Hangzhou Lectures on Eigenfunctions of the Laplacian (AM-188) [Course Book ed.] 9781400850549

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Annals of Mathematics Studies Number 188

Hangzhou Lectures on Eigenfunctions of the Laplacian

Christopher D. Sogge

PRINCETON UNIVERSITY PRESS PRINCETON AND OXFORD 2014

c Copyright 2014 by Princeton University Press 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 Sogge, Christopher D. (Christopher Donald), 1960Hangzhou lectures on eigenfunctions of the Laplacian / Christopher D. Sogge. pages cm Includes bibliographical references and index. ISBN 978-0-691-16075-7 (hardcover : alk. paper) – ISBN 978-0-691-16078-8 (pbk. : alk. paper) 1. Laplacian operator. 2. Eigenfunctions. I. Title. QA406.S66 2014 515’.3533–dc23 2013030692 British Library Cataloging-in-Publication Data is available The publisher would like to acknowledge the author of this volume for providing the camera-ready copy from which this book was printed. This book has been composed in LaTeX Printed on acid-free paper Printed in the United States of America 10 9 8 7 6 5 4 3 2 1

TO MY WIFE,

LIZ, MY CHILDREN,

LEWIS, SUSANNA AND WILL, MY PARENTS,

DON AND DONNA, AND MY LATE SISTER,

WENDY

Contents Preface

ix

1 A review: The Laplacian and the d’Alembertian 1.1 The Laplacian 1.2 Fundamental solutions of the d’Alembertian

1 1 6

2 Geodesics and the Hadamard parametrix 2.1 Laplace-Beltrami operators 2.2 Some elliptic regularity estimates 2.3 Geodesics and normal coordinates—a brief review 2.4 The Hadamard parametrix

16 16 20 24 31

3 The 3.1 3.2 3.3 3.4 3.5 3.6

39 39 48 53 55 58 65

sharp Weyl formula Eigenfunction expansions Sup-norm estimates for eigenfunctions and spectral clusters Spectral asymptotics: The sharp Weyl formula Sharpness: Spherical harmonics Improved results: The torus Further improvements: Manifolds with nonpositive curvature

4 Stationary phase and microlocal analysis 4.1 The method of stationary phase 4.2 Pseudodifferential operators 4.3 Propagation of singularities and Egorov’s theorem 4.4 The Friedrichs quantization

71 71 86 103 111

5 Improved spectral asymptotics and periodic geodesics 5.1 Periodic geodesics and trace regularity 5.2 Trace estimates 5.3 The Duistermaat-Guillemin theorem 5.4 Geodesic loops and improved sup-norm estimates

120 120 123 132 136

6 Classical and quantum ergodicity 6.1 Classical ergodicity 6.2 Quantum ergodicity

141 141 153

Appendix A.1 The Fourier transform and the spaces S(Rn ) and S 0 (Rn ) A.2 The spaces D0 (Ω) and E 0 (Ω) A.3 Homogeneous distributions A.4 Pullbacks of distributions A.5 Convolution of distributions

165 165 169 173 176 179

viii

CONTENTS

Notes

183

Bibliography

185

Index

191

Symbol Glossary

193

Preface

This monograph contains the material taught during a series of lectures at Zhejiang University in Hangzhou. The lectures were given during several visits spread out over a period of almost three years which started during the summer of 2010. I also presented large portions of the material in courses at the Johns Hopkins University during the spring semesters of 2010 and 2012. This endeavor was partially supported by the Chinese Ministry of Education, the NSF and the Simons Foundation. Since there were naturally people at Zhejiang University who could only attend, say, lectures for one of the three years, I attempted to split the material into three parts, each corresponding to one of those years. I also tried to organize the material and present it so that if somebody joined the course during one of the last two years it would be possible to follow the lectures needing minimal background from the earlier material, such as the statement of some of the missed theorems and not the proofs. The first part of the course, which corresponds to the first three chapters of the monograph, was mainly devoted to the proof of the sharp Weyl formula. The proof presented uses the Hadamard parametrix and much of this part of the course was devoted to its construction using elementary properties of the geodesic flow. In this part of the course I also showed that no improvements of the sharp Weyl formula are possible for the sphere equipped with the standard round metric due to the fact that, in this case, the eigenvalues repeat with the highest possible multiplicity. In the other direction, at the end of this part of the course, I showed that for manifolds with nonpositive curvature and especially for the n-torus one can make significant improvements for bounds for the remainder term in the Weyl law. An explanation is that in addition to having eigenvalues repeating with the highest possible multiplicity, the sphere has the largest possible collection of periodic geodesics in the sense that every geodesic on S n is periodic. In both cases, the torus and manifolds with nonpositive curvature have exactly the opposite type of behavior. I devoted the remaining part of the course to better understand this type of phenomena. To do so, I had to first present the material in the fourth chapter of the monograph, which corresponds to the second part of the course. This is a rapid (and admittedly sketchy) introduction to microlocal analysis and the theory of pseudodifferential operators. For the applications in this third part of the course, which I hope was its highlight, I needed to cover basics from the calculus of pseudodifferential operators and, more importantly, special cases of Egorov’s theorem and H¨ ormander’s theorem on propagation of singularities. My attempt in this part of the course (as well as in the first part) was to present the minimal amount of material that would be needed at the end. In the final part of the course I presented two main results that hearken back to the material presented at the end of the first part. I also chose to present them

x

PREFACE

since both have proofs that naturally bring together focal points of earlier parts of the course. The first of these, which is presented in the fifth chapter, is the DuistermaatGuillemin theorem which says that if the set of periodic geodesics in a given compact Riemannian manifold M is of measure zero then the remainder term in the Weyl law for M satisfies better bounds than the one for the sphere. The proof that I present uses an additional idea of Ivrii showing how one can use H¨ormander’s theorem about propagation of singularities and a generalized version of the earlier Weyl law involving pseudodifferential operators to reduce everything to a calculation that only requires information about the wave kernel near the time t = 0. Thus, the punchline of the proof uses the Hadamard parametrix, just as in the proof of the sharp Weyl formula from the first part of the course. The proof of the “generalized Weyl formula” that is presented is just a small variation on this earlier result as well. The Duistermaat-Guillemin theorem says that the favorable (but generic) assumption that the set of periodic geodesics is of measure zero leads to more favorable estimates for the Weyl law. The last main theorem that I presented in the course is of a similar nature. Roughly speaking, it says that if the long-term geodesic flow is uniformly distributed than (most) eigenfunctions exhibit a similar behavior in the sense that their L2 -mass becomes equidistributed as their energies go to infinity. More specifically, if the geodesic flow is ergodic than there always exists a subsequence {λjk } of full density so that, for the corresponding subsequence of L2 -normalized eigenfunctions {ejk }, the associated unit probability measures |eλjk |2 dVg tend in the weak∗ topology to the uniform probability measure dVg /Volg (M ). A stronger theorem is valid as well, which is the main result in the sixth chapter, which says that the eigenfunctions {eλjk } become uniformly distributed in phase space as well. This result is due in varying degrees of generality to Snirelman, Zelditch and Colin de Verdi´ere. Like the Duistermaat-Guillemin theorem, the proof brings together highlights from earlier parts of the course—in this case Egorov’s theorem, the generalized Weyl formula and von Neumann’s mean ergodic theorem. It is a pleasure to thank everyone who helped me with this project. First, I would like to thank my host at Zhejiang University, Doayuan Fang, as well as members from his group, including Chengbo Wang, Qidi Zhang and Ting Zhang. The hospitality that they showed and the assistance that they provided during my visits to Hangzhou were invaluable. I am also grateful for the feedback, which helped shape the manuscript, that they and the undergraduate and graduate students at Zhejiang University who attended my lectures provided. I would similarly like to thank the students at Johns Hopkins who took one or both of the courses there that I gave on portions of the material. I am also deeply indebted to my former Hopkins colleagues Bill Minicozzi and Steve Zelditch for helpful advice, answering many questions and numerous helpful conversations. Last, but certainly not least, I am extremely grateful for the support of my family during my numerous visits to China and the writing of the manuscript.

Hangzhou Lectures on Eigenfunctions of the Laplacian

Chapter One A review: The Laplacian and the d’Alembertian

1.1

THE LAPLACIAN

One of the main goals of this course is to understand well the solution of wave equation both in Euclidean space and on manifolds and then to use this knowledge to derive properties of eigenfunctions on Riemannian manifolds. This is a very classical idea. A key step in understanding properties of solutions of wave equations on manifolds will be to compute the types of distributions that include the fundamental solution of the wave operator in Minkowski space (d’Alembertian),  = ∂ 2 /∂t2 − ∆, with ∆=

n X

∂j2 ,

∂j = ∂/∂xj ,

(1.1.1)

(1.1.2)

j=1

being the Euclidean Laplacian on Rn . In the next section we shall compute fundamental solutions for , which is central to our goal. Here, though, since it will serve for us as a good model, we shall compute the fundamental solution for ∆. that the fundamental solution of a partial differential operator P (D) = P Recall aα ∂ α is a distribution E for which P (D)E = δ0 ,

(1.1.3)

where δ0 is the Dirac-delta distribution hδ0 , f P i = f (0), f ∈ S(Rn ). Here α = α (α1 , . . . , αn ) is a multi-index of length |α| = = ∂1α1 ∂2α2 · · · ∂nαn , D = j αj , ∂ 1 n n i ∂/∂x, and S(R ) is the space of Schwartz class functions on R , whose dual is 0 n 0 n the space of tempered distributions, S (R ). If φ ∈ S (R ), then hφ, f i denotes the value of φ acting on f . For later use, we shall also set α! = α1 ! α2 ! · · · αn !. Using the fundamental solution, one can solve the equation P (D)u = F. In fact, by (1.1.3), u = E ∗ F satisfies

P (D) E ∗ F = P (D)E ∗ F = δ0 ∗ F = F. Here “∗” denotes convolution, initially defined for say f, g ∈ S(Rn ) by Z
f ∗ g (x) = f (x − y)g(y) dy,

(1.1.4)

(1.1.5)

Rn

and then extended to distributions by approximating them by functions. Also, you can justify the first equality in (1.1.4) by using (1.1.5).

2

CHAPTER 1

We shall assume basic facts about distributions. Besides S 0 (Rn ), there is also D0 (Rn ), the dual space of C0∞ (Rn ) (compactly supported C ∞ functions), and E 0 (Rn ) (compactly supported distributions), which is the dual of the dual of C ∞ (Rn ). In the appendix we review the basic facts that we use here and elsewhere. The reader can also refer to many texts that cover the theory of distributions, including [37], [61] and [73]. Let us now derive a fundamental solution of ∆. Thus, we seek a E ∈ S 0 so that ∆E = δ0 . Since both ∆ and δ0 are invariant under rotations in Rn , it is natural to expect that E also has this qPproperty. In other words, we expect that E(x) = f (|x|) = f (r), n 2 where r = |x| = j=1 xj . Assuming for now that f is smooth for r > 0, since δ0 is supported at the origin, we would have that ∆f (|x|) =

X

∂j ∂j f (|x|) =

X

j


∂j f 0 (|x|)xj /|x|

j

=

X

00


f (|x|)x2j /|x|2 + f 0 (|x|)(1/|x| − x2j /|x|3 ) = 0,

x 6= 0,

j

and therefore f 00 (r) +

n−1 0 f (r) = 0, r

r = |x| > 0.

(1.1.6)

The last equation suggests that E should be homogeneous of degree 2 − n. Recall that u ∈ S 0 is homogeneous of degree σ if hu, fλ i = λσ hu, f i, f ∈ S, where fλ (x) = λ−n f (x/λ), λ > 0 is the λ L1 -normalized dilate of f . This reasoning turns out to be correct for n ≥ 3, but not for n = 2 since then 2 − n = 0 and constant functions are the ones on R+ which are homogeneous of degree 0, and then cannot give us our fundamental solution of the Laplacian on R2 . But since ln r also solves (1.1.6) when n = 2 and r2−n is a solution of the equation for n ≥ 3, perhaps f (r) = an r2−n , n ≥ 3,

and f (r) = an ln r, n = 2,

will work for the appropriate constants an . Specifically, we claim that we can choose the an so that if E(x) = an r2−n , n ≥ 3,

and E(x) = an ln |x|, n = 2,

(1.1.7)

then for g ∈ S we have g(0) = hE, ∆gi,

g ∈ S,

(1.1.8)

since g(0) = hδ0 , gi = h∆E, gi = hE, ∆gi, for all g ∈ S if and only if δ0 = ∆E. To verify (1.1.8), we shall need to use the divergence theorem. Note that ν = −x/|x| is the outward unit normal for a point x on the complement of the sphere of radius ε > 0 centered at the origin. Thus, for n ≥ 3, if dσ denotes the induced

3

A REVIEW: THE LAPLACIAN AND THE D’ALEMBERTIAN

Lebesgue measure on this sphere, Z an hE, ∆gi = ∆g(x) dx n−2 |x| n R Z
an X = lim ∂j ∂j g(x) dx n−2 ε→0+ |x|≥ε |x| j h Z X an
= lim − ∂j ∂j g(x) dx ε→0+ |x|n−2 |x|≥ε j Z i an X xj ∂ g(x) dσ − j n−2 |x| |x|=ε |x| j hZ X an
g(x) dx = lim ∂j2 ε→0+ |x|n−2 |x|≥ε XZ an xj − ∂ g(x) dσ n−2 |x| j |x| |x|=ε j i XZ an
xj + ∂j g(x) dσ . |x|n−2 |x| |x|=ε j The first term in the right vanishes since, as noted above, ∆|x|−n+2 = 0, when x 6= 0. For a given ε > 0, the second term is bounded by Z sup |∇g(x)|an ε−n+2 dσ ≤ Cε, x

|x|=ε

and so its limit is 0. The last term is Z Z (n − 2)an (n − 2)an g(x) dσ = − g(x) dσ, − |x|n−1 εn−1 |x|=ε |x|=ε which, as ε → 0+ , tends to Z −(n − 2)an g(0)

dσ. |x|=1

R Therefore, if An = |S n−1 | = |x|=1 dσ denotes the area of the unit sphere in Rn , we have the desired identity (1.1.8) for n ≥ 3 if an =

−1 , (n − 2)An

n ≥ 3.

(1.1.9)

We leave it as an exercise for the reader that we also have (1.1.8) for n = 2 if we set 1 a2 = . (1.1.10) 2π The minus signs in (1.1.9) are due to the fact that the Laplacian has a negative symbol, −|ξ|2 . Let us record what we have done in the following. Theorem 1.1.1 If n ≥ 3 and E(x) = an |x|2−n or n = 2 with E(x) = an ln |x|, with an defined by (1.1.9)–(1.1.10), then E is a fundamental solution of ∆, i.e., ∆E = δ0 .

4

CHAPTER 1

The fundamental solution for ∆, (1.1.7), that we have just computed is not unique since others are given by E(x) + h(x) if h(x) is a harmonic function, i.e., ∆h(x) = 0 for all x ∈ Rn . E, though, for n ≥ 3 is the unique fundamental solution vanishing at infinity. Remark 1.1.2 To help motivate computations that we shall carry out in the next chapter and throughout the text, let us see how the fundamental solution E for ∆ in Theorem 1.1.1 is pulled back via a linear bijection T : Rn → Rn . If y = T x then Tt

∂ ∂ = ∂y ∂x

and so ∆y =

n X ∂2 ∂2 ∂ jk ∂ + · · · + = g = ∆g , ∂y12 ∂yn2 ∂xj ∂xk j,k=1

with g jk = T −1 (T −1 )t . (1.1.11) Then ∆y u(y) = F (y) is equivalent to ∆g u(T x) = F (T x). In other words, the if pullback of ∆y via T is ∆g . If n ≥ 3, since an |y|2−n = an |x|2−n g |x|2g =

n X

gjk xj xk ,

(gjk ) = (g jk )−1 = T t T,

j,k=1

we conclude that n X j,k=1



∂ jk ∂ an |x|2−n = ∆y an |y|2−n = δ0 (y) g g ∂xj ∂xk 1

= δ0 (T x) = |detT −1 | δ0 (x) = |g|− 2 δ0 (x), (1.1.12) if |g| = det (gjk ). Note that, by a theorem of Jacobi (Sylvester’s law of inertia), every symmetric positive definite matrix g jk can be written in the above form, g jk = LLt , where L is real and invertible. Consequently, if we take T = L−1 and T t T = (gjk ) = (g jk )−1 , then the above argument shows that if n ≥ 3, then X an |g|1/2 ( gjk xj xk )(2−n)/2 j,k

P is a fundamental solution for j,k g jk ∂j ∂k . This decomposition g jk = LLt is not unique; however, it is if we require that L is a lower triangular matrix with strictly positive diagonal entries (the Cholesky decomposition from linear algebra). We shall make this assumption in all such factorizations to follow. Note that the matrix L in this decomposition is, roughly speaking, the matrix analog of taking the square root of a positive number. Let us conclude this section by reviewing one more equation involving the Laplacian.

5

A REVIEW: THE LAPLACIAN AND THE D’ALEMBERTIAN

As we pointed out before, we can use E to solve Laplace’s equation ∆u = F . Let us briefly study one more important equation involving ∆, the Dirichlet problem for R1+n = [0, ∞) × Rn , since it will also have some relevance in our calculation of fundamental solutions for the d’Alembertian,  = ∂t2 − ∆. If (y, x) ∈ [0, ∞) × Rn , the Dirichlet problem for this upper half-space is to show that for a given f ∈ S(Rn ) on the boundary we can find a function u(y, x) that is harmonic in R1+n with boundary values f , i.e., a solution of +  2
∂ n   ∂y2 + ∆ u(y, x) = 0, (y, x) ∈ (0, ∞) × R (1.1.13)   u(0, x) = f (x). By using the fact that in polar coordinates ∆=

n−1 ∂ ∂2 + + ∆S n−1 , ∂r2 r ∂r

where r = |x| and ∆S n−1 is the induced Laplacian on S n−1 (see § 3.4), one easily checks that if y P (y, x) = 2 , (y + |x|2 )(n+1)/2 then
∂2 + ∆ P (y, x) = 0, (y, x) ∈ (0, ∞) × Rn . 2 ∂y Therefore, if bn is a constant and f ∈ S, Z u(y, x) = bn P (y, x − w) f (w) dw (1.1.14) Rn n

is harmonic in (0, ∞) × R , i.e., it satisfies the first part of (1.1.13). Since P (y, x) = y −n P (1, x/y) = y −n (1 + |x/y|2 )−(n+1)/2 , we also have that as y → 0+ Z u(y, x) = bn (1 + |w|2 )−(n+1)/2 f (x − yw) dw → f (x), Rn

provided that the constant bn is chosen so that Z bn (1 + |x|2 )−(n+1)/2 dx = 1, Rn

in other words bn = π −(n+1)/2 Γ with

Z Γ(z) =

n + 1
, 2

(1.1.15)

∞

e−s sz−1 ds,

Re z > 0,

(1.1.16)

0

being Euler’s gamma function. Thus we have shown that the Poisson integral of f , (1.1.14), with bn given by (1.1.15) solves the Dirichlet problem for the upper half-space, (1.1.13). Note also that the constant bn in (1.1.15) is also given by the formula 2 bn = , (1.1.17) An+1 where, as before, Ad =

2π d/2 Γ(d/2)

denotes the area of the unit sphere, S d−1 , in Rd .

6 1.2

CHAPTER 1

FUNDAMENTAL SOLUTIONS OF THE D’ALEMBERTIAN

In this section we shall compute fundamental solutions for the d’Alembertian in R1+n . Thus we seek distributions E for which we have E = δ0,0 (t, x),

 = ∂t2 − ∆, (t, x) ∈ R1+n .

(1.2.1)

Here δ0,0 is the Dirac delta distribution centered at the origin in space-time. Recall that the Lorentz transformations are the linear maps from R1+n to itself preserving the Lorentz form Q(t, x) = t2 − |x|2 ,

(1.2.2)

which is the natural quadratic form associated with . Since both  and δ0,0 are invariant under these transformations, we expect E to also enjoy this property. Thus, we expect it to be of the form E = f (t2 − |x|2 ) where f is some distribution. If we plug this into our equation (1.2.1) and we use the polar coordinates formula for ∆, we can see that if E were of this form then we would have to have that f 00 (ρ) +

n+1 0 f (ρ) = 0, 2ρ

when ρ = t2 − |x|2 6= 0.

From this we expect f to be homogeneous of degree (1 − n)/2. If ( 1, s ≥ 0 H(s) = 0, s < 0,

(1.2.3)

(1.2.4)

denotes the Heaviside step function on R, then we can write down solutions of (1.2.3) with this homogeneity. Specifically, when n ≤ 3 the equation has the solution   c1 H(ρ), n = 1 f (ρ) = c2 H(ρ)ρ−1/2 , n = 2   c3 δ(ρ), n = 3. Thus, we expect that for appropriate constants cn the following are fundamental solutions for  in spatial dimensions n = 1, 2, 3   c1 H(t)H(t2 − |x|2 ), n = 1       E = c2 H(t)H(t2 − |x|2 )(t2 − |x|2 )−1/2 , n = 2 (1.2.5)       c H(t)δ(t2 − |x|2 ), n = 3. 3

We added the factor H(t) since, as we shall see below when we solve the Cauchy problem for , it is natural to want our fundamental solution to be supported in R1+n = [0, ∞) × Rn . When n = 3 our guess involves δ(t2 − |x|2 ), which is the Leray + measure in R1+3 associated with the function t2 − |x|2 (see Theorem A.4.1 in the appendix). Let us show that our guess is correct when n = 2, since this will serve as a model for arguments to follow. Thus, we wish to see that c2 can be chosen so that whenever F ∈ S(R1+2 ) we have Z
F (0, 0) = c2 H(t)H(t2 − |x|2 )(t2 − |x|2 )−1/2 F (t, x) dtdx. (1.2.6) R1+2

7

A REVIEW: THE LAPLACIAN AND THE D’ALEMBERTIAN

To simplify the integration by parts arguments, we note that we can regularize our guess by extending it into the complex plane and taking limits. Specifically, instead of truncating the distribution about its singularity as we did for the fundamental solution of the Laplacian, we shall use the fact that
−1/2 , H(t2 − |x|2 )(t2 − |x|2 )−1/2 = lim Im |x|2 − (t + iε)2 ε→0+

due to the fact that limε→0+ (|x|2 − (t + iε)2 )−1/2 is real when |x|2 > t2 , and of positive imaginary part if (t, x) is fixed with |x|2 < t2 and ε > 0 small. Therefore, (1.2.6) is equivalent to showing that c2 can be chosen so that we always have Z Z ∞
−1/2 2
F (0, 0) = c2 lim Im |x|2 − (t + iε)2 ∂t − ∆ F (t, x) dtdx. (1.2.7) ε→0+

R2

0

In addition to motivating what is to follow, the advantage of (1.2.7) over (1.2.6) is that the integration by parts arguments that we require are easy for the latter. If we use the polar coordinates formula for the Laplacian it is not difficult to check that
−1/2 (∂t2 − ∆) |x|2 − (t + iε)2 = 0, ε > 0, (t, x) ∈ R1+2 , and of course we also have that for ε > 0, Im (|x|2 − (t + iε)2 )−1/2 = 0 when t = 0. Therefore, if we integrate by parts, we find that when ε > 0 we have Z Z ∞
−1/2 2
c2 Im |x|2 − (t + iε)2 ∂t − ∆ F (t, x) dtdx R2 0 Z
−1/2
∂ Im |x|2 − (t + iε)2 F (0, x) dx = c2 t=0 2 ∂t ZR ε = c2 F (0, x) dx. 2 + ε2 )3/2 (|x| 2 R The last integral involves the Poisson integral of x → F (0, x), which, as we showed in the last section, will tend to F (0, 0) as ε → 0+ provided that c2 is the constant b2 in (1.1.15). Thus, we have verified our claim that for n = 2, (1.2.5) provides a fundamental solution for . Before proceeding further, we urge the reader to verify the assertion for the other two cases there, namely, n = 1 and n = 3, and compute c1 and c3 . Before we calculate fundamental solutions in other dimensions besides n = 2, let us try to add some perspective to what we have just done. Recall that for n ≥ 3 the fundamental solution of −∆ was a constant multiple of |x|2−n .P We can think of the latter as the pullback of H(r)r(2−n)/2 via the map x → r2 = x2j , and the latter is the metric form for ∆. Since the space-time dimension is larger by one, ideally we would like to obtain fundamental solutions for  by pulling back distributions like r(1−n)/2 via the Lorentz form Q(t, x) = t2 − |x|2 . The problem that arises, of course, is that since Q is a semidefinite form (unlike the one for ∆), we cannot do this directly. In our calculation for n = 2, we in effect did realize our goal of obtaining a fundamental solution of  via a regularization argument that produced a distribution which was the limit of smooth functions involving the appropriate power of complex extensions of Q. We shall do an analogous thing when n ≥ 3. Let us be more specific. We first define distributions Wa (t, x) = lim Im |x|2 − (t + iε)2 ε→0+


a

,

1 a ∈ − N, 2

(1.2.8)

8

CHAPTER 1

by which we mean that for F (t, x) ∈ S(R1+n ), Z
a hWa , F i = lim F (t, x) Im |x|2 − (t + iε)2 dtdx. ε→0+

R1+n

Note that if |x|2 > t2 > 0, limε→0+ (|x|2 − (t + iε)2 )a = (|x|2 − t2 )a is real for a = − 21 , − 32 , − 52 , . . . , while for a = −ν − 1, ν = 0, 1, 2, . . . we leave it as an exercise for the reader to see that Wa (t, x) is a constant multiple of sgn t δ (ν) (t2 − |x|2 ), with the constant being equal to π when ν = 1 (show this!, cf. also formula (3.2.10) in [37]). Therefore, supp Wa ⊂ { (t, x) ∈ R1+n : |x|2 ≤ t2 }, (1.2.9) which is the union of the forward and backward light cones through the origin. Note also that a simple integration by parts argument in the r = |x| variable shows that these distributions are well-defined. The distributions in (1.2.8) are the ones that will give us our fundamental solutions for  in R1+n , n ≥ 2 with a = −(n − 1)/2. To handle variable coefficient operators and construct the Hadamard parametrix we shall also require natural extensions of the above definitions to a ∈ 21 Z \ − 21 N. Based on (1.2.9) it is natural to set 1 Wa (t, x) = H(t2 − |x|2 )(t2 − |x|2 )a , a ∈ N, (1.2.10) 2 and W0 (t, x) = H(t2 − |x|2 ), (1.2.11) with H being the Heaviside function. Since ∆ = ∂ 2 /∂r2 + (n − 1)r−1 ∂/∂r + ∆S n−1 , the reader can verify that for a ∈ 21 Z and t > 0 Wa = 2|a|(n − 1 + 2a)Wa−1 ,

a 6= 0, −(n − 1)/2,

(1.2.12)

while for a = 0 and t > 0 W0 = 2(n − 1)π −1 W−1 .

(1.2.13)

The limit as ε → 0+ of the other exceptional case of a = −(n − 1)/2 (n ≥ 2) is a candidate for a fundamental solution since it is a distribution which is invariant under the Lorentz group, and, moreover, homogeneous of degree 2 − d, where d = n + 1 is the space-time dimension. What is more, we have  |x|2 − (t + iε)2


−(n−1)/2

= 0,

ε > 0, (t, x) ∈ R1+n .

(1.2.14)

Indeed, we claim that if cn is chosen appropriately, then the distribution E+ = cn H(t) lim Im |x|2 − (t + iε)2 ε→0+


−(n−1)/2

, n ≥ 2,

(1.2.15)

is a fundamental solution of , i.e., E+ = δ0,0 . Thus, we need to verify that cn , n ≥ 2, can be chosen so that whenever F (t, x) ∈ S(R1+n ), we have Z ∞Z
−(n−1)/2 F (0, 0) = lim cn Im |x|2 − (t + iε)2 F (t, x) dtdx, (1.2.16) ε→0+

0

Rn

9

A REVIEW: THE LAPLACIAN AND THE D’ALEMBERTIAN

assuming that n > 1. We just did this for n = 2 and the argument for higher dimensions is practically the same. Indeed, if we use (1.2.14) and the fact that
−(n−1)/2 Im |x|2 − (t + iε)2 = 0, t=0

we can integrate by parts as we just did in the 2-dimensional case to see that the right side of (1.2.16) equals Z cn lim

ε→0+

F (0, x) Rn


−(n−1)/2 ∂ dx Im |x|2 − (t + iε)2 ∂t t=0 Z ε F (0, x) 2 = (n − 1)cn lim dx. ε→0+ Rn (ε + |x|2 )(n+1)/2

For a given ε > 0, the last integral is our friend, the Poisson integral of x → F (0, x) that arose in the Dirichlet problem for the Laplacian. So if for n ≥ 2 we choose cn here to be cn = bn /(n − 1), where bn is as in (1.1.15) and (1.1.17), i.e., cn =

1 −(n+1)/2  n − 1  2 π Γ = , 2 2 (n − 1)An+1

(1.2.17)

then we conclude that (1.2.16) must be valid, i.e., E+ is a fundamental solution for . For n = 1 this calculation does not work, but based on what we just did we find that 1 E+ = H(t)H(t2 − |x|2 ), n = 1, 2 satisfies E = δ0,0 in R × R. We also notice that E+ =

1 H(t)H(t2 − |x|2 )(t2 − |x|2 )−1/2 , 2π

n = 2,

and in three spatial-dimensions we have the Kirchhoff formula E+ =

1 1 H(t)δ(t2 − |x|2 ) = H(t)δ(t − |x|), 2π 4πt

n = 3.

Therefore, when n ≤ 3, E+ is a measure supported in the forward light cone. When n ≥ 4 it is a more singular and complicated distribution than this. On the other hand, n = 2k + 1 is odd with k ≥ 1, then E+ involves (k − 1) derivatives of the δ-function on R pulled back via the Lorentz form (1.2.2), which is equivalent to the familiar spherical means formulae for solutions of the wave equation in odddimensions (see [62], formula (1.6)). By construction, the fundamental solution E+ is supported in the forward light cone, {(t, x) ∈ [0, ∞) × Rn : |x| ≤ t}. Let us now argue that it is the only such fundamental solution. Any other would have to be of the form E+ + u where u is a distribution satisfying u = 0 and supp u ⊂ [0, ∞) × Rn . But then u = δ0,0 ∗ u = (E+ ) ∗ u = E+ ∗ u = 0.1 So not only is E+ the only such fundamental solution, 1 Note

that these calculations are justified by the discussion at the end of §A.5 of the appendix.

10

CHAPTER 1

it actually is the only one supported in R1+n + . Because of this, it is often called the advanced fundamental solution. If E− is the reflection of E+ about the origin in R1+n , i.e., hE− , F i = hE+ , F− i,

F ∈ S,

where F− (t, x) = F (−t, −x), then clearly E− is a fundamental solution, which is called the retarded fundamental solution since it is supported in the backward light cone. Another fundamental solution then is given by E=

1 (E+ + E− ), 2

which is sometimes called the Feynman-Wheeler fundamental solution that makes use of both the advanced and retarded fundamental solutions, E+ and E− . Let us summarize what we have just done in the following. Theorem 1.2.1 The distribution E+ on R1+n defined by (1.2.15) and (1.2.19) is a fundamental solution, supported in the forward light cone, and is the unique fundamental solution supported in [0, ∞) × Rn . Its reflection through the origin, E− is the unique fundamental solution supported in the backward light cone through the origin, {(t, x) ∈ (−∞, 0] × Rn : t2 − |x|2 ≥ 0}, and E = 12 (E+ + E− ) is also a fundamental solution of . The construction of E+ using Wa , a = −(n − 1)/2, will turn out to be very useful when we replace ∆ by a variable coefficient operator later. Let us now see, though, that if we use the Fourier transform, we can write down a simple and natural formula for E+ . Recall that the Fourier transform is an isometry on S(Rn ) given by Z fˆ(ξ) = e−ix·ξ f (x) dx, f ∈ S(Rn ), (1.2.18) Rn

whose inverse is given by Fourier’s inversion formula Z −n f (x) = (2π) eix·ξ fˆ(ξ) dξ, f ∈ S(Rn ).

(1.2.19)

Rn

Note that if f, g ∈ S then Z

fˆ g dx =

Z f gˆ dx.

Rn

This allows us to define the Fourier transform of u ∈ S 0 (Rn ). To do so, we just define u ˆ by requiring that hˆ u, f i = hu, fˆi. Having recalled these definitions, we claim that we have the following natural formula for E+ : Z sin t|ξ| E+ (t, x) = H(t) × (2π)−n eix·ξ dξ = E+ (t), (1.2.20) |ξ| Rn where E+ (t) ∈ S 0 (Rn ) denotes the distribution, whose Fourier transform is sin(t|ξ|)/|ξ|, when t > 0 and 0 if t ≤ 0, i.e., Z sin t|ξ| ˆ −n hE+ (t), f i = (2π) H(t) f (ξ) dξ. |ξ| Rn

11

A REVIEW: THE LAPLACIAN AND THE D’ALEMBERTIAN

If (1.2.20) is valid it shows that E+ is a continuous function of t > 0 with values in E 0 (Rn ), but this follows from the earlier formula as well. Since the second term in (1.2.20) is a distribution supported in [0, ∞) × Rn , in view of Theorem 1.2.1, we would prove the formula if we could show that for all F ∈ S(R1+n ) we have Z ∞Z sin t|ξ| (F )∧ (t, ξ) dξdt, (1.2.21) F (0, 0) = (2π)−n |ξ| n 0 R where here we are taking the Rn Fourier transform so that (F )∧ (t, x) = (∂t2 + |ξ|2 )Fˆ (t, ξ). Since

sin t|ξ| = 0, and sin(t|ξ|) = 0, when t = 0, |ξ| if we integrate by parts we conclude that the right side of (1.2.21) equals Z Z ∂ sin t|ξ| −n −n ˆ F (0, ξ) dξ = (2π) Fˆ (0, ξ) dξ = F (0, 0), (2π) |ξ| Rn ∂t (∂t2 + |ξ|2 )

t=0

as desired, which proves (1.2.20). We also remark that if E is as in Theorem 1.2.1, then this also gives us that Z sin |tξ| 2E(t, x) = (2π)−n eix·ξ dξ. (1.2.22) |ξ| n R In addition to showing that hE+ (t), hi = (2π)−n

Z Rn

sin t|ξ| ˆ h(ξ) dξ, |ξ|

h ∈ S(Rn ),

0 (t) = ∂t E+ (t), which has a limit as t → 0+ , which is 0, (1.2.20) also shows that E+ is given by Z 0 ˆ dξ, hE+ (t), hi = (2π)−n cos(t|ξ|) h(ξ) Rn

0

has a limit in S , which is δ0 , the Dirac distribution for Rn . Using these facts we can solve the Cauchy problem:   u(t, x) = F (t, x), t > 0

(1.2.23)

  u(0, x) = f (x), ∂t u(0, x) = g(x). Corollary 1.2.2 Suppose that f, g ∈ C ∞ (Rn ) and that F ∈ C ∞ ([0, ∞) × Rn ) the Cauchy problem (1.2.23) has a unique solution u ∈ C ∞ ([0, ∞) × Rn ) given by Z t 0 u(t, · ) = E+ (t) ∗ f + E+ (t) ∗ g + E+ (t − s) ∗ F (s, ·) ds. (1.2.24) 0 n

∞

If we assume that f, g ∈ S(R ) and F ∈ C (R; S(Rn )), we also have Z Z sin t|ξ| u(t, x) = (2π)−n eix·ξ cos(t|ξ|)fˆ(ξ) dξ + (2π)−n eix·ξ gˆ(ξ) dξ |ξ| n n R R Z tZ sin(t − s)|ξ| ˆ −n F (s, ξ) dξ. (1.2.25) + (2π) eix·ξ |ξ| n 0 R

12

CHAPTER 1

We leave it up to the reader to show that (1.2.25) follows from (1.2.24) and the above Fourier transform calculations. By §A.5 of the appendix, if we set v(t, · ) = 0 E+ (t) ∗ f + E+ (t) ∗ g is in C ∞ ([0, ∞) × Rn ), then v ∈ C ∞ ([0, ∞) × Rn ), since, for each fixed t, E+ (t) ∈ E 0 (Rn ) and t → E+ (t) is a smooth function of t with 0 values in E 0 (Rn ). Since E+ (t) = 0 and E+ (t) = 0 for t > 0 and E+ (+0) = 0 0 limt→0+ E+ (t) = 0 and E (+0) = limt→0+ E+ (t) = δ0 , it is clear that v solves (1.2.23) with vanishing forcing term F . So what remains is to show that w(t, · ) = Rt E+ (t − s) ∗ F (s, · ) ds satisfies (1.2.23) with vanishing Cauchy data, i.e., f, g = 0, 0 since the same considerations that showed that v was smooth imply the same for w. But using once again that E+ (+0) = 0 and E 0 (+0) = δ0 we have Z t Z t 0 E+ (t − s) ∗ F (s, · ) ds, ∂t E+ (t − s) ∗ F (s, · ) ds = 0

0

and ∂t2

t

Z

Z E+ (t − s) ∗ F (s, · ) ds =

0

t 00 E+ (t − s) ∗ F (s, · ) ds + δ0 ∗ F (t, · ).

0

The first of these formulas tells us that ∂t w vanishes when t = 0 and clearly so does w. The second formula tells us that w = F when t > 0, since E+ = 0 when t > 0. This establishes the claim about w and finishes the proof of the existence results in the corollary. We need to prove the uniqueness results. The reader can verify that it just follows from the argument that showed that E+ was the unique fundamental solution of  supported in [0, ∞)×Rn . On the other hand, let us present an argument based on energy estimates which shows that under the stronger hypotheses in (1.2.25) there is uniqueness, since this will also serve as a model for material to follow. The energy inequality for  says that, if, say, u ∈ C 2 vanishes for large x, then Z t ku0 (t, · )kL2 (Rn ) ≤ ku0 (0, · )kL2 (Rn ) + ku(s, · )kL2 (Rn ) ds, (1.2.26) 0

where u0 = (∂t u, ∇x u) denotes the space-time gradient of u. Clearly (1.2.26) implies that there is a unique solution to the equation (1.2.23) with the given data (f, g) and forcing term F . For if u1 and u2 were two such solutions then their difference would be a solution of the equation (u1 − u2 ) = 0 with zero Cauchy data, and hence be identically zero in R1+n by (1.2.26). To prove (1.2.26) we use the identity 2∂t uu = ∂t |u0 |2 − 2

n X


∂j ∂t u∂j u ,

(1.2.27)

j=1

and integrate by parts to get 0

∂t ku (t,

· )k2L2

Z =

0 2

∂t |u | dx =

Z

0 2

∂t |u | dx − 2

n Z X


∂j ∂t u∂j u dx

j=1

Z =2

∂t uu dx ≤ 2ku0 (t, · )kL2 ku(t, · )kL2 .

A REVIEW: THE LAPLACIAN AND THE D’ALEMBERTIAN

13

Since this implies ∂t ku0 (t, · )kL2 ≤ ku(t, · )kL2 , we obtain (1.2.26) via integration from 0 to t. Remark 1.2.3 As noted in Theorem 1.2.1, E+ is supported in the forward light cone. Thus, its reflection about {t = 0}, E− , is a fundamental solution for the d’Alembertian supported in the backward light cone, {(t, x) ∈ R1+n : t ≤ 0, t2 − |x|2 ≥ 0}, and E = 12 (E+ + E− ) is another fundamental solution but with support in the full light cone {(t, x) ∈ R1+n : t2 − |x|2 ≥ 0}. By (1.2.22) we have Z 1 sin |tξ| −n E(t, x) = (2π) eix·ξ dξ. 2 |ξ| Rn With this in mind, it is not difficult to check that another natural fundamental solution to the d’Alembertian in R1+n , n ≥ 2, is given by the formula 1 EF (t, x) = (2π)−n 2

Z e

ix·ξ

Rn



 cos t|ξ| sin |tξ| −i dξ |ξ| |ξ| Z 1 dξ = (2π)−n eix·ξ ei|tξ| . (1.2.28) 2i |ξ| n R

We leave the verification of this assertion as a good exercise for the reader. This distribution, EF , is called the Feynman fundamental solution for the d’Alembertian. Unlike E+ , E− or E its support is not contained in the light cone. Instead supp EF = R1+n , which is a physical curiosity since, as noted above, the d’Alembertian has propagation speed one (see also Theorem 2.4.2 below). Another good exercise is to verify that if n ≥ 2 then by (1.2.15) and (1.2.17) EF (t, x) = lim

ε→0+


−(n−1)/2 1 |x|2 − (|t| + iε)2 , i(n − 1)An+1

(1.2.29)

with, as above, Ad denoting the area of the unit sphere in Rd . (See (6.2.1) in [37] for another proof of this fact.) Using (1.2.29) one finds that the difference between EF and E has support equal to {(t, x) ∈ R1+n : |x| ≥ |t|} (the exterior of the full light cone) when n is even, while for odd spatial dimensions the support of EF − E is all of Minkowski space. Let us conclude this section by using Corollary 1.2.2 to make some calculations regarding the distributions Wa (t, x) defined in (1.2.8), (1.2.10) and (1.2.12). As we saw above if we set n − 1 1 E0 = π −(n+1)/2 Γ H(t)W−(n−1)/2 (t, x), 2 2 then E0 = δ0,0 , and since, by (1.2.12) and (1.2.13), W−(n−1)/2+ν is a nonzero multiple of W−(n−1)/2+ν−1 , we conclude that for a given n ≥ 2 we can inductively choose nonzero constants αν so that if we set Eν = αν H(t)W−(n−1)/2+ν , then we have Eν = νEν−1 ,

ν = 1, 2, 3, . . . .

ν = 1, 2, 3, . . . ,

14

CHAPTER 1

Computing the constants αν is a bit tedious. Let us derive another formula that will be easier to use. We first note that if we set for ν = 0, 1, 2, . . . ZZ Eν (t, x) = lim ν! (2π)−n−1 eix·ξ+itτ (|ξ|2 − (τ − iε)2 )−ν−1 dξdτ (1.2.30) ε→0+

R1+n

in the sense of distributions, then supp Eν ⊂ [0, ∞) × Rn = R1+n + , by the PaleyWeiner theorem (see §7.3-7.4 of [37] or Theorem 19.2 of [53]).2 Since for F ∈ S(R1+n ), we have hE0 , F i = lim (2π)−n−1

ZZ

ε→0+

(|ξ|2 − τ 2 )Fˆ (ξ, τ ) dξdτ |ξ|2 − (τ − iε)2 ZZ = (2π)−n−1 Fˆ (ξ, τ ) dξdτ = F (0, 0),

where now Fˆ denotes the space-time Fourier transform, we conclude that E0 defined by (1.2.30) must be a fundamental solution for . Since it is supported in R1+n + , it must be the advanced fundamental solution E+ constructed before. Note that our original construction of E+ involved extending the real quadratic form |x|2 − t2 into the complex plane, while the one given by (1.2.30) for ν = 0 involves a similar construction, but this time on the Fourier transform side. Let us collect more facts about these distributions which will prove useful in the sequel. Proposition 1.2.4 Then

Let Eν , ν = 0, 1, 2, . . . be the distributions defined in (1.2.30).

supp Eν ⊂ {(t, x) ∈ [0, ∞) × Rn : |x| ≤ t},

ν = 0, 1, 2, 3, . . . ,

E0 = δ0,0 ,

(1.2.31) (1.2.32)

and Eν = νEν−1 ,

−2∂Eν /∂x = xEν−1 , 2∂Eν /dt = tEν−1 , ν = 1, 2, 3, . . . . (1.2.33)

Furthermore, Eν (+0) = lim Eν (t, · ) = 0, t→0+

ν = 0, 1, 2, 3, . . . ,

and ∂tk Eν (+0) = 0 for k ≤ 2ν. To verify this, we note that we have already proven the assertions for ν = 0. The Paley-Weiner theorem yields that supp Eν ⊂ R1+n + . Since (1.2.33) is easy to check directly from the definitions, we see from Corollary 1.2.2 that we have the stronger statement (1.2.31). By the discussion preceding the Proposition, Eν must be a nonzero multiple of W−(n−1)/2+ν for every ν = 0, 1, 2, . . . , which implies the last part of the proposition since it is easy to check the assertions for these distributions. 2 In the case of ν = 0, this can also be verified directly using the formula limε→0+ (|ξ|2 − R(τ − iε)2 )−1 = (|ξ|2 − τ 2 + i(sgn t)0)−1 , which leads to the expected formula limε→0+ (2π)−1 eitτ (|ξ|2 − (τ − iε)2 )−1 dτ = H(t) sin t|ξ|/|ξ|. The asserted support properties for Eν , ν = 1, 2, 3, . . . , follow from this and the fact that 2∂Eν /∂t = tEν−1 , ν = 1, 2, 3, . . . .

15

A REVIEW: THE LAPLACIAN AND THE D’ALEMBERTIAN

Remark 1.2.5 If we use (1.2.20), we can derive another formula for these distributions. Indeed, by using Corollary 1.2.2 we conclude that for ν = 1, 2, 3, . . . Z tZ

sin(t − s)|ξ| ˆ Eν−1 (s, ξ) dξds |ξ| 0 Z sin(t − sν )|ξ| sin(sν − sν−1 )|ξ| eix·ξ ··· |ξ| |ξ| 0≤s1 ≤s2 ≤···≤sν ≤t

Eν (t, x) = ν(2π)−n Z = ν! (2π)−n

eix·ξ

×

sin(s2 − s1 )|ξ| sin s1 |ξ| dξds1 . . . dsν . (1.2.34) |ξ| |ξ|

For instance, when ν = 1, we have Z  sin t|ξ|  dξ H(t) −n E1 (t, x) = . × (2π) eix·ξ − t cos t|ξ| 2 |ξ| |ξ|2

(1.2.35)

The reader can check that the factor inside the parenthesis vanishes to second order at ξ = 0, which cancels out the singularity coming from the factor 1/|ξ|2 . One can also prove (1.2.35) by using the last identity in (1.2.33). By an induction argument using either (1.2.34) or (1.2.33), one can show that Eν , ν = 1, 2, . . . , is a finite linear combination of Fourier integrals of the form Z H(t)tj (2π)−n eix·ξ sin t|ξ| |ξ|−ν−1−k dξ Rn

and j

−n

Z

H(t)t (2π)

eix·ξ cos t|ξ| |ξ|−ν−1−k dξ

Rn

with j and k being nonnegative integers satisfying j +k = ν. We can ignore the contributions from frequencies |ξ| ≤ 1 in our applications, since for any ν = 0, 1, 2, . . . , ˆν (t, ξ)1|ξ|≥1 the difference between Eν and the inverse Fourier transform of ξ → E will be a smooth function of t and x. Here 1|ξ|≥1 denotes the characteristic function of {ξ ∈ Rn : |ξ| ≥ 1}. Based on this, if we use Euler’s formula, we conclude that for any ν = 0, 1, 2, 3, . . . , modulo smooth functions of (t, x) ∈ Rn+ , Eν is a finite linear combination of terms of the form Z H(t) tj (2π)−n eix·ξ±it|ξ| |ξ|−ν−1−k dξ, |ξ|≥1 (1.2.36) for some j, k = 0, 1, 2, . . . , with j + k = ν.

Chapter Two Geodesics and the Hadamard parametrix

2.1

LAPLACE-BELTRAMI OPERATORS

A main goal of this course will be to study the spectrum of Laplace-Beltrami operators on compact manifolds. For the sake of simplicity, though, throughout this chapter we shall consider Laplace-Beltrami operators defined on an open subset Ω of Rn . We then will easily be able to lift results presented here to corresponding ones on compact manifolds, which among other things will provide us with the tools that we need to prove the sharp Weyl formula in the next chapter. Let us start our discussion by defining a metric on an open subset Ω ⊂ Rn . Everything that we shall review is standard and the reader seeking additional details can consult a number of standard texts on Riemannian geometry, such as [43]. Recall that if p ∈ Ω, the tangent space, Tp Ω, at p is the space of all tangent vectors at p. Specifically, Xp is in Tp Ω if it is a continuous linear map on C ∞ (Ω) having the property that there is a real vector X = (X 1 , . . . , X n ) in Rn such that Xp (u) =

n X

Xj

j=1

∂ u(p), u ∈ C ∞ (Ω). ∂xj

Thus, Xp annihilates constant functions, and the space of all tangent vectors at p, Tp Ω has dimension n. The tangent bundle, T Ω, is the union over p ∈ Ω of Tp Ω. A metric at p is a real-valued function on Tp Ω × Tp Ω , gp (Xp , Yp ), Xp , Yp ∈ Tp Ω, having the following properties: • gp is symmetric, meaning that if Xp , Yp ∈ Tp Ω then gp (Xp , Yp ) = gp (Yp , Xp ). • gp is bilinear, meaning that if Xp , Yp , Zp ∈ Tp Ω and a, b ∈ R, then gp (aXp + bYp , Zp ) = agp (Xp , Zp ) + bgp (Yp , Zp ). • gp is positive definite, meaning that gp (Xp , Xp ) > 0,

if 0 6= Xp ∈ Tp Ω.

If we let gij (p) = gp (Xi , Xj ), Xi = ∂/∂xi , Xj = ∂/∂xj , then the n × n positivedefinite matrix g(p) = (gijP (p)) represents the P components of the metric at p. By this, we mean that if v = v i Xi and w = wi Xi are in Tp Ω, then we have gp (V, W ) =

n X i,j=1

i

j

v w gp (Xi , Xj ) =

n X i,j=1

v i wj gij (p).

17

GEODESICS AND THE HADAMARD PARAMETRIX

Let us now compute how the metric changes via a change of coordinates, y = T x, where T : Rn → Rn is a linear bijection, and, say, p = 0 so that it is fixed by T . Since n X ∂ ∂xk ∂ = , ∂yi ∂yi ∂xk k=1

we have that g

n    ∂ X ∂  ∂xk ∂xl = , , gkl = (T −1 )t g T −1 ∂yi ∂yj ∂yi ∂yj ij k,l=1

which is consistent with the calculation we made in the remark following Theorem 1.1.1. A metric on our open subset Ω ⊂ Rn assigns in a smooth manner to each p ∈ Ω a metric if X and Y are two smooth vector fields on Ω, P gip on Tp Ω × Tp Ω.PThus, X (p)∂/∂xi , Y = Y i (p)∂/∂xi , with p → X i (p), Y i (p) being real and in X= C ∞ (Ω) for every i = 1, . . . , n, the real function p → g(X, Y )(p) = gp (Xp , Yp ) is in C ∞ (Ω). By the above discussion, to g there is assigned a unique real-valued symmetric positive-definite matrix gij (p) which depends smoothly on p ∈ Ω and gives the components of the metric g at p. If say, κ : Ω → Ω is a smooth diffeomorphism, the pushforward of g via κ, κ∗ g, will be the metric at y = κ(x) whose components are  
κ∗ g(y) ij = ((κ0 (x))−1 )t (g(x))(κ0 (x))−1 , y = κ(x), (2.1.1) ij

with κ0 (x) being the Jacobian of the map y = κ(x). Let us now work towards defining the Laplace-Beltrami operator Pnassociated to our metric g = (gij (p)). Recall that the Euclidean Laplacian ∆ = j=1 ∂ 2 /∂x2j is the Euclidean divergence of the Euclidean gradient, i.e., ∆f = div(grad f ).

(2.1.2)

We shall define the Laplace-Beltrami operator, ∆g , associated with g by this sort of formula where div = divg and grad = gradp are associated with g in a natural way which will be consistent with the above change of variables formula and also be consistent with other formulas in Euclidean space involving the Euclidean version of these operators. If, as above, X is a smooth vector field on Ω, we define its divergence by setting n X


div X (p) = divg X (p) = |g|−1/2 ∂i |g|1/2 X i (p),

(2.1.3)

i=1

with |g| = |g(p)| denoting the determinant of the components of the metric at p, i.e., |g| = |g(p)| = det (gij (p)). (2.1.4) Another property of the Euclidean Laplacian that we wish to preserve for ∆g besides (2.1.2) is the formula −h∆u, vi = hgrad u, grad vi.

(2.1.5)

18

CHAPTER 2

In the Euclidean case the left side is just Z − (∆u) v dx. Ω

Because we want our formulas for ∆g to be covariant (i.e., behave well with respect to changes of variables), the natural L2 -inner product that will be the one corresponding to ∆g will be the one with respect to the volume element dVg = |g|1/2 dx1 · · · dxn .

(2.1.6)

In other words, if u and v are scalar valued Z hu, vig = u v dVg . Ω

We leave it as a straightforward exercise for the reader to check that this formula is well behaved with respect to the above change of variables κ : Ω → Ω, i.e., h u(κ( · )), v(κ( · )) ig = hu, viκ∗ g , if, as above, κ∗ g, is the pushforward of g. Besides (2.1.2) and (2.1.5), another Euclidean formula that we want to generalize is hX, grad ui = −hdiv X, ui, (2.1.7) assuming that u ∈ C ∞ (Rn ) is real-valued and that X is a smooth real vector field on Rn . In the present case we assume that X is a smooth real vector field on Ω, as is the vector field grad u, that we have yet to define, assuming as above that u is smooth and real-valued. Since divg X and u are scalar valued then, by the above, Z −hdivg X, uig = − (divg X)(p) u(p) dVg (p) ZΩ X =− |g|−1/2 ∂j (|g|1/2 X j ) u |g|1/2 dx Ω Z X = X j (p) ∂j (u(p)) dVg (p). Ω

We also naturally define the analog of the left side of (2.1.7) using the metric as Z hX, grad uig = g(X, grad u)(p) dVg (p). Ω

Thus, for (2.1.7) to be valid, we need to define the vector field grad u = gradg u in such a way that at every point p ∈ Ω, we have X g(X, grad u)(p) = X i (p)∂i u(p). (2.1.8) But this will be the case if we define Y = grad u,

if Y j (p) =

n X k=1

g jk (p)∂k u(p),

(2.1.9)

19

GEODESICS AND THE HADAMARD PARAMETRIX

with (g jk ) being the inverse of the matrix (gjk ), i.e., (g jk (p)) = (gjk (p))−1 .

(2.1.10)

For then we have g(X, grad u)(p) =

X

X i gij (p)Y j =

i,j

X

X i gij (p)g jk (p)∂k u

i,j,k

=

X

X i δki ∂k u =

i,k

X

X i ∂i u,

i

which establishes (2.1.8) and consequently (2.1.7). Here, δji denotes the Kronecker delta function, which equals 1 for i = j and 0 otherwise. Now that we have defined the divergence and gradient operators associated with g, we define the Laplace-Beltrami operator ∆g u = div (grad u) = |g|−1/2

n X


∂j |g|1/2 g jk ∂k u,

(2.1.11)

j,k=1

according to (2.1.2) Recall that, in addition to (2.1.2), we want that the analog of (2.1.5) holds in our context, i.e., −h∆g u, vig = h gradg u, gradg v ig ,

u, v ∈ C0∞ (Ω),

(2.1.12)

assuming, say, that u and v are real-valued. We are in luck. For if we expand the left side and integrate by parts we obtain Z X −h∆g u, vig = − |g|−1/2 ∂j (|g|1/2 g jk )∂k u v |g|1/2 dx Ω

=

j,k

Z X X Ω

j


g jk ∂k u ∂j v|g|1/2 dx

k

Z =

g(grad u, grad v)(p) dVg Ω

= hgrad u, grad vig , as desired. In addition to (2.1.12), it is clear that ∆g is self-adjoint with respect to the above inner product. By this we mean that h ∆g u, v ig = h u, ∆g v ig , u, v ∈ C0∞ (Ω),

(2.1.13)

just as the Euclidean Laplacian is in Rn . Later on, when the context is clear, we might drop the subscript g in the inner-product. Note also that if we take u = v and combine the last two identities, we get Z X (2.1.14) g jk (x)∂j u(x)∂k u(x) dVg = −h∆g u, ui, u ∈ C0∞ (Ω). Ω j,k

Remark 2.1.1 Let us see how Remark 1.1.2 carries over to the present context. Since we have defined the Laplace-Beltrami operator in a coordinate-free manner

20

CHAPTER 2

we can write down formulas using (2.1.1) for pullbacks from one coordinate system to another. Specifically, if y = κ(x), κ : Ω → Ω is a diffeomorphism, then we have that the Laplace-Beltrami operator in the y-variables ∆g(y) = |g|−1/2

n X j,k=1


∂ ∂ |g|1/2 g jk (y) ∂yj ∂yk

pulls back via κ to the following one in the x-variables ∆g∗ (x) = |g ∗ |−1/2

n X j,k=1


jk
∂ ∂ |g ∗ |1/2 g ∗ (x) , ∂xj ∂xk

where (g ∗ (x))jk = (κ0 (x))t (g(y)) (κ0 (x))


jk

,

y = κ(x),

∗ and of course |g ∗ | = det(gjk ), and ((g ∗ (x))jk ) = ((g ∗ (x))jk )−1 . In other words, if ∆g(y) and ∆g∗ (x) are as above, we have that ∆g(y) u(y) = f (y) when ∆g∗ (x) u∗ (x) = f ∗ (x), where u∗ (x) = u(κ(x)) and f ∗ (x) = f (κ(x)). Note that while the principal symbol of ∆g(y) is

p(y, ξ) =

n X

g jk (y)ξj ξk ,

(2.1.15)

j,k=1

that of its pullback is p(κ(x), ((κ0 (x))t )−1 ξ).

(2.1.16)

As we shall see in §2.3, the map (x, ξ) → (κ(x), ((κ0 (x))t )−1 ξ) is consistent with the formula in the cotangent bundle. Recall that if P (x, D) = P change of variables α a (x)D is a differential operator of order m, its principal symbol is |α|≤m αP α p(x, ξ) = |α|=m aα (x)ξ . Thus, p(x, D) is the top-order part of P (x, D), and so P (x, D) − p(x, D) is of order m − 1. 2.2

SOME ELLIPTIC REGULARITY ESTIMATES

Let us start this section by seeing how we can use (2.1.14) to prove a simple elliptic regularity estimate for ∆g . Like (gjk (x)), its inverse (g jk (x)) is positive definite. Therefore if K is a compact subset of Ω, n X g jk (x)∂j u(x)∂k u(x) cK ≤

j,k=1 n X

≤ c−1 K ,

x ∈ K,

2

|∂j u(x)|

j=1

for some constant cK > 0 depending on K. Therefore, there is a uniform constant AK < ∞ so that X Z |∂ α u|2 dVg ≤ AK h−∆g u, uig , u ∈ C0∞ (Ω). (2.2.1) |α|=1

K

From this, we can deduce the following result.

21

GEODESICS AND THE HADAMARD PARAMETRIX

Proposition 2.2.1 Let K b Ω be fixed. Then there is a constant BK depending on K so that Z X Z |∂ α u(x)|2 dVg + |u|2 dVg ≤ BK h(−∆g + 1)u, uig , u ∈ C0∞ (Ω). (2.2.2) |α|=1

K

Ω

To prove (2.2.2), we note that Z h(−∆g + 1)u, uig = −h∆g u, uig +

|u|2 dVg ,

Ω

and, consequently, we get (2.2.2) from (2.2.1) if Bk = Ak + 1. Note that an application of the Schwarz inequality shows that (2.2.1) implies that X Z |∂ α u|2 dVg ≤ Bk k(−∆g + 1)uk2L2 (Ω) . (2.2.3) |α|≤1

K

Let us now see that we can strengthen (2.2.3) in some sense by controlling L2 norms of second derivatives at the expense of a lower error term in the right. Before we state this result, let us recall the corresponding result for the Euclidean case since we shall need this result in the proof. The simple result that we require says that n Z X j,k=1

|∂j ∂k u|2 dx ≤

Z

Rn

u ∈ C0∞ (Rn ).

|∆u|2 dx,

Rn

(2.2.4)

This is easy to prove. For if we integrate by parts, we find that the left side equals Z XZ XZ ∂j2 u ∂k2 u dx = |∆u|2 dx. ∂j ∂k u ∂j ∂k u dx = j,k

Rn

j,k

Rn

Rn

Let us state (2.2.4) in an equivalent manner. If E is the fundamental solution for ∆ that we computed in Theorem 1.1.1, let E0 f = E ∗ f, We then have

( ∆E0 f = f, E0 ∆u = u,

f ∈ S(Rn ). f ∈ L2 (Rn ), u ∈ S(Rn ),

(2.2.5)

and, by (2.2.4), X

k∂ α E0 hkL2 (Rn ) ≤ CkhkL2 (Rn ) .

(2.2.6)

|α|=2

Using these facts, we can prove the following. Proposition 2.2.2 Let K b Ω be fixed. Then there is a uniform constant CK so that Z  X Z X Z |∂ α u|2 dVg ≤ CK |∆g u|2 dVg + |∂ α u|2 dVg , |α|=2

K

Ω

|α|≤1

Ω

u ∈ C0∞ (Ω). (2.2.7)

22

CHAPTER 2

Proof. Using a partition of unity argument and the compactness of K, we see that in order to prove (2.2.7) it suffices to assume that u is supported in a small neighborhood of a given point x0 . Since dVg is comparable to dx on compact sets, it then also suffices to prove the analog of P (2.2.7) where dVg is replaced by dx throughout. As a final reduction, since ∆g = j,k g jk (x)∂j ∂k plus lower order terms, which can be absorbed in the last term of (2.2.7), we conclude that it suffices to show that when u ∈ C0∞ is supported in a small neighborhood N0 of x0 , we have Z X X Z 2 (2.2.8) g jk (x)∂j ∂k u(x) dx, |∂ α u|2 dx ≤ C |α|=2

j,k

assuming for convenience that X

g jk (x0 )∂j ∂k = ∆.

(2.2.9)

j,k

We can achieve (2.2.9) if we make a linear change of variable so that in the new coordinates gjk (x0 ) = 1 for j = k and 0 for j = 6 k. P If j,k g jk (x)∂j ∂k u = f, then if χ ∈ C0∞ (Rn ) satisfies 0 ≤ χ ≤ 1, χ = 1 on N0 , but vanishes outside a small neighborhood, N , of N0 , then we have X  X
f= g jk (x)∂j ∂k u(x) = ∆u(x) + χ(x) g jk (x) − g jk (x0 ) ∂j ∂k u(x) . j,k

j,k 2

If h = ∆u ∈ L , then we know from (2.2.5) that u = E0 h. Thus, we want to solve for h in the following X 
h(x) + χ(x) g jk (x) − g jk (x0 ) ∂j ∂k E0 h = f. (2.2.10) j,k

By (2.2.6) and the continuity of the coefficients g jk (x), we have that

X
1

(g jk (x) − g jk (x0 ))∂j ∂k E0 h 2 ≤ khkL2 (dx) ,

χ(x) 2 L (dx) j,k

assuming, as we may, that supp χ is small enough. Consequently, if X
R = χ(x) (g jk (x) − g jk (x0 ))∂j ∂k E0 , j,k

then I + R has an inverse with L2 (Rn ) → L2 (Rn ) norm of at most 2. Thus, (2.2.10) has a unique solution given by h = (I + R)−1 f. If we set E1 = E0 (I + X R)−1 , then E1 f = E0 h, where h satisfies (2.2.10). If, as above, f = g jk (x)∂j ∂k u, then E1 f = E0 h = u, and therefore, by j,k

(2.2.6), we have X X k∂ α ukL2 = k∂ α E0 (I + R)−1 f kL2 ≤ Ck(I + R)−1 f kL2 |α|=2

|α|=2

X jk

g (x)∂j ∂k u L2 , ≤ 2Ckf kL2 = 2C j,k

as desired.



23

GEODESICS AND THE HADAMARD PARAMETRIX

Remark 2.2.3 We also remark that, just as the identity (2.1.14) implied inequality (2.2.2), we could have used another identity from Riemannian geometry, the Bochner identity, to more directly prove Proposition 2.2.2 in the spirit of the easy proof of the corresponding Euclidean estimate (2.2.4). The Bochner identity for ∆g says that 1 ∆g |∇g u|2 = |∇2g u|2 + h∇g u, ∇g ∆g ui + Ric(∇g u, ∇g u) , 2 where Ric is the Ricci curvature and all operators are with respect to the metric g. (See, e.g., [43].) Integrating this identity leads to (2.2.7). However, since one of our goals is to present the material that we need requiring the minimal background of the reader, we chose instead to prove Proposition 2.2.2 less directly by using the above freezing of the coefficients argument. Another (and related) proof can be obtained by using the Hadamard parametrix construction given at the end of this chapter. Let us conclude this section by proving a stronger version of (2.2.7) involving Sobolev spaces. Recall that in Rn the L2 -Sobolev spaces are defined by the norms −n 2

Z

2

2 s

1/2

|ˆ g (ξ)| (1 + |ξ| ) dξ

kgkH s (Rn ) = (2π)

.

(2.2.11)

Rn

If m is a nonnegative integer, then Plancherel’s theorem implies that X Z kgk2H m (Rn ) ≈ |∂ α g|2 dx. |α|≤m

We say that a given u ∈ H 1 supported in Ω solves the equation ∆g u = f ∈ L2 in the sense of distributions (i.e., is a weak solution) if Z Z f v dVg = u ∆g v dVg , ∀v ∈ C0∞ (Ω), Ω

Ω

or equivalently, by (2.1.12), Z

Z f v dVg = −

Ω

n X

g ij (x)∂i u(x) ∂j v(x) dVg .

Ω i,j=1

We then have the following. Proposition 2.2.4 Suppose that u ∈ H 1 is supported in Ω and ∆g u ∈ H m for a given m = 0, 1, 2, . . . . Then if K b Ω there is a constant C = C(K, m) so that
(2.2.12) kukH m+2 (K) ≤ C k∆g ukH m + kukL2 . To prove this we first note that if (2.2.12) has been established for integers < m, we get the result for m by applying the inequality with m replaced by m − 1 to ∂j u, j = 1, 2, . . . , n, due to the fact that the commutator [∆g , ∂j ] = ∆g ∂j − ∂j ∆g is a m−1 second order differential operator and thus maps H m+1 to Hloc . Thus, we need only to prove (2.2.12) for m = 0.

24

CHAPTER 2

By (2.2.7) and (2.2.3), we know that the estimate R holds with uniform bounds if u ∈ C0∞ (Ω). To use this, choose φ ∈ C0∞ (Rn ) with φ dx = 1 and let Z ε u (x) = u(x − εy)φ(y) dy. Then uε ∈ C0∞ (Ω) if ε > 0 is small enough. Since (2.2.12) must hold for the uε with uniform bounds, we would get the same estimate when m = 0 for u if we knew that for f = ∆g u we have k∆g uε − f ε kL2 → 0,

as ε → 0.

This follows from Friedrich’s lemma. Lemma 2.2.5 Let v ∈ L2 (Rn ) and suppose that a is a Lipschitz function on Rn satisfying |a(x) − a(y)| ≤ M |x − y|, x, y ∈ Rn . Then if φ ∈ C0∞ (Rn ) and φε (x) = ε−n φ(x/ε), ε > 0, then for j = 1, 2, . . . , n, we have

(a∂j v) ∗ φε − a(∂j v ∗ φε ) 2 n L (R ) Z  ≤M (|φ| + |y| |∂j φ| ) dy kvkL2 (Rn ) . (2.2.13) Rn

Proof. We may assume that v ∈ C0∞ (Rn ) since C0∞ (Rn ) is dense in L2 (Rn ), and for v ∈ C0∞ (Rn ) the limit as ε → 0 of the left side of (2.2.13) is clearly zero. We then wish to estimate the L2 (dx) norm of Z
a(x − y) − a(x) (∂j v)(x − y) φε (y) dy Z
= a(x − y) − a(x) v(x − y) ∂j φε (y) dy Z − (∂j a)(x − y) v(x − y) φε (y) dy Z
≤ M |v(x − y)| |y| |∂j φε (y)| + |φε (y)| dy. The estimate (2.2.13) now follows from Minkowski’s integral inequality due to the fact that Z Z

|y| |∂j φε (y)| + |φε (y)| dy = |y| |∂j φ(y)| + |φ(y)| dy.  . One can also show that Proposition 2.2.2 implies Proposition 2.2.4 by using difference quotients instead of Friedrich’s lemma. See, e.g., [22], §6.3.1. 2.3

GEODESICS AND NORMAL COORDINATES—A BRIEF REVIEW

Given our Laplace-Beltrami operator X ∆g = |g|−1/2 ∂/∂yj (|g|1/2 g jk (y))∂/∂yk ,

25

GEODESICS AND THE HADAMARD PARAMETRIX

the purpose of this section is to show that, given any point y0 in Ω, we can choose a natural local coordinate system y = κ(x) vanishing at y0 so that the quadratic form associated with the metric takes a special form. In particular, in these coordinates, rays through the origin will be geodesics for the metric and the associated geodesic distance from y0 will correspond to lengths along these rays. The square of the geodesic distance will equal this quadratic form, as in Remark 1.1.2. Geodesics connecting nearby points are curves γi = γi (t), i = 1, . . . , n, of shortest length measured by the metric. Therefore, let us consider the variational problem of looking for the critical points of the length functional Z 1p g(γ 0 , γ 0 ) dt, `(γ) = 0

for curves with fixed endpoints γ(0) = x0 and γ(1) = x1 , with x0 close to x1 . If γ were a minimizer, it would then follow that if γε (t) = γ(t) + εη(t) with η(0) = η(1) were a one-parameter family of curves with γε (0) = x0 and γε (1) = x1 , then we would have `(γ) ≤ `(γε ) = `(γ + εη). Consequently, d `(γε ) = 0. dε ε=0 If we set L(x, z) =

X

gij (x)zi zj ,

i,j

with zi = γi0 (t), then the preceding identity becomes Z 0= 0

1

1 −1/2 h ∂L ∂L 0 i L · η(t) + · η (t) dt 2 ∂x ∂z  Z 1 1 −1/2 ∂L d  1 −1/2 ∂L  = L − L · η dt, 2 ∂x dt 2 ∂z 0

integrating by parts and using our assumption that η(0) = η(1) = 0 in the last step. Thus, since the last term must vanish for all such choices of η, we conclude that extremals of our problem are solutions of the associated Euler-Lagrange equations 1 −1/2 ∂L d  1 −1/2 ∂L  L L − = 0, 2 ∂xi dt 2 ∂zi

i = 1, . . . , n.

If we parameterize the curve by arclength Z tp s= g(γ 0 , γ 0 ) dt, 0

then L ≡ 1, and so the equation simplifies to the “geodesic equation,” ∂L d  ∂L  − = 0, ∂xi dt ∂zi

i = 1, . . . , n,

which is the Euler-Lagrange equation for the energy functional Z E(γ) = a

b

g(γ 0 , γ 0 ) dt.

(2.3.1)

26

CHAPTER 2

This remains true if the curve is parameterized by a multiple of arclength. Thus, minimizing E(γ) is the same as minimizing `(γ), assuming that the square of the speed, s → g(γ 0 (s), γ 0 (s)), is constant. Minimizers (keeping the endpoints fixed) are geodesics, and we call the length `(γ) of this geodesic, dg (x0 , x1 ), the geodesic distance between x0 and x1 . Let us now show that the second order system (2.3.1) of n-variables xi is equivalent to a first order system of 2n-variables. We first note that if γ(t) = (x1 (t), . . . , xn (t)), then we can rewrite (2.3.1) as Xh ∂gjk dxj dxk i d X dxk  − 2gik = 0, ∂xi dt dt dt dt

i = 1, . . . , n.

(2.3.2)

k

j,k

If we define ξ = ξ(t), by setting X dxk = g kj (x)ξj , dt j

k = 1, . . . , n,

(2.3.3)

where, as before, (g ij (x)) = (gij (x))−1 and substitute this into (2.3.2), we get i  X h ∂gjk d X g ja ξa g kb ξb − 2gik g ka ξa = 0, ∂xi dt

j,k,a,b

i = 1, . . . , n,

k,a

which yields −

dξi 1 X ∂g jk ξj ξk = , 2 ∂xi dt

i = 1, . . . , n,

(2.3.4)

j,k

P since j g ij gkj = δki , and, if G = (g jk (x)), then ∂i (G−1 ) = −G−1 (∂i G)G−1 . The system (2.3.3)-(2.3.4) involves (half of) the principal symbol of ∆g , i.e., p(x, ξ) =

1 X jk g (x)ξj ξk . 2 j,k

The Hamilton vector field associated with p is Hp =

∂p ∂ ∂p ∂ − . ∂ξ ∂x ∂x ∂ξ

The integral curves are (x(t), ξ(t)), where

and

X dxk ∂p = x˙ k (t) = = g kj ξj , dt ∂ξk j

(2.3.5)

∂p dξi 1 X ∂g jk = ξ˙i (t) = − =− ξj ξk , dt ∂xi 2 ∂xi

(2.3.6)

j,k

with g jk and ∂g jk /∂xi being evaluated at x(t), and ξ = ξ(t). If x(0) = x0 and x(1) = x1 , then, by the above discussion, γ(t) = x(t) is the geodesic with endpoints x0 and x1 , which satisfies X γk0 (0) = g kj (x0 )ξj (0). j

GEODESICS AND THE HADAMARD PARAMETRIX

27

Let Φt = exp tHp so that Φt (x0 , ξ0 ) = (x(t), ξ(t)) denotes the Hamilton flow starting at Φt (x0 , ξ0 )|t=0 = (x0 , ξ0 ) = (x(0), ξ(0)). Note that p must be constant on the flow since n  X ∂p dξj  d ∂p dxj = 0, p(x(t), ξ(t)) = + dt ∂xj dt ∂ξj dt j=1

by (2.3.5) and (2.3.6). Thus, if γ(t) = x(t), the square of the speed, X t→ gjk (x(t))γj0 (t)γk0 (t), jk

P has constant value 2p(x0 , ξ0 ), since γk0 (t) = x˙ k (t) = j g jk (x(t))ξj (t). Let us assume for now for simplicity that x0 = 0 and that gij (0) = δji , meaning that at x0 = 0 the metric is the standard one. We can always achieve this by making a linear change of variables by Jacobi’s theorem about diagonalizing quadratic forms. By homogeneity, we have that Φt (0, ξ) = (x(t), ξ(t)) =⇒ Φst (0, ξ/s) = (x(t), ξ(t)/s).

(2.3.7)

Let Π denote the projection onto the x-variable. Because of (2.3.7), we have that ΠΦ1 (0, η) = η + O(|η|2 ), which means that near η = 0, the map η → ΠΦ1 (0, η)

(2.3.8)

is a diffeomorphism whose Jacobian is the identity when η = 0. Therefore, near x = 0 we can change coordinates so that the map (2.3.8) is the identity map. These are the local normal coordinates near x0 vanishing there. By (2.3.7), in these coordinates we have that ΠΦt (0, η) = ΠΦ1 (0, tη) = tη. Thus, if x(t) is as in (2.3.7), we have that x(t) = tη, and therefore x(t) ˙ ≡ η,

if ξ(0) = η.

(2.3.9)

Note that the straight lines through the origin x(t) = tη are geodesics (when tη is small), since (x(t), ξ(t)) solves Hamilton’s equations (2.3.5)-(2.3.6). Given our formula for x(t), we can write down P a formula for ξ(t) using Hamilton’s equation. Specifically, since ηj = x˙ j (t) = k g jk (tη)ξk (t), we have that X ξj (t) = gjk (tη)ηk . (2.3.10) k

Since, as noted above, t → p(x(t), ξ(t)) must be constant, we obtain X p(x(t), ξ(t)) ≡ p(0, η) = 21 x2j , since ξ(0) = η = x.

28

CHAPTER 2

Therefore, since ΠΦ1 (0, η) = x(1) = η = x, and ξ(1) is given by (2.3.10), we have the fundamental identity that, in our coordinates, 1X 2 1 X jk 1X xj = g (x)ξj (1)ξk (1) = gjk (x) xj xk . 2 j 2 2 j,k

(2.3.11)

j,k

Recalling the definition of the geodesic distance, dg , given above, this equation says P that in these coordinates we have that its square satisfies, d2g (0, x) = j x2j . We claim that in our normal coordinate system we have a stronger formula. Namely, n X xj = gjk (x)xk , j = 1, 2, 3, . . . , n. (2.3.12) k=1

After relabeling, it suffices to check this formula when x = (x1 , 0, . . . , 0) is on the 1st coordinate axis. In that case, for j 6= 1, the formula would follow if g1k , k 6= 1, vanished identically near the origin on the x1 -axis. This is a consequence of Gauss’s lemma (see [16], p. 69) since here this quantity denotes the angle measured by the metric between the tangent vector of the geodesic, ∂/∂x1 , and the tangent vector, ∂/∂xk , k = 6 1, to the geodesic sphere here, which must be zero here by Gauss’s lemma. (Note that the geodesic sphere of radius r in our is P coordinates just the usual sphere of radius r centered at the origin, i.e., {x : x2j = r2 }, if r is small.) We still need to check that the formula is valid for j = 1, but this trivially follows from (2.3.11), since on the 1st coordinate axis, the identity is just that 1 2 2 x1

= 12 g11 (x1 , 0, . . . , 0) x21 .

This means that g11 ≡ 1 on this axis (for small x), which means that (2.3.12) is also valid here for j = 1. This finishes the proof of the claim, which means that we have established the following. Theorem 2.3.1 that

Let gij (y) be a metric on Ω and assume that at y0 ∈ Ω we have gij (y0 ) = δji .

(2.3.13)

Then we can find local coordinates κ(y) = x vanishing at y0 with Jacobian equal to the identity P there so that (2.3.12) is valid. Furthermore, in these coordinates, d2g (0, x) = j x2j . If (2.3.13) is not satisfied, one can achieve this by making an initial linear change of variables. As a result, we have the following. Corollary 2.3.2 If Y is a fixed relatively compact subset of Ω, then for z ∈ Y there are coordinates κz (y) = x vanishing at z satisfying κ0z (z) = In and X X gij (x)xj = gij (0)xj , i = 1, . . . , n, (2.3.14) j

j

and (dg (0, x))2 =

X i,j

gij (0)xi xj .

(2.3.15)

29

GEODESICS AND THE HADAMARD PARAMETRIX

One can choose δ > 0 so that for every z ∈ Y , κz is a diffeomorphism from all points y ∈ Ω with dg (z, y) < δ to points x = (x1 , x2 , . . . , xn ) ∈ Rn satisfying P ( i,j gij (0)xj xk )1/2 < δ. The map (z, y) → κz (y) depends smoothly on such z, y ∈ Ω. This result follows from Theorem 2.3.1 and a linear change of variables. Indeed if LLt = g ij (z) is the Cholesky decomposition of the cometric at z, and if T = L−1 , we can use the affine bijection y → T (y − z) sending z to the origin. For in these new variables we have that gij (0) = δji , which provides one way of seeing how the corollary follows from Theorem 2.3.1. Alternatively, if z = 0 and we set G0 = (g jk (0)), then one notes that the coordinates are satisfied when the map G0 η → ΠΦ1 (0, η) is the identity. Since in the initial coordinate system the Jacobian matrix of this map is the identity when η = 0, one can change variables near z = 0 to achieve this condition. The details are left to the reader. Remark 2.3.3 Following the reasoning in Remark 1.1.2, we can use Corollary 2.3.2 to describe another natural coordinate system vanishing at z. If, as above g ij (z) = LLt is the Cholesky decomposition of the cometric at z and we set T = L−1 , then the normal coordinates given by κg (y, z) = T κz (y), where κz (y) is as in the corollary, will also vanish at z, and, moreover, will satisfy n X i=1

vi2 =

n X


2 gij (x)xi xj = dg (y, z) , if x = κz (y),

and v = κg (y, z).

i,j=1

Additionally, since κ0z (y)|y=z = In , we have that κg (y, z) = T (y − z) + O(|y − z|2 ). Finally, if we write for y 6= z r > 0, ω ∈ S n−1 ,

κz (y, z) = rω,

then (r, ω) are called the geodesic polar coordinates of y about z, and, as noted above, we then have r = dg (y, z). Starting from z one arrives at y after traveling a distance r along the geodesic whose initial direction is ω. We remark that the map in (2.3.3) which identified the tangent vector x(t) ˙ in Tx Ω at x = x(t) with the cotangent vector ξ(t) ∈ Tx∗ Ω (the dual of Tx Ω) at x is the standard one, which is called the musical isomporphism. This map assigns to every tangent vector v ∈ Tx Ω the cotangent vector v[ = ξ = ιTx Ω→Tx∗ Ω (v) ∈ Tx∗ Ω given by X ξj = gij (x)v i (2.3.16) i

(lowering indices using the metric), so that X X ιTx Ω→Tx∗ Ω (v)(w) = ξj wj = gij (x)v i wj . j

i,j

30

CHAPTER 2

The inverse isomorphism, which is Pmap in the opposite direction, assigns to ξ ∈ Tx∗ Ω, ξ ] = v ∈ Tx Ω, where v j = i g ij (x)ξi (raising indices using the cometric). Note that this map first arose for us in the definition of the gradient, (2.1.9). The cotangent bundle, T ∗ Ω, is the union over all x ∈ Ω of Tx∗ Ω. Note that since κ0 (x)t ∂/∂y = ∂/∂x, if y = κ(x) is a diffeomorphism, we see from this that the pullback formula for the cotangent bundle is the map (κ(x), ξ) → (x, κ0 (x)t ξ). (2.3.17) P j For the pullback of the tangent vector v = v ∂/∂yj at y = κ(x) is the tangent P vector w = wj ∂/∂xj ∈ Tx Ω, where v = κ0 (x)w, which implies (2.3.17) because of (2.1.1) and (2.3.16). Note that (2.3.17) is exactly the transformation that occurred in formula (2.1.16) for the pullback of the principal symbol of ∆g under changes of coordinates. Note also that if, as in (2.3.17), (x, ξ) → (y, η) = (κ(x), (κ0 (x)t )−1 ξ) then the Liouville measure dxdξ on T ∗ Ω just gets transformed to the corresponding measure dydη in the new coordinates. We also implicitly used in the proof of Theorem 2.3.1 that Hamilton’s equations are preserved under this change of variables. In general, a transformation χ(x, ξ) = (y, η) of the cotangent bundle that preserves Hamilton’s equations x˙ i =

∂p ξ˙i = − , ∂xi

∂p , ∂ξi

i = 1, . . . , n,

is called a canonical transformation. If we call Y = (x, ξ), then we can write the above system more succinctly as ∂p Y˙ = J , ∂Y

(2.3.18)

where J is the symplectic matrix 

0 −In

In 0

 .

(2.3.19)

If Z = χ(x, ξ) = (y, η), then in order for Hamilton’s equation to be valid in the new coordinates we need that ∂p , Z˙ = J ∂Z where we are expressing p here in the Z coordinates. If we use the chain rule relating the Y and Z coordinates, we get from (2.3.18) that ∂p Z˙ = M JM t , ∂Z where M is the Jacobian matrix of χ. Thus, the condition for Hamilton’s equation to be valid in the new coordinates is M JM t = J.

(2.3.20)

The reader can check that the map (2.3.17) is a canonical transformation since it satisfies this condition. See, e.g., Chapter 10 of [27] for more details. Finally, in view of the above discussion, we see that the proof of Theorem 2.3.1 and Corollary 2.3.2 shows that the existence of normal coordinates is the same

31

GEODESICS AND THE HADAMARD PARAMETRIX

thing as the existence of a natural exponential map from the tangent space at each point z ∈ Ω. Specifically, if v is a tangent vector with small norm |v|g(z) = P ( j,k gjk (z)v j v k )1/2 , then one defines expz v to be the point x on the geodesic with tangent vector v at z satisfying dg (z, x) = |v|g(z) . In normal coordinates κz (y) = x vanishing at z, this map is the identity map. Consequently, the last part of Corollary 2.3.2 follows from the fact that the map (z, v) → (z, expz v) is a diffeomorphism of a neighborhood of the zero section in the cotangent bundle of Ω, T Ω, to a neighborhood of the diagonal in Ω × Ω, { (x, y) ∈ Ω × Ω : x = y }. 2.4

THE HADAMARD PARAMETRIX

Let us start this section by going over a variation of the example in Remark 1.1.2. We shall assume for the moment that (g jk ) is a real symmetric positive definite n×n matrix. As usual, let (gjk ) denote its inverse. The advanced forward fundamental solution for  = ∂t2 − ∆, E+ (t, x), is a radial function of x and so, depending on the context, we shall write it either as E+ (t, x) or E+ (t, |x|). By Jacobi’s theorem for quadratic forms, there must be a linear bijection T : Rn → Rn so that (gjk ) = T t T , which means that the pullback of ∆ = ∂ 2 /∂y12 + · · · + ∂ 2 /∂yn2 via y = T x is n X j,k=1

∂ jk ∂ g . ∂xj ∂xk

Therefore, since (∂t2 − ∆)E+ (t, |x|) = δ0,0 (t, x), it follows that 

∂t2 −

n X j,k=1

∂ jk ∂  g E+ (t, dg (0, x)) = δ0,0 (t, T x) ∂xj ∂xk = det gjk


−1/2

δ0,0 (t, x), (2.4.1)

pP gjk xj xk is the distance from the origin with respect to the conif dg (0, x) = stant coefficient metric (gjk ). Note that p dg (0, x) = |T x|, where | · | denotes the Euclidean distance, and also det T = det gij . We claim that if local coordinates are chosen as in Corollary 2.3.2 and we take g jk = g jk (0), then the above calculationPremains valid when is small and we P t jk replace the constant coefficient operator ∂j g jk (0)∂k by ∂j g (x)∂k . We are thus assuming that in our coordinate system (2.3.14) is valid for x near 0, i.e., X X gij (x)xj = gij (0)xj , i = 1, . . . , n, (2.4.2) j

j

and that the geodesic distance from the origin is given by X
1/2 dg (0, x) = gij (0)xi xj .

(2.4.3)

i,j

Our claim then is that for small t X ∂
−1/2 ∂
g jk (x) E+ (t, dg (0, x)) = det gjk (0) δ0,0 (t, x). ∂t2 − ∂xj ∂xk j,k

(2.4.4)

32

CHAPTER 2

Recall that E+ (t, |x|) = 0 if |x| > t, and therefore by requiring that t be small in (2.4.4), we ensure that (2.4.2) and (2.4.3) are valid on suppx E+ (t, dg (0, x)). We also remark that the factor (det gjk )−1/2 in (2.4.4) is natural since (det gjk (x))−1/2 δ0,0 (x − y, t − s) is the Schwartz kernel for the identity operator on R×Ω with respect to the measure dtdVg . To verify this claim, by (2.4.1) and the way that E+ was first constructed as a limit of smooth functions of |x|2 , it suffices to show that if F ∈ C ∞ (R) then for x near 0 X


X
∂j g jk (0)∂k F glm (0)xl xm

j,k

l,m

X

=


X
∂j g jk (x)∂k F glm (0)xl xm . (2.4.5)

j,k

l,m

But this follows from the stronger identity X X

g jk (0)∂k F glm (0)xl xm k

(2.4.6)

l,m

=

X


X
g jk (x)∂k F glm (0)xl xm

k

l,m

= 2xj F

0

X


glm (0)xl xm ,

j = 1, . . . , n.

l,m

Note that, by the chain rule, X X

g jk (0)∂k F glm (0)xl xm k

l,m

= F0

X

X
glm (0)xl xm × 2 g jk (0)gkl (0)xl

l,m

= 2xj F

0

k,l

X




glm (0)xl xm .

l,m

Similarly, if we use (2.4.2) we get X
X
g jk (x)∂k F glm (0)xl xm k

l,m

= F0

X

0

X

X
glm (0)xl xm × 2 g jk (x)gkl (0)xl

l,m

=F

k,l




glm (0)xl xm × 2

l,m

= 2xj F 0

X

g jk (x)gkl (x)xl

k,l

X


glm (0)xl xm .

l,m

Combining this with the preceding inequality gives us (2.4.6), and consequently our claim that (2.4.4) is valid for small t has been established.

33

GEODESICS AND THE HADAMARD PARAMETRIX

P Note that ∂j g jk (x)∂k agrees with the Laplace-Beltrami operator ∆g associated with our metric gjk (x) up to lower order terms. Further care in the Hadamard parametrix will be needed to provide corrections for the lower order terms, but the fact that (2.4.4) and (2.4.5) are valid is the main idea behind the parametrix construction. To deal with the contributions of the lower order terms we shall use the distributions Eν , ν = 1, 2, 3 . . . defined in (1.2.30) to provide correction terms in the parametrix. Since, by Proposition 1.2.4, (∂t2 − ∆)Eν = νEν−1 , ν = 1, 2, 3, . . . , the proof of (2.4.4) yields n X

∂t2 −

j,k=1

∂
∂ jk g (x) Eν (t, dg (0, x)) ∂xj ∂xk = νEν−1 (t, dg (0, x)),

ν = 1, 2, 3, . . . , (2.4.7)

while, since E+ = E0 , we can rewrite (2.4.4) in the current notation as 

∂t2 −

n X j,k=1


−1/2 ∂ jk ∂  E0 (t, dg (0, x)) = det gjk δ0,0 . g (x) ∂xj ∂xk

(2.4.8)

Also, if for fixed t we let Eν (t, |x|) = F (|x|2 ) and use (2.4.6), we get that X

g jk (x)∂k Eν (t, dg (0, x)) = 2xj ∂r F (d2g (0, x)), k

where the last factor on the right denotes the derivative of F evaluated at d2g (0, x). Since (2r)−1 (∂r Eν )(t, r) = (∂r F )(r2 ), and since, by Proposition 1.2.4, −2r−1 (∂r Eν )(t, r) = Eν−1 , we get that X
− 2 g jk (x)∂k Eν (t, dg (0, x)) = xj Eν−1 (t, dg (0, x)),

ν = 1, 2, 3, . . . .

(2.4.9)

k

The substitute for ν = 0 of this is that if for fixed t we let E0 (t, x) = F0 (|x|2 ), then X
g jk (x)∂k E0 (t, dg (0, x)) = 2xj F00 (d2g (0, x)). (2.4.10) k

Note that, by (2.4.2), the identities (2.4.9) and (2.4.10) are equivalent to ∂k Eν (t, dg (0, x)) = −

 1 X gjk (0)xj Eν−1 (t, dg (0, x)), 2 j k = 1, . . . , n, ν = 1, 2, 3, . . . , (2.4.11)

and ∂k E0 (t, dg (0, x)) = 2

X j

respectively.

 gjk (0)xj F00 (d2g (0, x)),

k = 1, . . . , n,

(2.4.12)

34

CHAPTER 2

We now have the tools that we need to write down the Hadamard parametrix. Note first that n X

−∆g = −

j,k=1

n

X ∂ ∂ jk ∂ g (x) + , ak (x) ∂xj ∂xk ∂xk k=1

where −1/2

k

a (x) = −|g(x)|

n X


g jk (x)∂j |g(x)|1/2 .

j=1 ∞

Thus, if α0 ∈ C , by (2.4.8), (2.4.10) and (2.4.12) (∂t2 −∆g )α0 E0 (t, dg (0, x))

(2.4.13)

−1/2

= α0 (0)|g| δ0,0 − (∆g α0 )E0 (t, dg (0, x)) n n X X +2 gjk (0)ak (x)xj α0 F00 (d2g (0, x)) − 4 xj ∂j α0 F00 (d2g (0, x)) j=1

j,k=1 −1/2

δ0,0 − (∆g α0 )E0 (t, dg (0, x))
+ 2 ρα0 − 2hx, ∇x α0 i F00 (d2g (0, x))

= α0 (0)|g|

if ρ(x) =

n X

gjk (0)ak (x)xj =

j,k=1

n X

gjk (x)ak (x)xj .

j,k=1

Similarly, by using (2.4.7), (2.4.9) and (2.4.11) we get for ν = 1, 2, 3, . . . and αν ∈ C∞ (∂t2 − ∆g )αν Eν (t, dg (0, x)) = ναν Eν−1 (t, dg (0, x)) − (∆g αν )Eν (t, dg (0, x))
1 − ραν − 2hx, ∇x αν i Eν−1 (t, dg (0, x)). (2.4.14) 2 If we are able to choose α0 (x) so that it solves the transport equation ρα0 = 2hx, ∇x α0 i,

α0 (0) = 1,

(2.4.15)

and the αν (x), ν = 1, 2, 3, . . . , so that 2ναν − ραν + 2hx, ∇x αν i − 2∆g αν−1 = 0,

(2.4.16)

then we would have (∂t2 − ∆g )

N X

αν Eν (t, dg (0, x)) = |g|−1/2 δ0,0 − (∆g αN )EN (t, dg (0, x)).

(2.4.17)

ν=0

Let us first see that (2.4.15) has a unique solution in a ball, {x ∈ Rn : |x| < δ} in which ρ is defined. If we use polar coordinates, x = rω, ω ∈ S n−1 , then for r = |x| > 0, the first part of the equation says that along rays through the origin ∂r α0 =

ρα0 . 2r

35

GEODESICS AND THE HADAMARD PARAMETRIX

Since ρ(0) = 0, we conclude that the unique solution of (2.4.15) is given by ! Z 1  Z |x| 1 1 ds ds = exp α0 (x) = exp ρ(sω) ρ(sx) . 2 s s 0 2 0 Clearly, α0 is a smooth positive function, since s → ρ(sx)/s does not have a singularity at s = 0. Using the definition of ρ, one checks that if Θ(x) = (det (gjk (x)))1/2 then ρ = −r Θ−1 ∂r (Θ) = −r ∂r (ln Θ). Consequently, α0 (x) = Θ−1/2 (x)Θ1/2 (0) = |g(x)|−1/4 |g(0)|1/4 ,

(2.4.18)

if |g(x)| = det gij (x). We leave the details as an exercise for the reader. We can solve (2.4.16) in a similar manner. Let us assume inductively that αν−1 ∈ C ∞ has been constructed. If we let u = αν /α0 and f = (∆g αν−1 )/α0 , then we can rewrite (2.4.16) as νu + r∂r u = f, and so ∂r (rν u) = rν−1 f . Consequently, Z |x| Z rν u(rω) = sν−1 f (sω)ds = rν 0

1

sν−1 f (sx) ds.

0

In other words Z αν (x) = α0 (x) 0

1

tν−1

∆g αν−1 (tx) dt, α0 (tx)

ν = 1, 2, 3, . . . .

By induction, the αν given by this formula are smooth, and they are the unique solutions of (2.4.16). Note that we solved the transport equations (2.4.15)-(2.4.16) by integrating along the rays through the origin, which, in our normal coordinate system are geodesics. In view of their solution, we have shown that (2.4.17) is valid, which is the main step in the Hadamard parametrix construction. In carrying out the various calculations that lead to (2.4.17), we assumed that we were working in normal coordinates about a point y ∈ Ω. As we saw in §2.3, the Jacobian of the transformation to these coordinates is the identity at y. So if we pulled back the functions αν to the original coordinates we would obtain functions of x which we denote as αν (x, y) so that the variant of (2.4.17) holds for small t where δ0,0 is replaced by δ0,y and Eν by Eν (t, dg (x, y)) with dg (x, y) being the geodesic distance between x and y. (We are abusing notation a bit since we called x the normal coordinates about y before, while now we are letting x denote the original coordinates, but this inconsistency should not cause any confusion.) The coefficients αν (x, y) then are obtained by integrating the pullbacks of the equations (2.4.15) and (2.4.16) along the geodesic rays starting at y. Since we have seen in Corollary 2.3.2 that the geodesic coordinates vary smoothly with y, we thus obtain the following important result. Theorem 2.4.1 Let Y be a fixed open and relatively compact subset of Ω. Then there is a δ > 0 and functions αν ∈ C ∞ (Ω × Y ) so that when t < δ we have (∂t2 − ∆g )

N X


−1/2 αν (x, y)Eν (t, dg (x, y)) = det gjk (y) δ0,y (t, x)

ν=0


− ∆g αN (x, y) EN (t, dg (x, y)),

(x, y) ∈ Ω × Y, (2.4.19)

36

CHAPTER 2

with ∆g = |g|−1/2

n X j,k=1


∂ ∂ |g|1/2 g jk (x) ∂xj ∂xk

denoting the Laplace-Beltrami operator in the x-variable. The principal coefficient, α0 , satisfies α0 (y, y) = 1, y ∈ Y. The last term in (2.4.19) should of course be thought of as the error term in the parametrix. Assuming that δ is small, we can make it arbitrarily smooth by choosing N to be sufficiently large. For instance, since EN is a constant multiple of the distribution WN −(n−1)/2 defined in §1.2, the error term is bounded and continuous if N > (n − 1)/2 and t is as above. It will be in C α ((−∞, δ) × Ω × Y ) if N > α + (n − 1)/2. Also, since the first term in the right side of (2.4.19) is the Schwartz kernel at time zero for the identity operator on R × Y with respect to our measure dtdVg , as we shall see thepleft sidepof (2.4.19) provides an approximation to the kernel of the operator sin(t −∆g )/ −∆g that will be defined in the next chapter. Let us conclude this section by using geodesic normal coordinates to establish a uniqueness theorem for the Cauchy problem  2  (∂t − ∆g )u(t, x) = 0, x ∈ Ω (2.4.20)   u|t=0 = f, ∂t u|t=0 = g. Specifically we shall establish the following result which shows that g = ∂t2 − ∆g has finite (i.e., unit) propagation speed. Theorem 2.4.2 Let x0 ∈ Ω and assume that u ∈ C ∞ ([0, δ] × Bg (x0 , δ)), where Bg (x0 , δ) is the geodesic ball {y : dg (x0 , y) ≤ δ}, which is assumed to lie in Ω. Then if δ is sufficiently small and 0 < T < δ it follows that ∂t u(T, x) = u(T, x) = 0 for x ∈ Bg (x0 , δ − T ) if ∂t u(0, x) = u(0, x) = 0 for x ∈ Bg (x0 , δ). Proof. We may assume that δ is small enough so that we can work in geodesic normal coordinates in Bg (x0 , δ) vanishing at x0 with gij (0) = δij . Thus, by Theorem 2.3.1 we have n X

g ij (x)xj = xi ,

1 ≤ i ≤ n, if |x| ≤ δ,

(2.4.21)

j=1

where |x| denotes the Euclidean length of x = (x1 , . . . , xn ). We then must show that if u is smooth on [0, δ] × {|x| ≤ δ}, (∂t2 − ∆g )u = 0 there and u(0, x) = ∂t u(0, x) = 0,

|x| ≤ δ,

(2.4.22)

|x| ≤ δ − T.

(2.4.23)

with 0 < T < δ, then u(T, x) = ∂t u(T, x) = 0,

To prove this, we shall use the following analog of (1.2.27)

37

GEODESICS AND THE HADAMARD PARAMETRIX

n h i X 2∂t u (∂t2 − ∆g )u = ∂t (∂t u)2 + g ij ∂i u∂j u i,j=1

− 2|g|−1/2

n X

∂i ∂t u

i=1

n X


|g|1/2 g ij ∂j u , (2.4.24)

j=1

where g ij = g ij (x) and |g| = det(gij (x)). We shall integrate this identity over the region Γ = {(t, x) : |x| ≤ δ − t, 0 ≤ t ≤ T }, noting that the boundary consists of ΣT = {(T, x) : |x| ≤ δ − T }, Σ0 = {(0, x) : |x| ≤ δ} and Σ = {(t, x) : |x| = δ − t, 0 < t < T }. The Euclidean outward unit normal on ΣT is (1, 0), and (∂t u, ∇x u) = 0 on Σ0 by (2.4.22). If ν = (ν t , ν x ) denotes the outward unit normal to a point (t, x) ∈ Σ then |ν t | = |ν x |,

and

√ x 2 ν = x/|x|.

(2.4.25)

Therefore since (∂t2 − ∆g )u = 0 in Γ, by the Euclidean divergence theorem and (2.4.22), if we integrate (2.4.24) over Γ with respect to the measure dt|g|1/2 dx we get Z

h

0=

n X

(∂t u(T, x))2 +

|x|≤δ−T

i g ij (x)∂i u(T, x)∂j u(T, x) |g|1/2 dx

i,j=1

Z  n X   (∂t u(t, x))2 + g ij (x)∂i u(t, x)∂j u(t, x) |g|1/2 ν t + Σ

i,j=1

− 2|g|1/2 ∂t u

n X

 g ij (x)νix ∂j u(t, x) dS,

i,j=1

with dS being Lebesgue surface measure on Σ. Note that by (2.4.21) and (2.4.25) we have n n n X 2  X  X  ij x ij x x g (x)νi ∂j u ≤ g (x)νi νj g ij (x)∂i u∂j u i,j=1

i,j=1

i,j=1

= |ν x |2

n X

 g ij (x)∂i u∂j u ,

i,j=1

which implies that the integrand in the second integral is nonnegative since ν t = |ν x |. Since the same is true for the first integrand we conclude that ∂t u(T, x) = 0 and ∇x u(T, x) = 0 if |x| ≤ δ − T. Since this argument also shows that the same is true when T is replaced by a smaller nonnegative value of time and since we are assuming (2.4.22), we get (2.4.23). 

38

CHAPTER 2

Note that the argument we just gave shows that there is a local energy inequality for g . Namely, if δ > 0 is small, 0 < T < δ, u ∈ C ∞ ([0, δ] × Bg (x0 , δ)) and (∂t2 − ∆g )u(t, x) = 0, then Z



(t, x) ∈ [0, δ] × Bg (x0 , δ),

 (∂t u(T, x))2 + g(gradg u(T, x), gradg u(T, x)) dVg

Bg (x0 ,δ−T )

Z ≤ Bg (x0 ,δ)



 (∂t u(0, x))2 + g(gradg u(0, x), gradg u(0, x)) dVg . (2.4.26)

Chapter Three The sharp Weyl formula

3.1

EIGENFUNCTION EXPANSIONS

From now on we shall mostly be concerned with analysis on compact boundaryless Riemannian manifolds. Let us quickly recall a few facts about this setting. First, M is a C ∞ manifold if it is a Hausdorff space for which there is a countable n e collection of S open sets Ων ⊂ M together with homeomorphisms κν : Ων → Ων ⊂ R satisfying Ων = M and κν 0 ◦ κ−1 ν : κν (Ων ∩ Ων 0 ) → κν 0 (Ων ∩ Ων 0 )

is C ∞ .

Note that the above mapping is between open subsets of Rn . We call y = κν (x) ∈ e ν ⊂ Rn the local coordinates of x in the coordinate patch Ων . Ω This C ∞ structure allows us to define C ∞ functions on M in a natural way. We e ν , u(κ−1 (y)) is say that u is C ∞ , or u ∈ C ∞ (M ), if for every ν the function on Ω ν e ν ). in C ∞ (Ω Next, t is called a tangent vector at x if t is a continuous linear operator on C ∞ , sending real-valued functions to R, and having the property that if x ∈ Ων then there is a vector tν ∈ Rn so that t(u ◦ κν ) =

n X j=1

tνj

∂ u(y)|y=κν (x) ∂yj

e ν ). whenever u ∈ C ∞ (Ω

Thus, as was the case for tangent vectors in the Euclidean case, t annihilates constant functions and the vector space of all tangent vectors at x, Tx M , has dimension n. Note that if x ∈ Ων ∩ Ων 0 and if we set κ = κν 0 ◦ κ−1 ν : κν (Ων ∩ Ων 0 ) → κν 0 (Ων ∩ Ων 0 ), then, by the chain rule, t(u ◦ κν 0 ) =

n X j=1

where

tνj

0

∂ u(Y )|Y =κν 0 (x) ∂Yj 0

tν = κ0 (y)tν ,

when u ∈ C ∞ (κν 0 (Ων ∩ Ων 0 )),

y = κν (x).

Thus, if we let TM =

[

Tx M

x∈M

and use the coordinates (x, t) → (κν (x), tν ), x ∈ Ων , the tangent bundle becomes a C ∞ manifold of dimension 2n.

40

CHAPTER 3

A vector field on M is a continuous assignment mapping each point of M to a tangent vector to M at that point. In other words it is a function x → t(x) ∈ Tx M for which x → tν (x) is a continuous function from Ων to Rn for every ν, if tν as above are the fiber coordinates. The vector field is said to be differentiable or C ∞ if this map has this property for every ν. In view of (2.3.17) and the remarks following Corollary 2.3.2, it is also natural to consider the cotangent bundle which we now define. For each x ∈ M , the dual of Tx M ,STx∗ M , is also a vector space of dimension n. The cotangent bundle, T ∗ M = Tx∗ M , is the C ∞ manifold of dimension 2n having the structure coming from the local coordinates (x, ξ) → (κν (x), ξ ν ) if x ∈ Ων and ξ ν is the unique vector in Rn such that ν

ν

h t, ξ i = h t , ξ i =

n X

tνj ξjν ,

j=1

whenever t is a tangent vector at x and (κν (x), tν ) are the corresponding local coordinates in T M . Note that this implies the following transformation formula for cotangent vectors: 0 κ0 (y)t ξ ν = ξ ν , y = κν (x), which is the analog of (2.3.17). This is because we require that 0

0

h tν , ξ ν i = h t, ξ i = h tν , ξ ν i. Thus, if (y, η) ∈ κν (Ων ∩Ων 0 )×Rn are the local coordinates of (x, ξ) ∈ T ∗ (Ων ∩Ων 0 ), it follows that (Y, ζ) = (κ(y), (κ0 (y)t )−1 η) are its local coordinates in κν 0 (Ων ∩Ων 0 )× Rn . Note also that the above transformation law allows us to define the Liouville measure on T ∗ M , which in local coordinates is just dxdξ. For this reason, the measure is also sometimes called the Lebesgue-Liouville measure. One can also define Riemannian metrics on M in analogy to what we did for ones on Euclidean space in the preceding chapter. A Riemannian metric on M is a family of positive definite inner products gx : Tx M × Tx M → R,

x ∈ M,

so that for all smooth vector fields Y and Z on M x → gx (Y (x), Z(x)) is a C ∞ function from M to R. In each system of local coordinates, y = κν (x), the vector fields ∂y∂ j , j = 1, . . . , n, give a basis of tangent vectors for each point in the coordinate patch. Relative to this coordinate system, the components of the metric at each point x ∈ Ων are given by  ∂   ∂   gij (y) = gx , , y = κν (x). ∂y i x ∂y j x Similarly, if we set |g| = det(gij (y)) then the Riemannian pvolume element, dVg , is the unique measure on M which in local coordinates is |g| dy, and the LaplaceBeltrami operator on M , ∆g , is the second order elliptic operator which in local coordinates is of the form n X ∂  1/2 jk  ∂ −1/2 |g| |g| g (y) , (3.1.1) ∂yj ∂yk j,k=1

41

THE SHARP WEYL FORMULA

if, as before, the matrix g jk (y) is the inverse of gjk (y). By this we mean that if e ν ) is defined by u u ∈ C0∞ (Ων ) and u e ∈ C0∞ (Ω e(y) = u(κ−1 ν (y)), then the value of ∆g u at x equals that of the operator in (3.1.1) applied to u e at y = κν (x). Note that ∆g is self-adjoint with respect to the inner product given by dVg , Z hu, vi = u v dVg , M

i.e., if u, v ∈ C ∞ (M ), then h∆g u, vi = hu, ∆g vi. One verifies this using a partition ∞ e of unity argument based on the fact that pthe operators in (3.1.1) on C0 (Ων ) are self-adjoint with respect to the measure |g| dy. If, as we shall assume from now on, M is compact there is a natural way of defining Sobolev spaces on M . If {φν } is a partition of unity subordinate to a finite S covering Ων = M by coordinate charts, then we set X kf kH s (M ) = kfν kH s (Rn ) , ν n where fν (y) = (φν f )(κ−1 ν (y)) and the R -Sobolev norms are given by (2.2.11). It is simple to check that different partitions of unity and such coordinate atlases give comparable norms. Using (2.1.12), Proposition 2.2.4 and a partition of unity argument one finds that if u ∈ H 1 and ∆g u ∈ L2 then

kuk2H 1 (M ) ≤ Ch (−∆g + 1)u, u i.

(3.1.2)

Note that given f ∈ L2 (M ) there is a unique u ∈ H 1 (M ) solving (−∆g + 1)u = f in the weak sense, by which we mean that hf, vi = Q(u, v),

∀v ∈ H 1 (M ),

(3.1.3)

if Q is the Dirichlet form Z g(gradg u, gradg v) dVg + hu, vi,

Q(u, v) = M

where, if u and v are supported in a coordinate work in local coR P jk patch and we ordinates, the first term on the right is g (x)∂j u ∂k v |g|1/2 dx (see (2.1.12)). Before verifying this claim, we note that since C ∞ (M ) is dense in H 1 (M ), (3.1.3) is equivalent to hf, vi = hu, (−∆g + 1)vi, v ∈ C ∞ (M ), p using (2.1.12) once again. To verify the claim we note that Q(u, u) is a norm that is equivalent to kukH 1 , and that, given f ∈ L2 (M ), we have p |hf, vi| ≤ kf kL2 kvkL2 ≤ kf kL2 Q(v, v), v ∈ H 1 . Therefore, since Q is an inner product on H 1 , by the Riesz representation theorem, as claimed, there must be a unique u ∈ H 1 (M ) satisfying (3.1.3).

42

CHAPTER 3

By (3.1.2) and the Schwarz inequality this implies that kukH 1 (M ) ≤ Ck (−∆g + 1)u kL2 (M ) .

(3.1.4)

This and Proposition 2.2.4 implies moreover that we also have for m = 0, 1, 2, . . . kukH m+2 ≤ Cm k (−∆g + 1)u kH m .

(3.1.5)

Using this estimate and an inductive argument we find that for m = 1, 2, . . . kukH 2m ≤ Cm k (−∆g + 1)m ukL2 .

(3.1.6)

Indeed the estimate for m = 1 is just (3.1.5), and if (3.1.6) has been established for smaller values of m we have that k(−∆g + 1)ukH 2m−2 ≤ Cm−1 k(−∆g + 1)m ukL2 , which yields (3.1.6) in view of (3.1.5). Using these estimates we can prove the following. Proposition 3.1.1 The spectrum of −∆g is discrete and nonnegative. If φµ is an eigenfunction with eigenvalue µ, i.e., −∆g φµ = µφµ , then φµ ∈ C ∞ (M ). Proof. By (2.1.12), we have that Z h−∆g u, ui = g(gradg u, gradg u) dVg ≥ 0, M

which implies that the spectrum must be nonnegative. Let Sλ be the orthogonal projection operator, which is given by the spectral theorem, onto the spectrum ≤ λ2 , with λ 6= 0. Then if u = Sλ u, i.e., the spectrum of u is ≤ λ2 , and so we have that k(−∆g + 1)ukL2 ≤ (λ2 + 1)kukL2 . Thus, by (3.1.4), kukH 1 ≤ C(λ2 + 1)kukL2 ,

if u = Sλ u.

Consequently, the set of all u with spectrum in [0, λ2 ] and having norms satisfying kukL2 ≤ 1 must be compact by the Rellich-Kondrachov compactness theorem (see [22], §5.7), since such functions have H 1 norms bounded by C(λ2 + 1). Thus, Sλ (L2 (M )) must be finite dimensional by a theorem of F. Riesz saying that the unit ball in a Banach space of infinite dimension cannot be compact. Therefore, the spectrum of −∆g must be discrete. The eigenfunctions must be smooth by (3.1.6) since they belong to H m for every m.  Since −∆g is a second order operator, as we momentarily shall see, it is convenient to label its spectrum counted with respect to multiplicity as 0 = λ20 < λ21 ≤ λ22 ≤ λ23 ≤ · · · . By the spectral theorem we then have −∆g =

∞ X

λ2j Ej

j=0

I=

∞ X j=0

Ej ,

43

THE SHARP WEYL FORMULA

with Ej denoting the orthogonal projection onto the jth eigenspace, E R j . In other words, if ej are eigenfunctions of −∆g with eigenvalue λ2j which satisfy |ej |2 dVg = 1 (i.e., L2 -normalized), then Z Ej f (x) = ej (x) hf, ej i = ej (x) f (y) ej (y) dVg . (3.1.7) M

The constant in this formula, hf, ej i, is the jth Fourier coefficient of f . Also, the projection operator Sλ mentioned above is the Fourier partial summation operator Z X Sλ f (x) = Ej f (x) = Sλ (x, y) f (y) dVg (y), (3.1.8) M

λj ≤λ

where the spectral function, Sλ (x, y) =

X

ej (x) ej (y),

(3.1.9)

λj ≤λ

is the kernel of Sλ . The main purpose of this chapter is to obtain sharp asymptotics for the Weyl counting function N (λ) = #{λj ≤ λ}, (3.1.10) p that is, the number of eigenvalues of −∆g counted with multiplicity which are ≤ λ. Note that since the ej are L2 -normalized we have Z N (λ) = Trace Sλ = Sλ (x, x) dVg . (3.1.11) M

The main goal of this chapter is to prove the sharp Weyl formula which says that there is a constant c, depending on (M, g) in a natural way, so that N (λ) = cλn + O(λn−1 ). Let us for now present some much cruder estimates for N (λ) and related estimates for eigenfunctions. p Lemma 3.1.2 If, as above, ej (x) are the L2 -normalized eigenfunctions of −∆g with eigenvalue λj then n

kej kL∞ (M ) ≤ C(1 + λj ) 2 +1 .

(3.1.12)

N (λ) ≤ C(1 + λ)n+2 .

(3.1.13)

Also, for λ > 0 Proof. If g ∈

S

λj ≤λ

Ej then by Sobolev’s theorem and (3.1.6) n

kgkL∞ ≤ CkgkH 1+[n/2] ≤ C(1 + λ) 2 +1 kgkL2 , which is stronger than (3.1.12). To prove (3.1.13), we note that the preceding inequality yields Z Sλ (x, y) f (y) dVg (y) = |Sλ f (x)| ≤ C(1 + λ) n2 +1 kf kL2 , and since this holds for all f ∈ L2 we have Z sup |Sλ (x, y)|2 dVg ≤ C(1 + λ)n+2 . x

44

CHAPTER 3

Since hej , ek i = 0, j 6= k, we have Z |Sλ (x, y)|2 dVg (y) = Sλ (x, x), and so we get (3.1.13) from (3.1.11) and the preceding inequality.



We shall prove the sharp Weyl formula by showing that Sλ (x, x) = c(x)λn + O(λn−1 ) and then integrating. We shall do this not by studying the function λ → Sλ (x, x) directly, but, rather, by studying its Fourier transform since this will allow us to use the wave equation, which in turn will allow us to do our calculation using the Hadamard parametrix. Let us be more specific. We first compute the Fourier transform of the characteristic function of [−λ, λ], λ > 0: Z λ Z λ sin λt e−iτ t dτ = 2 cos tτ dτ = 2 . t −λ 0 Therefore, by the Fourier inversion formula 1 lim N →+∞ π

Z

∞

sin tλ cos tτ dt = ρ(t/N ) t −∞ if ρ ∈

( 1 if |τ | < λ 0 if |τ | > λ

C0∞ (R)

(3.1.14)

and ρ(0) = 1.

The limit equals 21 if |τ | = λ and ρ is even. Thus, if λ is not in the spectrum, we have that for f ∈ C ∞ (M ) ∞ Z 1 X sin tλ Sλ f (x) = Ej f (x) cos tλj dt, (3.1.15) π j=0 t where, for each fixed j in the sum, we interpret the Fourier integral to be the above limit with τ = λj . Note that if f is smooth then its Fourier coefficients are rapidly decreasing since (1 + λ2j )−N h(1 − ∆g )N f, ej i = hf, ej i. Therefore by Lemma 3.1.2 ∞ X

p
Ej f (x) cos tλj = cos(t −∆g )f (x)

j=0

is an absolutely convergent series. Moreover, for such f this function solves the Cauchy problem (∂t2 − ∆g )u(t, x) = 0, (t, x) ∈ R × M, u(0, · ) = f, ∂t u(0, · ) = 0 p u(t, · ) = cos(t −∆g )f.

(3.1.16)

To calculate the kernel of Sλ and prove the sharp Weyl formula, we shall need to find a small-time parametrix for the solution operator. To do this, we shall use Theorem 2.4.1 and the following energy estimates which are analogous to the constant coefficient ones from §1.2, as well as (2.4.26). Lemma 3.1.3

Suppose that w ∈ C ∞ ([0, T ] × M ) solves ( (∂t2 − ∆g )w = F in [0, T ] × M w|t=0 = ∂t w|t=0 = 0.

45

THE SHARP WEYL FORMULA

Assume further that for a given nonnegative integer m we have ∂tj F = 0 when t = 0 and j < m. Then for 0 < t < T m+1 X

k∂tj w(t, · )kH m+1−j

j=0

 Z ≤ Cm,T 

t

k∂sm F (s, · )kL2 ds +

0

m−1 X

 k∂tj F (t, · )kH m−1−j  . (3.1.17)

j=0

Proof. Let us first prove (3.1.17) for m = 0 by using an argument which is similar to the proof of Theorem 2.4.2. We note that  d k∂s w(s, · )k2L2 − h∆g w(s, · ), w(s, · )i ds   = 2Re h∂s2 w, ∂s wi − h∆g w, ∂s wi + h∆g ∂s w, wi   = 2Re h∂s2 w, ∂s wi − h∆g w, ∂s wi = 2Re hF, ∂s wi, and therefore k∂t w(t, · )k2L2 − h∆g w(t, · ), w(t, · )i Z tZ ≤2 |F (s, x) |∂s w(s, x)| dVg ds 0 M Z t ≤ 2 max k∂s w(s · )kL2 kF (s, · )kL2 ds. 0≤s≤t

(3.1.18)

0

Since h−∆g w(s, · ), w(s, · )i ≥ 0, we conclude that Z t k∂t w(t, · )kL2 ≤ 2 kF (s, · )kL2 ds,

0 ≤ t ≤ T.

(3.1.19)

0 ≤ t ≤ T.

(3.1.20)

0

Since w(t, · ) =

Rt

∂s w(s, · ) ds, we also have Z t kw(t, · )kL2 ≤ 2t kF (s, · )kL2 ds,

0

0

Finally, since   kw(t, · )k2H 1 ≤ C −h∆g w(t, · ), w(t, · )i + kw(t, · )k2L2 , if we combine (3.1.18)-(3.1.20), we conclude that (3.1.17) is valid for m = 0. We can prove (3.1.17) for a given m ∈ N by induction, if we note that the right side of (3.1.17) majorizes the corresponding right side when m is replaced by a smaller value. To use this, we observe that if we assume (3.1.17) is valid with m replaced by m − 1 and if we apply this estimate to ∂t w then we can control all of the terms in the left side of (3.1.17) except for kw(t, · )kH m+1 . To handle this term, we use the fact that, by (3.1.5),
kw(t, · )kH m+1 ≤ C k∆g w(t, · )kH m−1 + kw(t, · )kH m−1
≤ C kF (t, · )kH m−1 + k∂t2 w(t, · )kH m−1 + kw(t, · )kH m−1 , to see that this term is also majorized by the right side of (3.1.17), which completes the proof. 

46

CHAPTER 3

We also require the following result which is just a variant of the standard Sobolev inequality. Lemma 3.1.4 and

If α = 0, 1, 2, . . . , f ∈ H m (Rn ) and m > α +

n 2

then f ∈ C α (Rn ),

t|β| k∂ β f kL∞ (Rn )
n n ≤ C tm− 2 kf kH˙ m (Rn ) + t− 2 kf kL2 (Rn ) ,

|β| ≤ α, t > 0. (3.1.21)

Here H˙ m (Rn ) denotes the homogeneous Sobolev space with norm defined by Z kf k2H˙ m (Rn ) = (2π)−n |fˆ(ξ)|2 |ξ|2m dξ. Proof. The first assertion is a special case of the Sobolev embedding theorem. To prove (3.1.21) we note that the inequality is scale-invariant and therefore it suffices to prove it for t = 1, i.e.,
k∂ β f kL∞ (Rn ) ≤ C kf kH˙ m (Rn ) + kf kL2 (Rn ) , |β| ≤ α. But, by (2.2.11) and Plancherel’s formula, the right side is comparable to kf kH m (Rn ) , and so this estimate just follows from the standard Sobolev estimate kukL∞ (Rn ) ≤ Cs kukH s (Rn ) ,

s>

n , 2

which completes the proof.

 p We can now present the parametrix for u(t, x) = cos t −∆g f , which is the solution of the Cauchy problem (∂t2 − ∆g )u = 0, u|t=0 = f , ∂t u|t=0 =p0. By Theorem 2.4.2, we know that the kernel, Kt (x, y), of f → u(t, x) = cos t −∆g f vanishes when the Riemannian distance, dg (x, y), is larger than t, provided that t > 0 is smaller than the injectivity radius of (M, g). In other words, g has unit propagation speed. We shall make this assumption that |t| is smaller than the injectivity radius throughout the rest of this section. Using this we can work in a local coordinate patch Ω and assume that f ∈ C0∞ (ω) with ω b Ω. Indeed, because of the finite propagation speed for g , u(t, · ) for small t will also be supported in a fixed relatively compact subset of Ω, which allows us to assume that ∆g is as in (3.1.1). It then is self-adjoint with respect to the measure dVg = |g|1/2 dy. We recall that the Hadamard parametrix inpformulap(2.4.19) of Theorem 2.4.1 provides an approximation of the operator sin t −∆g / −∆g , t ≥ 0. Since all the p terms in (2.4.19) have t ≥ 0 in their support and since cos t −∆g is even in t we are led to expecting that if we set  PN
 ∂t ν=0 αν (x, y)Eν (t, dg (x, y)) , t ≥ 0 KN (t, x; y) = (3.1.22) 
PN  −∂t α (x, y)E (−t, d (x, y)) , t < 0, ν g ν=0 ν then by taking N to be larger and larger the integral operator p with this kernel should provide a better and better approximation of f → cos t −∆g f . Specifically, we

47

THE SHARP WEYL FORMULA

expect that, for small |t|, p
cos t −∆g f (x) =

Z KN (t, x; y) f (y) dVg (y) Z + RN (t, x; y) f (y) dVg (y), (3.1.23)

where the kernel, RN , of the remainder term becomes increasingly smooth as N → +∞. Since we are truncating the series at the N th stage in (3.1.22), we might expect that the remainder should satisfy the same estimates as the next term in the expansion which involves ∂t EN +1 and so is O(|t|2(N +1)−n ), if N is large enough, by Proposition 1.2.4. By the above discussion, it suffices to verify these assertions for small positive t. To use our energy estimates, let w = wN be defined by Z p
w(t, x) = cos t −∆g f (x) − KN (t, x; y) f (y) dVg (y) Z = RN (t, x; y) f (y) dVg (y). Then w|t=0 = ∂t w|t=0 = 0 by Proposition 1.2.4, and, by (2.4.19), we also have that (∂t2 − ∆g )w(t, x) = FN (t, x) Z
= ∆g αN (x, y) ∂t EN (t, dg (x, y)) f (y) dVg (y). (3.1.24) Recall that the various terms in (3.1.22) and therefore the kernel in the right side of (3.1.23) have dg (x, y) ≤ t in their support. By Proposition 1.2.3, the kernel of the integral operator in the right side of (3.1.24) is C m if N > m + n+1 2 , and, for such m,
α |∂t,x,y ∆g αN (x, y)∂t EN (t, dg (x, y)) | ≤ Ct2N −|α|−n , |α| ≤ m. Indeed this estimate follows from the fact that ∂t EN (t, x) is homogeneous of degree 2N − n. Since ∂t EN (t, dg (x, y)) = 0 when dg (x, y) > t, by Minkowski’s inequality and the preceding estimate we conclude that

j α

∂t ∂x FN (t, · )kL2 ≤ Ct2N −j−|α|− n2 kf kL1 , |α| + j ≤ m, and, therefore, Z 0

t

k∂sm FN (s,

· )kL2 ds +

m−1 X

k∂tm−1−j FN (t, · )kL2

n

≤ Ct2N −m− 2 +1 kf kL1 .

j=0

As a result, the energy estimates in Lemma 3.1.3 yield n

kw(t, · )kH m+1 + k∂t w(t, · )kH m ≤ Ct2N −m− 2 +1 kf kL1 , assuming, as above, that N > m + n+1 2 . If we take m = 0 we find (cf. (3.1.20)) that the bound for the second term in the left yields n

kw(t, · )kL2 ≤ Ct2N +2− 2 kf kL1 .

48

CHAPTER 3

If we use Lemma 3.1.4, we conclude that if j + |α| + n + 1 < N that ∂tj ∂xα w is continuous and that |∂tj ∂xα w| ≤ Ct2N +2−|α|−j−n kf kL1 , provided that j ≤ 1. Using the equation ∂t2 w = FN − ∆g w we see by induction that the above bounds hold for all j < N − 1 − |α|. Thus, for small t ≥ 0 we have shown that Z α ∂t,x RN (t, x; y) f (y) dVg (y) ≤ Ct2N +2−n−|α| kf kL1 , |α| + n + 1 < N. If we replace f by ∂ β f , then the above arguments also yield Z α β ∂t,x RN (t, x; y) (∂ f )(y) dVg (y) ≤ Ct2N +2−n−|α|−|β| kf kL1 , provided that |α| + |β| + n + 1 < N . Since |g|1/2 is smooth, we can integrate by parts to conclude that Z α ∂t,x,y RN (t, x; y)f (y) dVg (y) ≤ Ct2N +2−n−|α| kf kL1 , if |α| ≤ N − (n + 2). By taking the supremum over all kf kL1 ≤ 1, we see that this implies that RN ∈ C N −(n+3) and α |∂t,x,y RN (t, x; y)| ≤ C|t|2N +2−n−|α| ,

if |α| ≤ N − (n + 2).

(3.1.25)

Thus, we have established the following. Theorem 3.1.5 Given a compact n-dimensional Riemannian manifold (M, g) there is a δ > 0 so that if N > n + 3 then, in local coordinates, (3.1.23) is valid with the remainder kernel RN being in C N −n−3 ([−δ, δ] × M × M ) and satisfying (3.1.25). 3.2

SUP-NORM ESTIMATES FOR EIGENFUNCTIONS AND SPECTRAL CLUSTERS

Recall that we are trying to prove that the Weyl counting function satisfies N (λ) = cλn + O(λn−1 ). We know from Lemma 3.1.2 that N (λ) is a tempered function of p λ ≥ 0, and since the spectrum of −∆g is discrete it is locally constant with jump discontinuities. Since λ → cλn is continuous, if the above Weyl formula is valid then we necessarily must have that the jumps are O(λn−1 ), i.e., N (λ + 0) − N (λ − 0) = O(λn−1 ),

λ≥1 (3.2.1)

N (λ + 0) = lim N (τ ), τ →λ+

N (λ − 0) = lim N (τ ). τ →λ−

This is one of the results that we shall establish in this section, and it will be needed for the proof of the Weyl formula. As we shall see below, (3.2.1) is actually a consequence of the following sup-norm estimates for eigenfunctions n−1

kej kL∞ (M ) ≤ Cλj 2 ,

λ ≥ 1,

(3.2.2)

49

THE SHARP WEYL FORMULA

p where, as before, the ej are L2 -normalized and are eigenfunctions of −∆g with eigenvalue λj , i.e., −∆g ej = λ2j ej . The bounds in (3.2.2) improve the crude ones in (3.1.12) that were just proved by using Sobolev estimates. We shall see in the next section that the bounds in (3.2.2) cannot be improved in the sense that they are sharp in some cases. The proof of either of these two estimates actually yields a stronger result that will be used. It says that the number of eigenvalues λj ∈ [λ, λ + 1], λ ≥ 1, is O(λn−1 ) and that L2 -normalized functions with spectrum in this range (spectral n−1 clusters) have sup-norms which are O(λ 2 ). Obtaining results like the latter are equivalent to studying the L2 (M ) → L∞ (M ) mapping properties of the spectral projection operators X χλ f = Ej f, (3.2.3) λj ∈[λ,λ+1]

where Ej is, as in (3.1.7), the projection onto the eigenspace, Ej , with eigenvalue λj . We can now state the main result of this section. Theorem 3.2.1 then have

Fix an n-dimensional compact Riemannian manifold (M, g). We N (λ + 1) − N (λ) = O((1 + λ)n−1 ),

λ ≥ 0,

(3.2.4)

and, moreover, there is a uniform constant C so that kχλ f kL∞ (M ) ≤ C(1 + λ)

n−1 2

kf kL2 (M ) ,

λ ≥ 0.

(3.2.5)

As we shall see in §3.3, these results are sharp in the sense that they cannot be improved when (M, g) is the n-sphere, S n , endowed with the round metric. Let us turn to the proof. We first observe that if {ej (x)}∞ j=0 as in (3.1.7) is our orthonormal basis of eigenfunctions with eigenvalues {λj }∞ j=0 , then the kernel of χλ is X χλ (x, y) = ej (x) ej (y). λj ∈[λ,λ+1] ∞

2

We recall that the L (M ) → L (M ) operator norm of χλ then is exactly equal to 1/2 Z , sup |χλ (x, y)|2 dVg (y) x

and since hej , ek i =

δjk ,

we conclude that the square of the operator norm equals X kχλ k2L2 →L∞ = sup |ej (x)|2 = sup χλ (x, x). (3.2.6) x

x

λj ∈[λ,λ+1]

The second estimate in Theorem 3.2.1 asserts that this is O((1 + λ)n−1 ). If we could establish this then the other estimate there, (3.2.4), would follow due to the fact that Z #{λj ∈ [λ, λ + 1]} = Trace χλ = χλ (x, x) dVg (x). (3.2.7) M

p The operator χλ is a multiplier operator for −∆g . By this we mean an operator of the form ∞ X p
m −∆g f = m(λj )Ej f, (3.2.8) j=0

50

CHAPTER 3

assuming that m(τ ), τ ≥ 0, is bounded. In this case the series converges absolutely for f ∈ C ∞ (M ) and extends to be a bounded operator on L2 (M ). Note that by p (3.1.7) the kernel of m( −∆g ) is ∞ X

m(λj )ej (x) ej (y).

j=0

The operator χλ is the multiplier operator with m(τ ) = 1[λ,λ+1] (τ ) if 1[λ,λ+1] denotes the indicator function of [λ, λ + 1]. In practice we should not expect to be able to compute the kernel of χλ directly due to the jumps in the spectrum. On the other hand, if we replace 1[λ,λ+1] by a smoother function then we might be able to do so and then use its L2 → L∞ bounds and orthogonality arguments to obtain (3.2.5). Since we want to use the Hadamard parametrix in Theorem 3.1.5, it will be convenient to use multiplier operators involving a function χ ∈ S(R) having the properties that χ ≥ 0,

χ(0) = 1, and χ(t) ˆ = 0,

|t| ≥ δ/2,

(3.2.9)

if δ is as in Theorem 3.1.5. Such functions exist for if 0 6= ρ ∈ C0∞ ((−δ/4, δ/4)) is an even nonnegative function then a constant multiple of the inverse Fourier transform of (ρ ∗ ρ)(τ ) will have this property. We then claim that we can use orthogonality to see that the operators χ eλ f = χ(λ −

∞ X p −∆g )f = χ(λ − λj )Ej f, j=0

provide an approximation to the spectral projection operators in (3.2.3). Specifically, we claim that the estimates ke χλ f kL2 (M ) ≤ C(1 + λ)

n−1 2

kf kL1 (M )

(3.2.10)

kf kL1 (M ) ,

(3.2.11)

would imply that kχλ f kL2 (M ) ≤ C(1 + λ)

n−1 2

which, by duality, is equivalent to (3.2.5). To verify this, we note that ke χλ f k2L2 =

∞ X


2 χ(λ − λj ) kEj f k2L2 .

j=0

Also, by (3.2.9), χ(0) = 1 and therefore there must be a δ0 > 0 so that (χ(τ ))2 ≥ 1/2 if |τ | ≤ δ0 . Therefore, if (3.2.10) were valid, we conclude that X kEj f k2L2 ≤ 2ke χλ f k2L2 ≤ C(1 + λ)n−1 kf k2L1 , |λj −λ|≤δ0

which clearly yields (3.2.11) as kχλ f k2L2 =

X λj ∈[λ,λ+1]

kEj f k2L2 .

51

THE SHARP WEYL FORMULA

We have therefore reduced matters to proving (3.2.10). To establish it, we note that we have the following estimate for the L1 → L2 operator norm Z 2 ke χλ kL1 →L2 = sup |e χλ (x, y)|2 dVg (x) y∈M

= sup

M ∞ X

χ(λ − λj )


2

|ej (y)|2

y∈M j=0

≤ kχkL∞ (R) · sup χ eλ (y, y), y∈M

using the fact that, by (3.2.9), χ(τ ) ≥ 0 in the last step. This means that we would be done if we had the following estimates for the restriction of the kernel of χ eλ to the diagonal |e χλ (y, y)| ≤ C(1 + λ)n−1 , λ ≥ 0. (3.2.12) To prove this, we first note that, by Euler’s formula, Z 1 −itτ itλ χ(t)e ˆ e dt χ(λ − τ ) = 2π Z 1 itλ = χ(t)e ˆ cos tτ dt − χ(λ + τ ), π and consequently, if Rλ f =

∞ X

χ(λ + λj )Ej f,

j=0

we have that, for f ∈ C ∞ (M ), 1 χ eλ f + Rλ f = π

Z

1 π

Z

=

itλ

χ(t)e ˆ

∞ X

cos tλj Ej f



dt

(3.2.13)

j=0

p itλ χ(t)e ˆ cos t −∆g f dt.

Since χ ∈ S(R), we have that χ(λ + λj ) = O((1 + λ + λj )−N ), λ, λj ≥ 0, for any N . Therefore, by Lemma 3.1.2, the kernel of Rλ satisfies Rλ (y, y) =

∞ X

χ(λ + λj )|ej (y)|2 ≤ CN (1 + λ)−N , λ ≥ 0,

j=0

for any N . Therefore, we would have (3.2.12) if we could show that we have the uniform bounds Z p
1 (3.2.14) χ(t) ˆ eitλ cos t −∆g (y, y) dt = O((1 + λ)n−1 ), λ ≥ 0, π p p with (cos t −∆g )(x, y) denoting the kernel of the operator cos t −∆g defined in the last section. To prove this estimate we may use Theorem 3.1.5 since χ(t) ˆ = 0, |t| ≥ δ. Indeed, if N > n + 3 and KN is as in (3.1.22), we have that the left side equals Z 1 itλ χ(t)e ˆ KN (t, y, y) dt + O(1), π

52

CHAPTER 3

because the remainder term in (3.1.23) is bounded for such N . Since the coefficients αν , ν = 0, 1, 2, . . . , in (3.1.22) are bounded and do not depend on t, we would get (3.2.14) if we could show that Z
1 χ(t) ˆ ∂t Eν (t, 0) − ∂t Eν (−t, 0) eitλ dt π
= O( 1 + λ)n−1 , λ ≥ 0, ν = 0, 1, . . . , N, (3.2.15) with Eν (t, x) being defined in (1.2.30) or the formula preceding it. The term with ν = 0 is the most singular by Proposition 1.2.4, and so let us start by focusing on the worst case in (3.2.15) where ν = 0. If we recall that E0 is the forward fundamental solution, E+ , for the constant coefficient wave operator, , i.e., Z sin t|ξ| H(t) eix·ξ dξ, E0 (t, x) = (2π)n Rn |ξ| then we find that the term in (3.2.15) corresponding to ν = 0 is just ZZ −n −1 itλ (2π) π χ(t)e ˆ cos t|ξ| dξdt X ZZ it(λ±|ξ) = (2π)−n−1 χ(t)e ˆ dξdt ±

= (2π)−n

Z

χ(λ − |ξ|) dξ + (2π)−n

Rn

Z χ(λ + |ξ|) dξ. Rn

Since χ(λ + |ξ|) = O((1 + λ + |ξ|)−N ) for all N the last term is rapidly decreasing in λ ≥ 0, and so we get the desired bound in (3.2.15) for ν = 0 as if N > n we have Z Z χ(λ − |ξ|) dξ ≤ CN (1 + |λ − |ξ| |)−N dξ Rn Rn Z n = CN λ (1 + λ| 1 − |ξ| |)−N dξ ≈ λn−1 , λ ≥ 1. Rn

If one uses the fact that, by Proposition 1.2.4, t ∂Eν = Eν−1 , ν = 1, 2, 3, . . . , ∂t 2 then one can see via integration by parts and an induction argument that for ν = 1, 2, 3, . . . the terms in (3.2.15) are O((1 + λ)n−1−2ν ). One could also deduce this from the argument that we have just given for ν = 0 by using Remark 1.2.5. For instance the bounds for ν = 1 follow from (1.2.35). The bounds for general ν = 1, 2, 3, . . . can be proven using the fact that, modulo smoothing a smooth function, Eν (t, x) is a finite linear combination of terms of the form (1.2.36). Alternatively, one could obtain (3.2.15) from the fact that
t → ∂t Eν (t, 0) − ∂t Eν (−t, 0) is a distribution in S 0 (R) which is homogeneous of degree 2ν − n, and, therefore, its Fourier transform is a distribution which is homogeneous of degree (n − 1) − 2ν. Since the terms in (3.2.15) are the convolution with this distribution and the Fourier transform of 0 6= χ ∈ S(R) and χ ≥ 0, we can deduce that the terms in (3.2.15) have size which is actually asymptotic to a constant multiple of λn−1−2ν as λ → +∞, which is stronger than (3.2.15).

53

THE SHARP WEYL FORMULA

3.3

SPECTRAL ASYMPTOTICS: THE SHARP WEYL FORMULA

The purpose of this section is to prove the sharp Weyl formula. Theorem 3.3.1 Then for λ ≥ 1

Fix a compact n-dimensional Riemannian manifold (M, g). N (λ) = (2π)−n ωn Volg (M )λn + O(λn−1 ), n 2

(3.3.1)

n 2)

denotes the volume of the Euclidean unit ball in Rn if ωn = An /n = π /Γ(1 + and Volg (M ) denotes the Riemannian volume of M : Z Volg (M ) = dVg (y). M

Moreover, there is a uniform constant C so that Sλ (x, x) − (2π)−n ωn λn ≤ C(1 + λ)n−1 , λ > 0,

(3.3.2)

if Sλ (x, y) is the spectral function (3.1.9). Note the remarkable fact that (3.3.2) says that the sum of squares of the values of the eigenfunctions whose eigenvalues are ≤ λ is, up to a lower order error, independent of M , the metric and even the point x! Remark 3.3.2 Note also, that we could rewrite (3.3.1) using the Liouville measure dxdξ on T ∗ M as follows: ZZ −n n n−1 ) N (λ) = (2π) λ n X dxdξ + O(λ  jk ∗ g (x)ξj ξk ≤ 1 (x, ξ) ∈ T M : j,k=1

= (2π)−n Volg (B ∗ M ) λn + O(λn−1 ), if B ∗ (M ) ⊂ T ∗ M denotes the ball bundle associated with the cometric, {(x, ξ) :

n X

g jk (x)ξj ξk ≤ 1},

j,k=1

and ∗

ZZ

Volg (B M ) = {(x, ξ) ∈ T ∗ M :

n X

dxdξ, g jk (x)ξj ξk ≤ 1 }

j,k=1

is its Liouville measure. In the next section we shall see that both (3.3.1) and (3.3.2) are sharp for the sphere S n with the standard metric. Also, (3.3.2) is one version of a local Weyl formula. We shall encounter a more refined version later on. Proof. Since the above local Weyl formula is stronger than its integrated version, (3.3.1), we just need to prove (3.3.2). Also, by p (3.2.4), we may make the technical assumption that λ is not in the spectrum of −∆g so that we can use the formula (3.1.14).

54

CHAPTER 3

To use this formula we note that since the Fourier transform of 1[−λ,λ] (τ ) is sin λt 2 , we conclude that if we fix a function β ∈ C0∞ (R) with β(0) = 1, and let t Z sin λt 1 β(t) rλ (τ ) = 1[−λ,λ] (τ ) − cos tτ dt, π t then for every N = 1, 2, 3, . . . there is a constant CN so that
−N |rλ (τ )| ≤ CN 1 + | |τ | − λ | ,

λ ≥ 1.

(3.3.3)

One proves this either by a simple integration by parts argument using (3.1.14), or by realizing that, by this identity, the second term in the definition of rλ is the R∞ convolution of 1[−λ,λ] with a function K ∈ S(R) satisfying −∞ K(τ ) dτ = 1. p In order to use the parametrix for cos t −∆g , we shall also assume that β(t) = 0,

|t| ≥ δ,

(3.3.4)

where δ is as in Theorem 3.1.5. Next, we note that by (3.1.15), 1 Sλ (x, x) − π

∞

Z β(t)

X p
sin tλ rλ (λj ) |ej (x)|2 . cos t −∆g (x, x) dt = t j=0

By (3.3.3), (3.2.5) and (3.2.6) we have for any N , ∞ ∞ X X rλ (λj )|ej (x)|2 ≤ CN (1 + |λ − λj |)−N |ej (x)|2 j=0

j=0 0 ≤ CN

∞ X

(1 + |λ − k|)−N

00 ≤ CN

X

|ej (x)|2



λj ∈[k,k+1]

k=0 ∞ X



(1 + |λ − k|)−N (1 + k)n−1

k=0

≈ λn−1 ,

λ ≥ 1,

assuming, as we may, in the last step that N > n. From this we conclude that in order to prove (3.3.2) it is enough to prove that there is a uniform constant C (depending on β) so that for λ > 0 and x ∈ M , Z p 1
sin tλ −n n n−1 β(t) cos t −∆ (x, x) dt − (2π) ω λ . (3.3.5) g n π ≤ C(1 + λ) t To p prove this, as in the proof of Theorem 3.2.1, we shall use the parametrix for cos t −∆g in Theorem 3.1.5. If we choose the N there to be larger than n + 3, then by (3.1.25), we have RN (t, x, x) = O(|t|), |t| ≤ δ, and so Z sin tλ RN (t, x, x) dt = O(1). β(t) t Therefore, since, by Theorem 2.4.1, the leading coefficient in the parametrix, α0 , satisfies α0 (x, x) ≡ 1 and since the others, αν , ν = 1, 2, 3, . . . , are bounded, we

55

THE SHARP WEYL FORMULA

conclude that (3.3.5) would be a consequence of Z Z 1
sin tλ −n β(t) ∂t E0 (t, 0) − ∂t E0 (−t, 0) dt − (2π) dξ π t n {ξ∈R : |ξ|≤λ} ≤ Cλn−1 , λ ≥ 1, (3.3.6) and Z 1
sin tλ ∂t Eν (t, 0) − ∂t Eν (−t, 0) dt π β(t) t ≤ Cν λn−2ν , λ ≥ 1,

ν = 1, 2, 3, . . . , (3.3.7)

where, as in the proof of Theorem 3.2.1, the Eν (t, x) are defined in (1.2.30). If we repeat the argument that we used to establish (3.2.15) for ν = 0, we see that Z
sin tλ 1 β(t) ∂t E0 (t, 0) − ∂t E0 (−t, 0) dt π t Z Z sin tλ = (2π)−n π −1 β(t) cos t|ξ| dξdt t n R Z Z = (2π)−n dξ + (2π)−n rλ (|ξ|) dξ {ξ∈Rn : |ξ|≤λ} Rn Z
= (2π)−n dξ + O (1 + λ)n−1 , {ξ∈Rn : |ξ|≤λ}

using (3.3.3) with N > n in the last step. Thus, we have established (3.3.6). Since inequality (3.3.7) follows from the arguments at the end of the proof of Theorem 3.2.1, the proof is complete.  3.4

SHARPNESS: SPHERICAL HARMONICS

Let us conclude this chapter by showing that Theorem 3.3.1 is sharp in the sense that neither (3.3.1) nor (3.3.2) can be improved when (M, g) is the n-sphere with the metric induced by the Euclidean metric in Rn+1 . In that case the Laplace-Beltrami operator, we denote by ∆S n , is the one induced by the Rn+1 Laplacian, Pn+1which 2 ∆ = j=1 ∂ /∂x2j . In other words, if f ∈ C ∞ (S n ), we have

∆S n f (x) = ∆ f (x/|x|) , x ∈ S n , where f (x/|x|) denotes the extension of f defined on S n to a function on Rn+1 \0 which is homogeneous of degree zero. The volume element that is associated with ∆S n is the measure on S n induced by the Euclidean volume element on Rn+1 . In other words, dVS n is just the induced Lebesgue measure on S n . We shall note S n endowed with this metric as (S n , can), the so-called “round sphere.” Recall that we have the related formula ∆=

∂2 n ∂ 1 + + ∆S n . ∂r2 r ∂r r2

56

CHAPTER 3

From this we see that if uk is a homogeneous polynomial of degree k, i.e., uk (x) =

X

α

n+1 1 α2 aα xα , xα = xα 1 x2 · · · xn+1 ,

αj = 0, 1, 2, . . . ,

|α| =

n+1 X

αj ,

1

|α|=k

which is harmonic in Rn+1 , i.e., ∆uk (x) ≡ 0, then its restriction ek (ω) to S n must be an eigenfunction of −∆S n with eigenvalue µk = k(k + n − 1).

(3.4.1)

For uk (x) = rk ek (ω), ω = x/|x| and thus,  2  ∂ n ∂ ∆uk (x) = ek (ω) + rk + rk−2 ∆S n ek (ω) ∂r2 r ∂r h i
= rk−2 k(k + n − 1)ek (ω) + ∆S n ek (ω) = 0. We claim that the eigenvalue µk of −∆S n repeats with multiplicity ≈ k n−1 , which would show that the Weyl formula (3.3.1) cannot be improved here since √ µk ≈ k. Thus, we wish to show that the resulting eigenspace, Hk , has dimension ≈ k n−1 . One calls Hk the space of spherical harmonics of degree k since it is the restriction to S n of homogeneous harmonic polynomials of degree k. To prove this claim, let us consider the space Pk of all homogeneous polynomials of degree k in Rn+1 with complex coefficients, i.e., polynomials of the form X P (x) = a α xα , |α|=k

with aα ∈ C. For later use, let us compute the dimension of Pk . This clearly is the same as the number of multi-indices α = (α1 , α2 , . . . , αn+1 ), with |α| = k, since the monomials {xα }, |α| = k, form a basis for Pk . Clearly the number of these multiindices is the same as the number of ways that one can place k identical coins in n + 1 slots. To calculate this imagine n + k vertical segments arranged horizontally, left to right. From these remove n of them, thus leaving k. This procedure gives us n + 1 slots: the first being everything to the left of the first removed, the second slot being everything between the first and second removed segment, etc., up to the last slot (the (n + 1)-st), which is everything to the right of the last removed segment. Place a coin on each of the k remaining vertical segments which were not removed. We then have 0 ≤ α1 ≤ k coins in the first slot, 0 ≤ α2 ≤ k in the second and so on. We also automatically have α1 + α2 + · · · + αn+1 = k. This represents one way of distributing the k coins in the n + 1 slots. Since it was obtained by choosing n vertical segments from the n + k ones
in our original array, we conclude that the number of ways of doing this is n+k n , and thus   n+k (n + k)! dim Pk = = . n n! k! To P we associate the constant coefficient differential operator on Rn+1 given by P (∂) =

X |α|=k

aα

∂α . ∂xα

57

THE SHARP WEYL FORMULA

We then assert that (P, Q)k = P (∂)Q =

X

aα

|α|=k

∂α Q, ∂xα

P, Q ∈ Pk

(3.4.2)

is an inner product on Pk . We first note that if P, Q ∈ Pk then the last term in (3.4.2) is a constant function, and, therefore, of course P this is to be the value of (P, Q)k . To verify this assertion, we note that if Q(x) = |α|=k bα xα , then X

(P, Q)k =

α! aα bα .

|α|=k

Thus, (P, Q) P k = (Q, P )k , and since it is clearly a bilinear form satisfying 0 = (P, P )k = |α|=k α! |aα |2 = 0 if and only if P ≡ 0, the assertion follows. If we let Uk denote the set of all polynomials in Pk which are harmonic in Rn+1 , then our next assertion is that, with respect to this inner product if k ≥ 2, we have the direct sum decomposition Pk = Uk ⊕ Vk ,

(3.4.3)

where Vk = {R ∈ Pk : R(x) = |x|2 Qk−2 (x), Qk−2 ∈ Pk }. In other words, Vk is the polynomials in Pk which are divisible by |x|2 . To see (3.4.3), we note that if R(x) = |x|2 Qk−2 (x) is as above then R(∂) = ∆Qk−2 (∂) = Qk−2 (∂)∆. Consequently, if P ∈ Pk satisfies (R, P )k = 0 for all such R, we have 0 = (R, P )k = Qk−2 (∂)∆P = (Qk−2 , ∆P )k−2 ,

for all Qk−2 ∈ Pk−2 .

Thus, ∆P ∈ Pk−2 must be orthogonal to every element of Pk−2 with respect to the (k − 2) inner product, which therefore implies that ∆P = 0. Thus, we have shown that if (R, P )k = 0 for all R ∈ Vk , then P ∈ Uk , which is our assertion (3.4.3). We now note that, by (3.4.3), we have that dim Hk = dim Uk = dim Pk − dim Pk−2     n+k n+k−2 = − n n
(n + k − 2)(n + k − 3) . . . (k + 1) = 2nk + n(n − 1) × n! 2k n−1 n−2 = + O(k ). (n − 1)! Therefore, if N (λ) is the Weyl counting function for (S n , can), then we have the √ following fact about its jumps at µk , with µk as in (3.4.1),
λ−(n−1) N (λ + 0) − N (λ − 0)

√ λ= µk

−→

2 , (n − 1)!

as k → ∞.

This means that, for (S n , can), there cannot be an asymptotic formula for N (λ) with a continuous main term and an error term which is o(λn−1 ).

58

CHAPTER 3

Let us now argue that the bounds in (3.2.5) (and consequently (3.3.2)) cannot √ be improved. If λ = µk , then the operator χλ there is the orthogonal projection k operator L2 → Hk . If we choose an orthonormal basis of Hk , {Ymk }dm=1 , dk = √ dim Hk , then, as we observed in §3.2, the kernel of χλ , λ = µk , can be written as χλ (x, y) =

dk X

Ymk (x) Ymk (y),

x, y ∈ S n .

m=1

Since ∆S n and dVS n are invariant under rotations, we see that if T : Rn+1 → Rn+1 is a rotation, then χλ (T x, T y) = χλ (x, y), x, y ∈ S n . Consequently the restriction to the diagonal, χλ (x, y), must be a constant, which by (3.2.7), is given by the formula χλ (x, x) ≡

dim Hk , |S n |

λ=

√ µk ,

with |S n | being the surface area of S n , i.e., its volume with respect to dVS n . Therefore, by (3.2.6), kχλ k2L2 →L∞ =

dim Hk ≈ λn−1 , |S n |

λ=

√ µk ,

showing, as asserted, that (3.2.5) cannot be improved for (S n , can). In general sup-norm estimates for the spectral projection operators χλ are stronger than the sup-norm estimates (3.2.2) for individual eigenfunctions. For (S n , can) even the latter cannot be improved. For, by what we have just shown, if x ∈ S n is fixed then dk X y→ Ymk (x)Ymk (y) = eλ (y) is an eigenfunction of

√

m=1

−∆S n with eigenvalue λ = |eλ (y)| ≈ λn−1 ,

and keλ kL2 =

dk X

√ µk , satisfying

y = x,

!1/2 |Ymk (x)|2

= χλ (x, x)


1/2

≈λ

n−1 2

,

m=1

which means that (3.2.2) cannot be improved. This eigenfunction on (S n , can) is called a zonal function about x, and one usually takes x to be either the north or south pole on S n . 3.5

IMPROVED RESULTS: THE TORUS

Although, as we showed in the last section, the bounds for the error term in the Weyl formula in Theorem 3.3.1 cannot be improved for S n , there are many cases where one can obtain improved bounds for the error term. We shall explore several instances of this, but we start with the simplest one, which is the n-torus, n ≥ 2, where one can make power improvements.

59

THE SHARP WEYL FORMULA

Recall that the n-torus, Tn = S 1 × · · · × S 1 = {(e2πix1 , e2πix2 , . . . , e2πixn ) : (x1 , x2 , . . . , xn ) ∈ Rn } is naturally identified with Rn /Zn via the map (x1 , x2 , . . . , xn ) → (e2πix1 , e2πix2 , . . . , e2πixn ).

(3.5.1)

A function on Rn is called periodic if f (x + j) = f (x) whenever j ∈ Zn . In view of the map (3.5.1) there is a natural identification of periodic functions on Rn with functions on Tn . The torus can also be identified with the unit cube Q = {x ∈ Rn : −

1 1 ≤ xk < , k = 1, 2, . . . , n}, 2 2

if we identify opposite faces of Q. The space of continuous functions on Tn , C(Tn ), does not correspond to the space of continuous functions on Q, but rather the space of functions f whose periodic extensions f˜ to Rn are continuous. Similar remarks apply to the space of smooth functions, C ∞ (Rn ), on Tn . The Euclidean metric gkl = δlk on Rn corresponds with these identifications to one on Tn which is also flat. The n-torus, Tn , endowed with this metric is called the flat torus. The corresponding volume element then is just the measure dx on Tn so that for f ∈ C(Tn ) we have Z Z f dx = f˜ dx, Tn

Q

where in the right side f˜ denotes the periodic extension of f to Rn , which we are integrating over Q with respect to Lebesgue measure. The Laplace-Beltrami operator on Tn similarly arises from the Euclidean one. If f ∈ C ∞ (Tn ),Pthen ∆f , n ∆ = ∆Tn , corresponds to the restriction to Q of ∆Rn f˜, where ∆Rn = k=1 ∂k2 is the Euclidean Laplacian and f˜, as above, is the periodic extension of f . The standard orthonormal basis for this Laplacian then is {e2πij·x : j ∈ Zn }.

(3.5.2)

Indeed, clearly, for every j = (j1 , . . . , jn ) ∈ Zn , e2πij·x = eigenfunction of −∆Tn with eigenvalue λ2j =

n X

(2πjk )2 = (2π)2

k=1

n X

Qn

k=1

e2πijk xk is an

jk2 = (2π)2 |j|2 .

k=1

Since (3.5.2) is also an orthonormal basis for L2 (Tn ), we conclude that for the flat torus we have N (λ) = #{j ∈ Zn : |j| ≤ λ/2π}. (3.5.3) By Theorem 3.3.1 we know that N (λ) = (2π)−n ωn λn + O(λn−1 ); however, unlike for the sphere, we can improve this result considerably. Theorem 3.5.1 If ∆ is the Laplacian on flat torus Tn = Rn /Zn , then we have that its Weyl counting function satisfies 2

N (λ) = (2π)−n ωn λn + O(λn−2+ n+1 ),

λ > 1.

(3.5.4)

60

CHAPTER 3

As (3.5.3) indicates, (3.5.4) is really just one about counting lattice points in Rn , saying that the number of integer lattice points inside a ball of radius R ≥ 1 is 2 equal to the volume of this ball modulo an error term which is O(Rn−2+ n+1 ). This theorem is due to Hlawka [34], but we shall give a somewhat different proof from the original one using the wave equation. Let us now turn to the proof of Theorem 3.5.1. In what follows, ∆Tn and ∆Rn shall denote the Laplacians on Tn and Rn , respectively. We shall exploit their relationship, described above, and the corresponding relationship between solutions of the wave equation ∂t2 − ∆Tn on Tn and periodic solutions of the d’Alembertian, ∂t2 − ∆Rn , on Rn . As in the proof of Theorem 3.5.1, a key step in proving (3.5.3) will be to obtain an appropriate estimate for spectral projection operators. Before, to obtain O(λn−1 ) error bounds for the Weyl formula, we needed to prove sup-norm estimates for functions whose spectrum was contained in a unit band. Now we need to prove sharper estimates in the case of Tn for functions with spectrum contained in small bands. Specifically, we shall use the following. Proposition 3.5.2 Fix n ≥ 2 and let Ej : L2 (Tn ) → L2 (Tn ) denote the projection onto the eigenspace spanned by e2πij·x , j ∈ Zn , i.e., Ej f (x) = fˆ(j)e2πij·x , Z fˆ(j) = f (y)e−2πij·y dy. Tn

Let
χελ f (x) =

X

Ej f (x).

λj ∈[λ,λ+ε]

Then there is a uniform constant C so that for 1 ≤ ε−1 ≤ λ p n−1 i h√ n+1 kf kL2 (Tn ) . ελn−1 + ε− 4 ελ 2 kχελ f kL∞ (Tn ) ≤ C

(3.5.5)

n−1

In particular, if we take ε = λ− n+1 , we have

X n−1 n−1

Ej f ≤ Cλ 2 − 2(n+1) kf kL2 (Tn ) .

∞ n λj

− n−1 ∈[λ,λ+λ n+1

L

(3.5.6)

(T )

]

Note that one can repeat the arguments showing that (3.2.5) implies (3.2.4) to see that (3.5.6) implies that n−1

2

N λ + λ− n+1 ) − N (λ) = O(λn−2+ n+1 ),

(3.5.7)

which would necessarily hold if (3.5.3) were valid. Since (3.5.5) implies (3.5.6) we shall just prove the former. To do so we shall use the approximate spectral projection operators defined by X p

χ ˜ελ f = χ ε−1 (λ − −∆Tn ) f = χ ε−1 (λ − λj ) Ej f, j∈Zn

where χ ∈ S(R) is as in (3.2.9) with δ = 1. Then if χ ˜ελ (x, y) denotes the kernel of ε χ ˜λ , we can repeat the arguments showing that (3.2.12) implies (3.2.11) to conclude

61

THE SHARP WEYL FORMULA

that, in order to establish (3.5.5), it is enough to prove that we have the following estimate for the kernel restricted to the diagonal  n−1  |χ ˜ελ (y, y)| ≤ C ελn−1 + (λ/ε) 2 , 1 ≤ ε−1 ≤ λ. (3.5.8) Furthermore, since we are assuming that ε−1 ≤ λ, we can also repeat the arguments showing that (3.2.12) is a consequence of (3.2.14) to see that (3.5.8) would follow from the uniform estimates Z p 1 
n−1  iλt ˆ cos(t −∆Tn ) (y, y) e dt ≤ C ελn−1 + (λ/ε) 2 , π εχ(εt) 1 ≤ ε−1 ≤ λ, (3.5.9) due to the fact that the Fourier transform of the even function τ → χ(ε−1 τ ) is the even function t → χ(εt). ˆ √ As noted before, u = cos t −∆Tn f , t > 0, is the solution of the Cauchy problem ( (∂t2 − ∆Tn )u = 0, on R+ × Tn (3.5.10) u|t=0 = f, ∂t u|t=0 = 0. Let us identify f with the corresponding function on Q, and let f˜ denote the periodic extension√to Rn , which was described at the beginning of the section. We then let u ˜ = cos t −t∆Rn f˜, t > 0, be the solution of the Cauchy problem ( (∂t2 − ∆Rn )˜ u = 0, on R+ × Rn (3.5.11) u ˜|t=0 = f˜, ∂t u ˜|t=0 = 0. By the translation invariance of the Laplacian on Rn , it is clear that u ˜(t, x + j) = u ˜(t, x) for j ∈ Zn since f˜(x + j) = f˜(x) for such j. Thus, u ˜ is periodic in the spatial variable. We therefore conclude that if we restrict u ˜ to R+ × Q and let u be the corresponding function on R+ × Tn then u must solve (3.5.10). In other words, all solutions of the Cauchy problem (3.5.10) come from periodic solutions of the Cauchy problem (3.5.11) and vice versa. Therefore, by Theorem 2.4.2, we must have that if we identify Q with Tn as above then X p p

cos(t −∆Tn ) (x, y) = cos(t −∆Rn ) (x − (y + j)), x, y ∈ Q. (3.5.12) j∈Zn

As a result, proving (3.5.9) is equivalent to showing that X Z p
1 itλ n ε χ(εt) ˆ cos(t −∆ ) (j) e dt R j∈Zn π  n−1 n−1  ≤ C ελ + (λ/ε) 2 ,

1 ≤ ε−1 ≤ λ. (3.5.13)

Note that if E+ is the forward fundamental solution for the d’Alembertian then p
cos(t −∆Rn ) (x) = ∂t E+ (t, x) − ∂t E+ (−t, x). Since, by Theorem 1.2.1, E+ (t, x) = 0 when |x| > t, and since supp χ ⊂ [−1, 1], we conclude that all the summands in (3.5.13) with |j| > ε−1 must vanish.

62

CHAPTER 3

We claim that for j = 0 we have Z 1
itλ ε χ(εt) ˆ ∂ E (t, 0) − ∂ E (−t, 0) e dt t + t + π 1 ≤ ε−1 ≤ λ, (3.5.14)

≤ Cελn−1 , while for x ∈ Rn , |x| ≥ 1, we also have the uniform bounds Z 1
itλ ε χ(εt) ˆ ∂ E (t, x) − ∂ E (−t, x) e dt t + t + π ≤ Cε(λ/|x|)

n−1 2

,

1 ≤ ε−1 ≤ λ, {x ∈ Rn : |x| ≥ 1}. (3.5.15)

Note that if these two estimates were valid then we would have (3.5.13) since then its left side would be majorized by X n−1 n−1 n−1
ελn−1 + ελ 2 |j|− 2 ≤ C ελn−1 + (λ/ε) 2 , {j∈Zn : 1≤|j|≤ε−1 }

as desired. To prove (3.5.14), we repeat the arguments from §3.2 to see that the term inside the absolute value equals Z Z (2π)−n χ(ε−1 (λ − |ξ|)) dξ + (2π)−n χ(ε−1 (λ + |ξ|)) dξ = O(ελn−1 ), Rn

Rn

since χ ˆ ∈ S(R) and 1 ≤ ε−1 ≤ λ. To prove (3.5.15) we recall that if n is odd then there is a Kirchhoff-type formula which says that ∂t E+ (t, x) = an H(t)|x|−

n−1 2

 ∂  n−1 2 δ(t − |x|), n = 3, 5, 7, . . . , ∂t

(3.5.16)

since then E+ is a multiple of the Heaviside function times δ ((n−3)/2) (t2 − |x|2 ). Using (3.5.16) and the fact that χ is even we conclude that for odd n we have 1 π

Z

εχ(εt) ˆ (∂t E+ (t, x) − ∂t E+ (−t, x))eitλ dt = a0n |x|−

n−1 2

 ∂  n−1   2 εχ(εt) ˆ cos tλ ∂t t=|x| = O(ε (λ/|x|)

n−1 2

),

1 ≤ ε−1 ≤ λ, |x| ≥ 1,

as desired. Thus, we have established (3.5.15) for odd n, which completes the proof of Proposition 3.5.2 for these dimensions. The case of even dimensions n = 2, 4, 6, . . . is a bit more technical due to the lack of Kirchhoff-type formulae, which of course is a reflection of the lack of strong Huygens’s principle for these cases. Notwithstanding, since E+ (t, x) is homogeneous of degree 1 − n, after a change of variables t → |x|t, one finds that (3.5.15) for even (or odd) spatial dimensions n is a consequence of the following estimates which are stated, for future use, in more generality than needed now.

63

THE SHARP WEYL FORMULA

Lemma 3.5.3

For ν = 0, 1, 2, . . . we have

Z Z cos tλ ∂t Eν (t, 1) dt + sin tλ ∂t Eν (t, 1) dt t ≤ Cν,n λ

n−1 2 −ν

,

λ ≥ 1. (3.5.17)

The bound for the first term in the left of (3.5.17) implies (3.5.15). We shall use the bound for the second term in the left with ν = 0 later in this section, while the bounds for the other ν will be needed in the next section. Proof. To prove (3.5.17), we recall that Eν (t, 1) is supported in [1, +∞) and that for 2ν < n − 1 ∂t Eν (t, 1) = cν tH(t) lim Im 1 − (t + iε)2


1−n 2 +ν−1

ε→0+

= cν

tH(t) (1 + t)

n+1 2 −ν

Im (1 − t) − i0


− n+1 2 +ν

,

where (s − i0)a = lim (s − iε)a . ε→0+

Since the Fourier transform of (s − i0)a is a constant times H(λ)λ−1−a if a < 0 (see Theorem A.3.2), we obtain (3.5.17) when 2ν < n − 1. Similar arguments apply to the cases where 2ν ≥ n − 1 since then Eν is a multiple of H(t)W−(n−1)/2+ν , with W−(n−1)/2+ν , ν ≥ (n − 1)/2 defined by (1.2.10) and (1.2.12). This completes the proof of Lemma 3.5.3 and hence that of Proposition 3.5.2.  As we shall see in the next chapter, one can also prove (3.5.15) and the more general estimates in Lemma 3.5.3 using stationary phase and formula (1.2.20) and Remark 1.2.5. End of proof of Theorem 3.5.1. We shall prove that if N (λ) is the Weyl counting function for Tn , n ≥ 2, then we have N (λ) = (2π)−n ωn λn + R(λ), where |R(λ)| ≤ C ελn−1 + (λ/ε)

n−1 2


,

1 ≤ ε−1 ≤ λ,

(3.5.18)

− n−1 n+1

in (3.5.18) so that the last for some uniform constant C. By choosing ε = λ two terms agree, we obtain (3.5.3). To proceed, as in the first step in the proof of Theorem 3.3.1, let us fix a function β ∈ C0∞ (R) with β(0) = 1. We shall also assume for convenience that β(t) = 0 for |t| ≥ 1. If we then let Z 1 sin λt ε rλ (τ ) = 1[−λ,λ] (τ ) − β(εt) cos tτ dτ, π t then it follows from (3.3.3) and a change of variables that for every N = 1, 2, 3, . . . there is a constant CN so that
−N |rλε (τ )| ≤ CN 1 + ε−1 | |τ | − λ | , λ, ε ≥ 1. (3.5.19)

64

CHAPTER 3

We then have 1 Sλ (x, x) − π

Z

∞

X p
sin tλ β(εt) cos t −∆Tn (x, x) dt = rλε (λj )|ej (x)|2 , t j=0

where because of (3.5.5) X ∞ ε n−1
2 rλ (λj )|ej (x)| ≤ C ελn−1 + (λ/ε) 2 , j=0

1 ≤ ε−1 ≤ λ,

by the same argument from §3.3 that established the estimate when ε = 1. As a result, it suffices to show that there is a uniform constant C so that Z p 1
sin tλ −n n cos t −∆Tn (x, x) dt − (2π) ωn λ π β(εt) t n−1
≤ C ελn−1 + (λ/ε) 2 , 1 ≤ ε−1 ≤ λ. If we recall (3.5.12), we see that this would follow from showing that Z Z 1 p sin tλ β(εt) cos(t −∆Rn )(0) dt − (2π)−n dξ π t n {ξ∈R : |ξ|≤λ} ≤ Cελn−1 ,

1 ≤ ε−1 ≤ λ, (3.5.20)

and X 1 Z p sin tλ π β(εt) t cos(t −∆Rn )(j) dt n

06=j∈Z

≤ C(λ/ε)

n−1 2

,

1 ≤ ε−1 ≤ λ. (3.5.21)

If we repeat the arguments that were used in §3.3 for the case of ε = 1, we see that the left side of (3.5.20) equals Z −n rλε (|ξ|) dξ ≤ Cελn−1 , 1 ≤ ε−1 ≤ λ, (2π) using (3.5.19). To prove the remaining inequality, (3.5.21), we first note that if n is odd, then by the proof of (3.5.15) for such n, we have Z p sin λt cos(t −∆Rn )(x) dt β(εt) t X  ∂  n−1  sin λt  2 − n−1 β(εt) = cn |x| 2 ∂t t ±t=|x| ± ≤ C|x|−

n+1 2

λ

n−1 2

,

if |x| ≥ 1.

This leads to (3.5.21) for odd n = 3, 5, 7, . . . since the summands all vanish when |j| > ε−1 as β(t) = 0, |t| ≥ 1, and, therefore, by the preceding inequality, the left side of (3.5.21) is majorized by X n+1 n−1 n−1 n−1 λ 2 |j|− 2 ≤ Cλ 2 ε− 2 , {j∈Zn : 0