1 , T2 ] ⊂ {t1 ≤ t2 }, we then see that the Newton polyhedron N (φ f ) contains a compact edge of the form [T1 , T˜2 ] ⊂ L, where T1 is the left endpoint of the principal edge π(φ) and thus lies above the bisectrix and where the second coordinate of the point T˜2 is still ≤ d(φ), so that T˜2 again lies on or below the bisectrix. This implies that we indeed have Lf = L0 = L and d f (φ) = d(φ). Let us finally assume that the coordinates (x1 , x2 ) are not adapted to φ and that in adapted coordinates, φ is represented by φ a . Recall also that m0 = m. We then write φ f (y1 , y2 ) = φ (y1 , y2 + ψ(y1 ) + (f (y1 ) − ψ(y1 ))) = φ a (y1 , y2 + f1 (y1 )), where f1 (y1 ) := f (y1 ) − ψ(y1 ), that is, φ f = (φ a )f1 . If f1 is flat, then the Newton polyhedra of φ f and φ a are the same, which clearly implies that hr (φ) = hf (φ). Let us therefore assume that f1 is nonflat. Then f1 has a formal Puiseux series expansion m cj y1 j f1 (y1 ) ∼ j ≥1

with leading term c1 y1m1 , where c1  = 0. Then m = m0 ≤ m1 < m2 < · · · are rational numbers with a fixed common denominator. (i) Consider first the case where the principal face of the Newton polyhedron N (φ a ) is a compact edge whose slope has modulus 1/a. If m1 ≥ a, then we have m1 ≥ a > m0 = m. It is then easily seen that Lf = L and that the augmented Newton polyhedra N r (φ a ) and N f (φ f ) do agree in the half space bounded from below by the bisectrix, since the “perturbation” of φ a by f1 (y1 ) will have no effect on the Newton polyhedron of φ a within this half space. Consequently, we find that hf (φ) = hr (φ). Indeed, if m1 > a, then the claim about the effect of the “perturbation” of φ a by f1 (y1 ) is obvious. And, if m1 = a, then let us denote by κ pr the principal weight associated to the principal face π(φ a ) of the Newton polyhedron of φ a , so that π(φ a ) is contained in the line pr

pr

Lpr := {(t1 , t2 ) ∈ R2 : κ1 t1 + κ2 t2 = 1}. Since the coordinates of φ a are adapted to φ a , arguing in a similar way as in the case where the coordinates (x1 , x2 ) were adapted to φ, we then see that the Newton polyhedron N ((φ a )f1 ) contains a compact edge of the form [T1 , T˜2 ] ⊂ Lpr , where the left endpoint T1 lies above the bisectrix and where the second coordinate of

PROOFS OF PROPOSITIONS 1.7 AND 1.17

249

T˜2 is still ≤ h, so that T˜2 lies on or below the bisectrix. This implies again that hf (φ) = hr (φ). Assume finally that m1 < a. Then m = m0 ≤ m1 < a. We shall then prove that, within the closed half space lying above the bisectrix, the augmented Newton polyhedron of φ a is contained in the one of φ f . To this end, observe first that again L = Lf . We shall argue in a somewhat similar way as in the case where [T1 , T2 ] ⊂ {t1 ≤ t2 }, but with φ a taking over the role of φ, f1 the role of f, and m1 the role of m0 . The basis for our arguments will be the identity φ f = (φ a )f1 . Indeed, denote by Lf1 the supporting line to N (φ a ) with slope 1/m1 , and denote by S1 = (a1 , b1 ) the vertex of N (φ a ) ∩ Lf1 with largest second coordinate. The same kind of reasoning that we had employed in the case where [T1 , T2 ] ⊂ {t1 ≤ t2 } (distinguishing between the cases where N (φ a ) ∩ Lf1 is a single point or a compact interval) then shows that indeed the principal face of (φ a )f1 will be an interval of the form [S1 , S˜2 ] contained in the line Lf1 . Moreover, within the half space where t1 ≥ a1 , the augmented Newton polyhedra N r (φ a ) and N f (φ f ) will agree since φ f = (φ a )f1 . And, since Lf1 is a supporting line to the convex set N (φ a ), which is less steep than the line L, all this together shows that we have N r (φ a ) ∩ {t2 ≥ t1 } ⊂ N f (φ f ) ∩ {t2 ≥ t1 }. But this clearly implies that the line (m) = (m0 ) will intersect the boundary of the augmented Newton polyhedron N f (φ f ) at some point whose second coordinate is less than or equal to the second coordinate of the point of intersection with the boundary of N r (φ a ), and thus hf (φ) ≤ hr (φ). (ii) Consider next the case where the principal face of N (φ a ) is a noncompact horizontal edge. This case can be viewed as the limiting case as a → ∞ of the previous case (i) when m1 < a and can thus be treated in a very similar way. We leave the details to the interested reader. (iii) Finally, we consider the case where the principal face N (φ a ) is a vertex (h, h). We then chose the index lpr as in the beginning of Chapter 6, so that (h, h) is the right endpoint (Alpr −1 , Blpr −1 ) of the compact edge γlpr −1 = [(Alpr −2 , Blpr −2 ), (Alpr −1 , Blpr −1 )], whose (modulus) of slope is given by 1/alpr −1 . We may then argue essentially in the same way as in case (i), by replacing the edge given by the principal face in (i) by the edge γlpr −1 and the exponent a by alpr −1 . Observe that if m1 = alpr −1 , then the change of coordinates given by the nonlinear shear defined by f1 will transform the edge γlpr −1 into another edge [(Alpr −2 , Blpr −2 ), (A˜ 2 , B˜ 2 )] lying on the same line as γlpr −1 , and the right endpoint (A˜ 2 , B˜ 2 ) will still lie on or below the bisectrix. This follows from the same kind of reasoning that we applied in the case where [T1 , T2 ] ⊂ {t1 ≤ t2 }. Again we leave the details to the interested reader. This concludes the proof of the inequality (9.1) and, hence, also our direct proof of Proposition 1.17.

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Index a (exponent associated to principal face), 108 Airy-type integral, 31 al (exponent associated to edge γl ), 12 B (multiplicity of root of ∂2 φκal ), 118, 136, 139 B0 , B1 , B3 , 78, 186 coordinates local, 5 adapted, 6 linearly adapted, 11 d(φ) (Newton distance), 5 (m) (line parallel to bisectrix at height m + 1), 15, 136 dh (P ) (homogeneous distance of P ), 7 exponent pB , 141 p˜ B , 141 pc (critical exponent), 10, 16 p˜c , 141 pH , 141 face (of a Newton polyhedron), 5 φ a (φ in adapted coordinates), 8 φ˜a (φ in modified adapted coordinates), 136, 179 φ f , 17 φκ (κ-principal part of φ), 7 φ (l) (φ in modified adapted coordinates given by x = s(l) (y)), 126, 133 finite-type function, 5 φpr (principal part of φ), 5 Fourier restriction, 1 fractional shear, 17 fractionally smooth, 17 γl (edge of the Newton polyhedron), 12 H := 1/κ˜ 2 , 141 h(φ) (height of φ), 5 hf (φ), 18 hl , 15, 136 f hl , 19

h lin (φ) (linear height of φ), 11 H ± (right/left half plane), 17 hr (φ) (r-height of φ), 15 h˜ r (φ), 23 hypersurface of finite type, 1 interpolation parameter θB , 141 θ˜B , 141 θc , 141 θ˜c , 141 θH , 141 [l] (cluster of roots), 13 Littlewood-Paley decomposition, 55 µF , 50 n(P ) (maximal order of roots of P ), 7 N (φ) (Newton polyhedron), 5, 18 Nd (φ) (Newton diagram), 5 N f (φ f ) (augmented Newton polyhedron), 19 (N LA) assumption, 11 N r (φ a ) (augmented Newton polyhedron), 15 νδ , 138 I , 216 νδ,Ai I I I , 217 νδ,Ai νδI I , 216 νδI V , 217 νδλ , 143 λ , 205 νδ,0 λ , 80, 189, 204 νδ,Ai λ , νλ , νλ νl,I l,I I l,I I I , 88, 194 λ , 85, 192, 205 νl,∞ λ , 80, 189, 204 νδ,l λ , 85,192, 193 νl,1 V νδ,j , 73 VI , 73 νδ,j V II , 73 νδ,j νjλ , 61 λ , ν λ , 207, 237 ν0,I 0,I I λ , νλ , νλ , νλ , νλ νl,∞ l,0,0 l,I l,I I l,I I I , 220 ν(φ) (Varchenko’s exponent), 9

258 oscillatory integral, 3, 198, 214 π(φ) (principal face), 5 principal line, 6 root, 8 root jet ψ, 8, 47 weight, 6 ψ, see principal root jet ψ, 8 Puiseux series, 13 (R) Condition, 19 ˜ (homogeneous gauge), 188 ρ = ρ(δ) root, 12 singularity of type A, 46 type D, 46 subdomain Dl , 108 Dla , 108 D(l) , 125, 128  , 126, 128, 133, 136, 179 D(l) a D(l) , 126, 134 D(l) (cl ), 125, 128, 134 Dpr , 108, 123, 133 D(c0 ), 116 D a (c0 ), 116 El , 108 Ela , 108

INDEX E(l) , 127, 128 a , 127 E(l) supporting line, 7 T (φ) (Taylor support), 5, 18 I , 216, 217 Tδ,Ai I I I , 217, 233 Tδ,Ai TδI I , 216, 220 TδI V , 217, 236 λ , 87 Tl,1 Tδλ , 144 λ , 205 Tδ,0 λ , 80, 189, 196, 204 Tδ,Ai λ , 86, 205 Tl,∞ λ , 80, 189, 196, 204, 209 Tδ,l V , 73 Tδ,j VI Tδ,j , 73 V II , 73 Tδ,j Tjλ , 62 van der Corput estimate, 30, 34 Varchenko’s algorithm, 8 weight κ, 6, 51 κ l , 12 κ (l) , 127, 128, 141 κ, ˜ 136