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THE GIBBS PHENOMENON, THE PINSKY PHENOMENON,AND VARIANTS FOR EIGENFUNCTION EXPANSIONSMichael E. Taylor aa Mathematics Department , University of North Carolina , Chapel Hill, NC, 27599, U.S.A.Published online: 07 Feb 2007.

To cite this article: Michael E. Taylor (2002) THE GIBBS PHENOMENON, THE PINSKY PHENOMENON, AND VARIANTS FOREIGENFUNCTION EXPANSIONS, Communications in Partial Differential Equations, 27:3-4, 565-605, DOI: 10.1081/PDE-120002866

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THE GIBBS PHENOMENON, THE PINSKY

PHENOMENON, AND VARIANTS FOR

EIGENFUNCTION EXPANSIONS

Michael E. Taylor

Mathematics Department, University of NorthCarolina, Chapel Hill, NC 27599, USA

ABSTRACT

We examine analogues of the Gibbs phenomenon for eigen-function expansions of functions with singularities acrossa smooth surface, though of a more general nature thana simple jump. The Gibbs phenomena that arise still have auniversal form, but a more general class of ‘‘fractional sineintegrals’’ arises, and we study these functions. We also makea uniform analysis of eigenfunction expansions in the presenceof the Pinsky phenomenon, and see an analogue of the Gibbsphenomenon there. These analyses are done on three classesof manifolds: strongly scattering manifolds, includingEuclidean space; compact manifolds without strongly focus-ing geodesic flows, including flat tori and quotients of hyper-bolic space, and compact manifolds with periodic geodesicflow; including spheres and Zoll surfaces. Finally, we uncovera new divergence phenomenon for eigenfunction expansionsof characteristic functions of balls, for a perturbation of theLaplace operator on a sphere of dimension �5.

565

Copyright & 2002 by Marcel Dekker, Inc. www.dekker.com

COMMUN. IN PARTIAL DIFFERENTIAL EQUATIONS, 27(3&4), 565–605 (2002)

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1. INTRODUCTION

The Gibbs phenomenon, for the partial sums of the Fourier series of afunction on the circle with a jump discontinuity, has been seen to havecounterparts in a variety of situations. Multi-dimensional analogues havebeen treated in several recent papers, including (1–4). These papers examine(amongst other things) the Gibbs phenomenon for a compactly supportedpiecewise smooth function with a simple jump across a smooth surface �, ina number of cases, including Fourier inversion on Euclidean space R

n, toriTn, spheres Sn, and other Riemannian manifolds. In this context, by Fourier

inversion of a function f we mean taking

SR f ¼ �Rðffiffiffiffiffiffiffiffi��p

Þ f ð1:1Þ

and passing to the limit R!1. Here � is an appropriate self-adjointextension of the Laplace operator and �Rð�Þ is the characteristic functionof the interval ½�R,R, set equal to 1=2 at the endpoints.

The Gibbs phenomenon is essentially a local effect, but in higherdimensions nonlocal effects also arise in the Fourier inversion of functionswith simple singularities. The following phenomenon was analyzed in (5).Suppose B � R

3 is a ball of radius a, centered at 0, g 2 C10 ðR3Þ, and

f ¼ g�B. Then, as R!1,

SR f ð0Þ ¼ f ð0Þ �2

�ðAvg@B gÞ sin aRþ oð1Þ: ð1:2Þ

In particular, SR f ð0Þ ! f ð0Þ if and only if the mean value of gj@B vanishes.This result, often called the Pinsky phenomenon, can be understood as afocusing effect.

In (2) the Pinsky phenomenon was treated via an analysis using thewave equation. We can write

SR f ðxÞ ¼1

�

Z 1�1

sinRt

tuðt,xÞ dt, ð1:3Þ

with

uðt, xÞ ¼ cos tffiffiffiffiffiffiffiffi��p

f ðxÞ, ð1:4Þ

uniquely defined as the solution to the following initial value problem forthe wave equation:

utt ��u ¼ 0, uð0, xÞ ¼ f ðxÞ, utð0, xÞ ¼ 0: ð1:5Þ

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If f is compactly supported on Rn, or in other cases where the singularities

of f scatter off to infinity and one has a mild control over local energy decay(when the Riemannian manifold is what we call a ‘‘strongly scattering mani-fold’’) then the behavior of (1.3) as R!1 depends on the behavior ofuðt, xÞ only for t in a bounded interval (given x restricted to a compact set).In (2) the phenomenon (1.2) is seen to result from a perfect focus caustic.The effects of other caustics on the pointwise behavior of Fourier inversionis also considered in (2).

The papers (1) and (2) also applied wave equation techniques to studythe Gibbs phenomenon, (1) for Fourier inversion on R

2 and (2) both for Rn

and other strongly scattering manifolds. While (1) used the formula for thefundamental solution of the wave equation on R�R

2, (2) used a progres-sing wave expansion.

This paper has the dual purpose of extending the scope of analyses ofGibbs-type phenomena and analyzing the effects of focusing phenomena,particularly the Pinsky phenomenon, not only at points of focus but also ina neighborhood of the focus, thus providing a precise, uniform analysis ofthe behavior of SR f ðxÞ as R!1 for a significant class of functions withsingularities of classical conormal type.

We consider Gibbs-type phenomena for functions having a y�aþ typesingularity on a smooth surface �¼f y ¼ 0g, with 0 � a < 1. As in the caseof a jump discontinuity, the Gibbs phenomenon takes a universal form. Wehave, as R!1,

S�R f ðxÞ �Xj�0

AjðxÞRa�j Fa�jðR ðxÞÞ, ð1:6Þ

on a neighborhood of �. Here S�R f is a ‘‘filtered’’ partial Fourier inverse off , defined by inserting a cut-off �ðtÞ into (1.3); see (1.11) below. Its purpose isto isolate the Gibbs effect. Other cut-offs might isolate various focusingeffects. The function vanishes on � and has derivative of norm 1. Wemention parenthetically that the analysis giving (1.6) is applicable outsidethe range a 2 ½0, 1Þ, but the resulting phenomena are not ‘‘Gibbs’’ phenom-ena. See §4 for further discussion of this point.

The sine-integral function appearing in the classical Gibbs phenom-enon is extended to a class of fractional sine-integrals FaðsÞ, appearing in(1.6), whose qualitative properties are explored in §3. We also discuss theefficient numerical evaluation of these functions. It appears that there is aninteresting dichotomy depending on whether a < ac or a > ac, withac � 0:79. See §3 for a further discussion of this.

If f has a classical conormal singularity on � and the wave uðt, xÞ ¼cos t

ffiffiffiffiffiffiffiffi��p

f ðxÞ has a perfect focus at t ¼ �a, at a point p =2�, the uniform

EIGENFUNCTION EXPANSIONS 567

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analysis of S�R f ðxÞ for x in a neighborhood of p, established in §8, takes thefollowing form. In exponential coordinates with center at p (so p ¼ 0), wehave, on a neighborhood B of p,

S�R f ðxÞ � f ðxÞ þX�¼�

e�iaRXj�0

R�þðn�3Þ=2�j ��j ðx,RxÞ, ð1:7Þ

where � is the order of singularity of f on �, e.g., � ¼ a in the casementioned above. The functions ��j ðx, zÞ are determined by the detailednature of the singularities of f on �. They are smooth and have the asymp-totic behavior

��j ðx, zÞ � jzj

�ðn�1Þ=2X ¼�

e ijzj �ðx, z=jzjÞ, ð1:8Þ

as jzj ! 1, with �� smooth on B� Sn�1.The Pinsky phenomenon is manifested in the critical case � ¼

�ðn� 3Þ=2, in which case

S�R f ðxÞ ¼ f ðxÞ þX�¼�

e�iaR��0 ðx,RxÞ þOðR

�1Þ, ð1:9Þ

uniformly on a neighborhood of p. Thus the behavior at the focus is given by

S�R f ð0Þ ¼ f ð0Þ þX�¼�

e�iaR��0ð0, 0Þ; ð1:10Þ

compare (1.2). We might have ��ð0, 0Þ ¼ 0, e.g., in the context of (1.2) withAvg@B g ¼ 0. However, if f actually has a singularity on � of order�ðn� 2Þ=2, then ��0 ð0, zÞ will not be identically zero. In such a case, onehas pointwise convergence S�R f ðxÞ ! f ðxÞ for each x in a neighborhood ofthe focus p, but the convergence will not be uniform. This can be regarded asa shade of the Gibbs phenomenon!

At this point we need to say more about the introduction of a cut-offinto (1.3) and the analysis of the difference. As mentioned earlier, for com-pactly supported f on R

n and certain other noncompact manifolds satisfyinga strong scattering condition, the contribution to (1.3) from large jtj isnegligible. Clearly, for compact manifolds, the strong scattering conditionfails. As shown in (2), the case M ¼ Sn can be treated, making use of theperiodicity of cos tA, where A ¼ ½��þ ððn� 1Þ=2Þ21=2. In that case one canexpress SR f ðxÞ as an integral over a circle, and one obtains results quiteparallel to the case of R

n. We will say more about that case in §9. In anycase, such a trick is not applicable to most compact Riemannian manifolds;notably it does not work for tori. Wave equation techniques can still yield a

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good bit of information about Fourier inversion, but it is convenient to usea cut-off. Pick an even function � 2 C10 ðRÞ, with �ðtÞ ¼ 1 for t close to 0,and set

S�R f ðxÞ ¼1

�

ZsinRt

t�ðtÞ uðt, xÞ dt: ð1:11Þ

It is elementary to show that S�R f ! f in L2-norm, for any f 2 L2ðMÞ. We

call this ‘‘filtered Fourier inversion.’’ The wave equation techniquesmentioned above are effective in analyzing the pointwise behavior ofS�R f ðxÞ in a very general setting. Then it remains to analyze the difference

T�R f ¼ SR f � S�R f ¼ �Rð


Þ f , ð1:12Þ

where

�Rð�Þ ¼ �Rð�Þ � �� Rð�Þ: ð1:13Þ

An attack on (1.12) initiated in (3) takes

T�R f ðxÞ ¼X�

�Rð�Þ ff ð�Þ’�ðxÞ, ð1:14Þ

where f’�: � 2 specffiffiffiffiffiffiffiffi��p

g is an orthonormal basis of L2ðMÞ consisting of

eigenfunctions of � with eigenvalue ��2, and ff ð�Þ ¼ ð f , ’�ÞL2 , and writes

jT�R f ðxÞj �X�

j�Rð�Þj�j ff ð�Þj2

( )1=2 X�

j�Rð�Þj�j’�ðxÞj2

( )1=2

: ð1:15Þ

Applying Hormander’s estimateXR��Rþ1

j’�ðxÞj2� CRn�1, ð1:16Þ

given in (6), with n ¼ dimM, one obtains

kT�R f kL1 � CR�þðn�3Þ=2

k f k�� , ð1:17Þ

where

��ðMÞ ¼ f 2 D0ðMÞ:

XR��Rþ1

j ff ð�Þj2 � CR2��2g:

(ð1:18Þ

This led to a number of results on pointwise Fourier inversion on compactmanifolds, in (3), (7), (4), and (8).


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Part of the work in applying (1.17) has been to establish that variousfamiliar function spaces embed into ��

ðMÞ. In particular there are embed-ding results for spaces of conormal distributions. We define I�1 ðM,�Þ toconsist of distributions of the form Pf , where f is piecewise smooth witha simple jump across the smooth surface � and P is a pseudodifferentialoperator of order � and type ð1, 0Þ; we write P 2 OPS�1, 0ðMÞ. This containsthe space I�ðM,�Þ of classical conormal distributions, defined as above butrequiring that P be a ‘‘classical’’ pseudodifferential operator of order �:P 2 OPS�ðMÞ. It was shown in (4) that when � is a smooth hypersurfacein a flat torus T

n,

I�ðTn,�Þ � ��ðT

nÞ: ð1:19Þ

In (7) the more general result

I�1 ðM,�Þ � ��ðMÞ ð1:20Þ

was established, for a general compact Riemannian manifold M. The result(1.20) was then extended in (8) to

I�� ðM,�Þ � ��ðMÞ, ð1:21Þ

given 1=2 � � � 1, for any smooth conic Lagrangian � � T�Mn0.The exponent of R in (1.17) vanishes at � ¼ �ðn� 3Þ=2, the point

where the Pinsky phenomenon arises. This should not be surprising, but itmeans further work is required to show that the Pinsky phenomenon isvisible on compact manifolds. In fact, if we take �ðtÞ ¼ 1 on an interval½�T0,T0 and know that uðt, xÞ ¼ cos t


f ðxÞ has no caustics as strongas the perfect focus type for jtj � T0, and if a mild geometrical restriction isplaced on M, one can strengthen (1.17) to

limR!1

R��ðn�3Þ=2 kT�R f kL1 ¼ 0, ð1:22Þ

for f 2 I�ðM,�Þ. We demonstrate this in §5; in fact the proof is a slightmodification of the proof of the basic case � ¼ �ðn� 3Þ=2 done in (8).

Our work in §§2–4 on the Gibbs phenomenon for filtered Fourierinversion deals with a variety of cases in which f 2 I�ðM,�Þ with� > �ðn� 3Þ=2. In such cases it is desirable to supplement (1.17) with Lp-estimates on T�R f , which involve lower powers of R. For example, it is clearfrom the definition (1.18) that

kT�R f kL2 � CR��1 k f k�� : ð1:23Þ

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One can interpolate between (1.17) and (1.23), but this does not produceoptimal results. Sharper estimates are produced in §5, using work of (9).Following (9), one obtains an Lpn-estimate on T�R f , for pn ¼ 2ðnþ 1Þ=ðn� 1Þ, and interpolates that with (1.17) and with (1.23). As seen later, in§8, examples involving (1.7) explicitly demonstrate the sharpness of theseestimates on T�R f in the range pn � p � 1, therefore giving a proof of thesharpness of the Sogge estimates in this range, different from that given in (9).

These estimates on T�R f together with the Gibbs phenomenon analysisyield sharp estimates on SR f in the weak-Lp space LpwðMÞ, given f 2I�ðM,�Þ, 0 < � < 1. In §6 it is shown that fSR f :R � 1g is bounded inL1=�w ðMÞ as long as either n � 3 or � � ðn� 3Þ=ð2n� 2Þ. The Pinsky focus

analysis done in §8 also demonstrates the sharpness of this result.Having described the scope of this paper, we briefly sketch the struc-

ture of the following sections. Sections 2–4 deal with the generalized Gibbsphenomenon and associated special functions FaðsÞ. In §5 we deriveLp-estimates on the remainder T�R f ¼ SR f � S

�R f , and in §6 we draw

conclusions on Lpw-estimates. Section 7 gives general results on perfectfocus caustics, in preparation for the detailed analysis in §8 of S�R f in aneighborhood of a perfect focus, elucidating the Pinsky phenomenon.

In §9 we discuss SR ¼ �Rðffiffiffiffiffiffiffiffi��p

Þ for the Laplace operator � onspheres, Zoll surfaces, and variants. The ‘‘mild geometrical restriction’’mentioned before (1.22) fails dramatically here, and we discuss a mechanismfor producing results that are just as sharp in this case as they are forEuclidean space, though with subtle differences in the answers, reflectingthe effect of the special geometry on the analysis. These results are producedfor SRf as R!1, avoiding the intervals in which the eigenvalues of


cluster. In §10 we consider an example of a perturbation of this operator forwhich the partial sums within clusters exhibit wilder divergence phenomena.

We close with one remark on notation. Above (1.19) we alluded tovarious classes of pseudodifferential operators. Our terminology here, forOPSm1, 0ðMÞ, etc., follows that used in (10) and in Chapter 13 of (11),terminology essentially derived from (12). The symbol class Sm is denotedSmphg in (13).

2. FILTERED FOURIER INVERSION OF FUNCTIONS

WITH CONORMAL SINGULARITIES

Here we analyze

S�R f ðxÞ ¼1

�

Z 1�1

sinRt

t�ðtÞ uðt, xÞ dt, ð2:1Þ


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where uðt, xÞ ¼ cos tffiffiffiffiffiffiffiffi��p

f ðxÞ and f ðxÞ has the form

f ðxÞ ¼ gðxÞ’ðxÞ�aþ , 0 � a < 1: ð2:2Þ

Here g,’ 2 C1ðMÞ and we assume ’ > 0 on O, ’ < 0 on MnO, andd’ðxÞ 6¼ 0 for x 2 � ¼ @O. The cut-off � 2 C10 ðRÞ is assumed to havesupport in an interval I ¼ ð�c, cÞ sufficiently small that uðt, xÞ develops nocaustics for t 2 I . We assume �ðtÞ ¼ 1 for t 2 I1 ¼ ð�c1, c1Þ.

For t 2 I , the method of geometrical optics produces a progressingwave expansion of uðt, xÞ, as a sum of two functions, with conormal singu-larities across surfaces of the form

t ¼ � ðxÞ, jd j ¼ 1, j� ¼ 0, ð2:3Þ

on a neighborhood O of �. We may assume j ðxÞj < c1 for x 2 O. Oneobtains uðt, xÞ as the even part of

vðt, xÞ ¼ A0ðt, xÞð ðxÞ � tÞ�aþ þ A1ðt, xÞð ðxÞ � tÞ

1�aþ þ � � �

þ ANðt, xÞð ðxÞ � tÞN�aþ þ RNðt, xÞ, ð2:4Þ

where Aj 2 C1ðI �MÞ are obtained from certain transport equations

and RN 2 CNðI �MÞ. One can simplify (2.4) as follows. Replace A0ðt, xÞ

by A0ð ðxÞ, xÞ, and absorb the difference into the second term in (2.4). Simi-larly alter the second term and continue. We obtain an expansion of the form

vðt, xÞ ¼ A0ðxÞð ðxÞ � tÞ�aþ þ A1ðxÞð ðxÞ � tÞ

1�aþ þ � � �

þ ANðxÞð ðxÞ � tÞN�aþ þ RNðt,xÞ, ð2:5Þ

with Aj 2 C1ðMÞ and a new RN 2 C

NðI �MÞ.

Thus

S�R f ðxÞ ¼XNj¼0

AjðxÞIjðx,RÞ þ JNðx,RÞ, ð2:6Þ

with

Ijðx,RÞ ¼1

�

Z 1�1

sinRt

t�ðtÞ ð ðxÞ � tÞ j�aþ dt, ð2:7Þ

and

JNðx,RÞ ¼1

�

Z 1�1

sinRt

t�ðtÞRNðt, xÞ dt: ð2:8Þ

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Note that if we rewrite (2.2) as

f ðxÞ ¼ ~ggðxÞ ðxÞ�aþ , ð2:9Þ

then

A0ðxÞ ¼ ~ggðxÞ, x 2 �: ð2:10Þ

To tackle I0ðx,RÞ, we first note that omitting �ðtÞ in the integrand justchanges the integral by a rapidly decreasing quantity in R, so we have

I0ðx,RÞ ¼ I#0 ðx,RÞ þOðR

�1Þ, ð2:11Þ

with

I#0 ðx,RÞ ¼1

�

Z 1�1

sinRt

tð ðxÞ � tÞ�aþ dt

¼1

�

Z 1�1

sin t

t ðxÞ �

t

R

� ��aþdt

¼ Ra FaðR ðxÞÞ, ð2:12Þ

where, for a 2 ½0, 1Þ,

FaðsÞ ¼1

�

Z 1�1

sin t

tðs� tÞ�aþ dt

¼1

�

Z s

�1

sin t

tðs� tÞ�a dt: ð2:13Þ

For a ¼ 0 this is the sine-integral one sees in treatments of the classicalGibbs phenomenon. We will make a detailed study of FaðsÞ for generala < 1 in the next section.

For j � 1, the terms Ijðx,RÞ given by (2.7) are amenable to a similaranalysis. As long as j ðxÞj < c1, we have thatZ 1

�1

sinRt

tð ðxÞ � tÞ j�aþ ð1� �ðtÞÞ dt ¼ OðR

�1Þ, ð2:14Þ

as R!1. Now, for j � 1, a 2 ½0, 1Þ, this is not an absolutely convergentintegral, but it is well defined as an oscillatory integral, and the validity ofEq. (2.14) is established by a standard integration by parts argument. Thus,parallel to (2.11)–(2.12), we have, for j � 1,

Ijðx,RÞ ¼ I#j ðx,RÞ þOðR

�1Þ, ð2:15Þ


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with

I#j ðx,RÞ ¼1

�

Z 1�1

sinRt

tð ðxÞ � tÞ j�aþ dt

¼ Ra�j Fa�jðR ðxÞÞ, ð2:16Þ

where Fa�j is given as in (2.13), with now Fb defined as an oscillatoryintegral, for b � 0.

3. QUALITATIVE BEHAVIOR OF THE

FRACTIONAL SINE-INTEGRALS

The functions FaðsÞ, for a < 1, arose in the last section to describe thefiltered Fourier inversion of functions with a singularity of type ðxÞ�aþacross a smooth surface S ¼ fx: ðxÞ ¼ 0g, assuming d 6¼ 0 on S. Thefunctions are given by

FaðsÞ ¼1

�

Z s

�1

sin t

tðs� tÞ�a dt

¼1

�

Z 1�1

sin t

tðs� tÞ�aþ dt: ð3:1Þ

If a 2 ð0, 1Þ this integral is absolutely convergent. For a � 0 it is interpretedas an oscillatory integral. The case a ¼ 0 gives the familiar sine-integralone sees in treatments of the Gibbs phenomenon for Fourier inversion offunctions with a simple jump discontinuity. We want to establish a numberof properties of FaðsÞ and discuss its numerical evaluation.

Note that (3.1) is a convolution. Since convolution by ðsin tÞ=�tcoincides with Fourier multiplication by �½�1, 1, we see that

FaðsÞ ¼ CðaÞF��½�1, 1ðtÞt

�ð1�aÞ�ðsÞ, ð3:2Þ

where t�ð1�aÞ is taken to be the boundary value on R of a functionholomorphic on Re t < 0. That is,

t�ð1�aÞ ¼ e�ið1�aÞ jtj�ð1�aÞ, t < 0: ð3:3Þ

If a 2 ð0, 1Þ, t�ð1�aÞ is locally integrable, and for a ¼ 0 it is a principal-valuedistribution. More generally, we define it by inductive application oftb�1 ¼ b�1ðd=dtÞtb. Since (3.2) gives Fa as the Fourier transform of adistribution with compact support, we see that FaðsÞ is a smooth functionof s for each a < 1.

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One can compute the factor CðaÞ by integrating both t�aþ and t�ð1�aÞ

against e�t2=2 and using the Plancherel identity. One obtains the identity

2CðaÞe�ið1�aÞ=2 ¼21=2�a

�� 1�

a

2

� ��

1� a

2

� : ð3:4Þ

An examination of (3.2) allows us to specify the asymptotic behaviorof FaðsÞ as s!�1.

Proposition 3.1. Given a < 1, there exists real-valued BðaÞ such that

FaðsÞ � s�aþ þ BðaÞ

sinðs� �ð1� aÞ=2Þ

sþOðs�2Þ, jsj ! þ1: ð3:5Þ

Proof. By (3.2), the asymptotic behavior of FaðsÞ for large jsj arises from thethree singularities of �½�1, 1ðtÞt

�ð1�aÞ. The power singularity at t ¼ 0 givesback the term s�aþ in (3.5). There are two jumps, of magnitude �CðaÞ at t ¼ 1and of magnitude CðaÞe�ið1�aÞ at t ¼ �1. One verifies that these give rise tothe second term on the right side of (3.5), with

BðaÞ ¼ 2CðaÞe�ið1�aÞ=2, ð3:6Þ

which in turn is given by (3.4). &

We next explore the properties of FaðsÞ for s of moderate size, at leastfor a not too negative.

Proposition 3.2. For �1 < a < 1, we have

Fað0Þ > 0: ð3:7Þ

Proof. This follows readily from the representation

Fað0Þ ¼1

�

Z 10

sin t

tt�a dt:

One sees that

I� ¼

Z �ð�þ1Þ

��

sin t

tt�a dt, � 2 Z

þ

is an alternating sequence whose absolute value is monotonically decreasing.(In fact, one can read off Fað0Þ from (3.15) below.) &


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It is well known that the sine-integral F0ðsÞ has the property

F0ðsÞ > 0, 8 s � 0: ð3:8Þ

This is not true for all FaðsÞ, a 2 ð0, 1Þ. In fact, we have the following.

Proposition 3.3. Assume s 2 ðð2kþ 1Þ�, ð2kþ 2Þ�Þ, k 2 Zþ. Then

lima%1

FaðsÞ ¼ �1: ð3:9Þ

Proof. Write t�aþ ¼ f1a þ f2a with f1a supported on ½0, 1 and f2a on ½1,1Þ.Then convolution by ð1� aÞ f1a is an approximate identity as a% 1.Meanwhile one verifies directly that

lima!1

1� a

�

Zsin t

tf2aðs� tÞ dt ¼ 0,

so we have

lima%1ð1� aÞFaðsÞ ¼

sin s

s: ð3:10Þ

&

On the other hand, using (3.8) and the asymptotic behavior (3.4), wehave the following:

Proposition 3.4. There exists ac 2 ð0, 1Þ such that

0 � a < ac, s � 0 ¼) FaðsÞ > 0: ð3:11Þ

Numerical evidence suggests that the optimal value is

ac � 0:79: ð3:12Þ

For an efficient numerical evaluation of FaðsÞ, with a 2 ð0, 1Þ, it is convenientto use the representation (3.2), i.e.,

FaðsÞ ¼CðaÞffiffiffiffiffiffi2�p

Z 1

0

t�ð1�aÞheits þ e�ið1�aÞe�its

idt: ð3:13Þ

Writing the expression in square brackets as

2 e�ið1�aÞ=2 cos ts��ð1� aÞ

2

� , ð3:14Þ

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we have

FaðsÞ ¼BðaÞffiffiffiffiffiffi2�p

Z 1

0

t�ð1�aÞ cos ts��ð1� aÞ

2

� dt, ð3:15Þ

with BðaÞ as in (3.6). Evaluating at s ¼ 0 and using (3.7), we verify thatBðaÞ > 0, consistent with (3.4). We write the integral on the right side of(3.15) as

cos�ð1� aÞ

2F1aðsÞ þ sin

�ð1� aÞ

2F2aðsÞ, ð3:16Þ

where

F1aðsÞ ¼

Z 1

0

t�ð1�aÞ cos ts dt,

F2aðsÞ ¼

Z 1

0

t�ð1�aÞ sin ts dt:

ð3:17Þ

Now we can expand cos ts and sin ts into power series and integrate term byterm, obtaining

F1aðsÞ ¼X1k¼0

ð�1Þk

ð2kÞ!

s2k

2kþ a, F2aðsÞ ¼

X1k¼0

ð�1Þk

ð2kþ 1Þ!

s2kþ1

2kþ 1þ a:

ð3:18Þ

If one wishes to graph FaðsÞ over s 2 ½�8�, 8�, it is adequate to sum theseseries over 0 � k � 40.

4. THE GIBBS PHENOMENON FOR FILTERED

FOURIER INVERSION

We bring together results of the last two sections to analyze conver-gence of S�R f to f near the singular set �. We recall our set-up. We assume

f ðxÞ ¼ ~ggðxÞ ðxÞ�aþ , 0 � a < 1, ð4:1Þ

with as in (2.3) and ~gg 2 C1. By (2.5) we also have

f ðxÞ ¼XNj¼0

AjðxÞ ðxÞj�aþ þ RNð0, xÞ, ð4:2Þ


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with smooth AjðxÞ, while (2.6) and (2.15)–(2.16) give

S�R f ðxÞ ¼XNj¼0

AjðxÞRa�jFa�jðR ðxÞÞ þ JNðx,RÞ þOðR

�1Þ, ð4:3Þ

with FaðsÞ analyzed in §3 and JNðx,RÞ given by (2.8), i.e.,

JNðx,RÞ ¼ SR�xNð0Þ, ð4:4Þ

the partial Fourier inversion of the following function on R:

�xNðtÞ ¼ �ðtÞRNðt, xÞ: ð4:5Þ

Since �xN 2 CN0 ðRÞ, we have j��xNð�Þj � Ch�i

�N , so a crude estimate yields

jJNðx,RÞj � CNR�ðN�1Þ, N � 2: ð4:6Þ

A sharper estimate will be given below.To analyze the sums in (4.2), we use (3.4) to write

Ra�jFa�jðR ðxÞÞ ¼ ðxÞj�aþ þ R

a�j�a�jðR ðxÞÞ, ð4:7Þ

where, for 0 � a < 1,

j�aðsÞj �Cs�aþ

1þ jsj1�aþ

C

1þ jsj, ð4:8Þ

and, for j � 1,

j�a�jðsÞj �C

1þ jsj, ð4:9Þ

so, for j � 1,��Ra�j�a�j

�R ðxÞ

�� CRa�j

1þ jR ðxÞj� CRa�j: ð4:10Þ

Thus comparing (4.2) and (4.3) gives

S�R f ðxÞ � f ðxÞ

¼ A0ðxÞhRaFaðR ðxÞÞ � ðxÞ

�aþ

iþXNj¼1

Ra�jAjðxÞ�a�jðR ðxÞÞ þ JNðx,RÞ þOðR�1Þ: ð4:11Þ

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So far we have the estimate (4.6) on jJNðx,RÞj, but taking the expansion outfor a few more terms and using estimates of the form (4.10) yields thesharper estimate

jJNðx,RÞj � CNRa�1�N , N � 0: ð4:12Þ

Taking the N ¼ 0 case of (4.11) now gives the following.

Proposition 4.1. Assume f is given by (4.1), i.e.,

f ðxÞ ¼ ~ggðxÞ ðxÞ�aþ , 0 � a < 1, ð4:13Þ

with ~gg 2 C1. Then

S�R f ðxÞ ¼ f ðxÞ þ A0ðxÞhRaFaðR ðxÞÞ � ðxÞ

�aþ

iþOðRa�1Þ, ð4:14Þ

uniformly for x 2 O, a neighborhood of �.

Recalling that A0ðxÞ ¼ ~ggðxÞ on �, we can write

f ðxÞ � A0ðxÞ ðxÞ�aþ ¼ hðxÞ ðxÞ

1�aþ , h 2 C1ðOÞ, ð4:15Þ

and rewrite (4.14) as

S�R f ðxÞ ¼ A0ðxÞRaFaðR ðxÞÞ þ hðxÞ ðxÞ

1�aþ þOðRa�1Þ, ð4:16Þ

uniformly for x 2 O.One useful feature of (4.14) is that we can go from

jFaðsÞ � s�aþ j �

Cs�aþ

1þ jsj1�aþ

C

1þ jsjð4:17Þ

to

jS�R f ðxÞ � f ðxÞj �C ðxÞ�aþ

1þ jR ðxÞj1�aþ

CRa

1þ jR ðxÞjþ CRa�1: ð4:18Þ

In particular this is OðRa�1Þ on any compact subset of O on which ðxÞ isbounded away from zero. But of course the greatest interest in (4.14) and(4.16) arises from the detailed behavior of FaðsÞ, as exposed in §3.

Remark. While we have phrased the results in this section for functions f ofthe form (4.1) with 0 � a < 1, the analytical techniques apply to generalf 2 IaðM,�Þ, even for a =2 ½0, 1Þ. If a < 0, almost no alterations are neededin the presentation. However, the phenomenon then captured by (4.3)


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involves uniform convergence, and it is unremarkable compared to theGibbs phenomenon. General f 2 I0ðM,�Þ might have a logarithmicsingularity, leading to a slight variation in the a ¼ 0 case of (4.3), whichwe leave to the reader.

The behavior of S�R f for f 2 IaðM,�Þ with a � 1 is also un-Gibbs-like,

but in the opposite way, exhibiting extreme non-localization. To illustratethis in the case a ¼ 1, we consider the case where f is a smooth multipleof surface measure on �. In that case, we replace (2.5) by vðt, xÞ ¼A0ðxÞ�ð ðxÞ � tÞ þ � � �, so we have an analysis of (2.6) with

I0ðx,RÞ ¼1

�

Z 1�1

sinRt

t�ðtÞ �ð ðxÞ � tÞ dt ¼

1

�

sinR ðxÞ

ðxÞ: ð4:19Þ

This exibits the same type of non-localization phenomenon as Fourierinversion of � 2 E0ðRÞ. For f 2 IaðM,�Þ with a > 1, matters can evidentlybe worse.

5. Lp-LOCALIZATION

If f is more singular than the critical degree at which the Pinskyphenomenon arises, for example if f 2 I�ðM,�Þ with � > �ðn� 3Þ=2,then we do not have L1-localization results. Here we will obtain estimatesthat imply Lp-localization results for � between �ðn� 3Þ=2 and 1. We beginby noting the following operator properties, noted in (1.17) and (1.23):

kT�R f kL1 � CR�þðn�3Þ=2

k f k�� ,

kT�R f kL2 � CR��1 k f k�� ,ð5:1Þ

One could apply interpolation to this, but a stronger bound is obtained byapplying results of (9). For this, take

pn ¼ 2nþ 1

n� 1, n ¼

n� 1

2ðnþ 1Þ: ð5:2Þ

It is shown in (9) that

kPR f kLpn � CR n k f kL2 , ð5:3Þ

where

PR ¼ �½R,Rþ1ðffiffiffiffiffiffiffiffi��p

Þ: ð5:4Þ

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We hence have

kT�R f kLpn � CR�þ n k f k�� : ð5:5Þ

Interpolating (5.5) with each part of (5.1) yields the following.

Proposition 5.1. Given even � 2 C10 ðRÞ, we have

kT�R f kLp � CR�þðn�5Þ=4�ðn�1Þ=2p

k f k�� , 2 � p � pn, ð5:6Þ

and

kT�R f kLp � CR�þðn�3Þ=2�n=p

k f k�� , pn � p � 1: ð5:7Þ

Let f 2 I�ðM,�Þ. Since I�ðM,�Þ � ��ðMÞ, the results (5.6)–(5.7) are

applicable to f . The estimates (5.6)–(5.7) yield Lp-localization as long as theexponent of R is negative. As we will discuss in Proposition 5.3 below, undercertain conditions we can say that

limR!1

R��ðn�3Þ=2kT�R f kL1 ¼ 0: ð5:8Þ

In such a case, we can interpolate (5.5) with (5.8), to get the followingrefinement of what one could obtain from (5.7):

Proposition 5.2. If f 2 I�ðM,�Þ and (5.8) holds, then, for pn < p � 1,

limR!1

R��ðn�3Þ=2þn=p kT�R f kLp ¼ 0: ð5:9Þ

We now give a general condition under which (5.8) holds. This result isbasically a corollary of Proposition 2.1 of (8).

Proposition 5.3. Assume that, for R � 1, " 2 ð0, 1,XR��Rþ"

j’�ðxÞj2� C�ð"ÞRn�1 þ C ð",RÞRn�1, ð5:10Þ

uniformly for x 2M, with

limR!1

ð",RÞ ¼ 0, 8" > 0; lim"!0

�ð"Þ ¼ 0: ð5:11Þ

Let � �M be a smooth ðn� 1Þ-dimensional surface. Assume that there existT0 2 ð0,1Þ and � > 0 such that for the geodesic flow-out of the unit normalbundle of � any caustics that occur for jtj � T0 have order � ðn� 1Þ=2� �.


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Assume

f 2 I�ðM,�Þ: ð5:12Þ

Then, for each even � 2 C10 ðRÞ such that �ð�Þ ¼ 1 for j�j � T0 þ 1, we have(5.8).

Proof. As in the proof of Proposition 2.1 in (8), we pick �1 2 C10 ðRÞ, equal

to 1 on ð�1, 1Þ, and to 0 outside ð�2, 2Þ, and set �"ðtÞ ¼ �1ð"tÞ, S"R ¼ S

�"R .

Following (2.7)–(2.11) of (8), and using the inclusion I�ðM,�Þ � ��ðMÞ,

we obtain from (5.10)–(5.11) that

kSR f � S"R f k

2L1 � C½�ð"Þ þ ð",RÞR

2�þðn�3Þ: ð5:13Þ

Next we have

S"R f ðxÞ � S�R f ðxÞ ¼

1

�

ZsinRt

t½�"ðtÞ � �ðtÞ uðt, xÞ dt, ð5:14Þ

where uðt, xÞ ¼ cos tffiffiffiffiffiffiffiffi��p

f ðxÞ. Whenever " < 1=ðT0 þ 1Þ, �" � � is sup-ported on T0 < jtj < T" for some T" 2 ðT0,1Þ. Hence our hypothesis onpossible caustics produced by the flow-out of the unit conormal bundle of �for jtj � T0 gives

jS"R f ðxÞ � S�R f ðxÞj � C" R

�þðn�3Þ=2��, ð5:15Þ

uniformly in x. Combining (5.13) and (5.15), we have

lim supR!1

R�2��ðn�3Þ kSR f � S�R f k

2L1 � C�ð"Þ, 8" 2 ð0, 1=ðT0 þ 1Þ,

ð5:16Þ

which gives (5.8). &

As discussed in §3 of (8), an argument in the spirit of (14) showsthat the hypothesis (5.10) holds provided the geodesic flow-out of theunit cotangent spaces S�xM have only stable caustics, for t 6¼ 0. Thisholds for generic compact Riemannian manifolds M. Clearly it holdsfor M ¼ T

n, a flat torus; in this case (5.8) was established in (5.25)–(5.30)of (4).

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6. THE GIBBS PHENOMENON AND

WEAK-Lp ESTIMATES

Let M be a compact, n-dimensional Riemannian manifold, � �M asmooth ðn� 1Þ-dimensional surface. Our goal in this section is to prove thefollowing.

Proposition 6.1. Assume 0 < � < 1 and f 2 I�ðM,�Þ. Then

SR f is bounded in L1=�w ðMÞ, for R � 1, ð6:1Þ

provided that either

n � 3 ð6:2Þ

or n � 4 and

� �n� 3

2n� 2: ð6:3Þ

Remark. As we will see in §8, boundedness in (6.1) can fail for n � 4 and� < ðn� 3Þ=ð2n� 2Þ.

Proof of Proposition 6.1. Pick � 2 C10 ðRÞ as in §2. Then the results of§§2–4 give

S�R f bounded in L1=�w ðMÞ, for R � 1: ð6:4Þ

It remains to consider T�R f . We set p ¼ 1=� 2 ð1,1Þ, and consider threecases.

First suppose pn � p <1, i.e., 0 < � � 1=pn. Then (5.7) gives

kT�R f kLp � CRðn�3Þ=2�ðn�1Þ=p

k f k�1=p : ð6:5Þ

This is bounded for R � 1 if and only if ðn� 3Þp � 2ðn� 1Þ. Clearly thisholds for all p if n � 3, and if n � 4 it holds if and only if � ¼ 1=p satisfies(6.3). Note that the right side of (6.3) is � ðn� 1Þ=ð2nþ 2Þ ¼ 1=pn.

Next, suppose 2 � p � pn, i.e., 1=pn � � � 1=2. Then (5.6) gives

kT�R f kLp � CRðn�5Þ=4�ðn�3Þ=2p

k f k�1=p : ð6:6Þ

This is bounded for R � 1 if and only if pðn� 5Þ � 2ðn� 3Þ. Clearly thisholds for all p if n � 5. For n � 6 we have the condition

p �2n� 6

n� 5: ð6:7Þ


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But the right side of (6.7) is seen to be � pn ¼ ð2nþ 2Þ=ðn� 1Þ, so in fact(6.7) holds whenever p � pn.

Finally suppose 1 < p � 2, i.e., 1=2 � � < 1. In this case we can use(5.1), which gives

kT�R f kL2 � Ck f k�1 : ð6:8Þ

This finishes the proof.

7. PERFECT FOCUS CAUSTICS

Here we study the asymptotic behavior as �!1 of

vðx, �Þ ¼

Zeit�uðt, xÞ dt, ð7:1Þ

where uðt, xÞ is a conormal distribution whose Lagrangian � �T�ðR�R

nÞn0 is that of the fundamental solution of the (flat-space) wave

equation. Note that if

Rðt, xÞ ¼sin t

ffiffiffiffiffiffiffiffi��pffiffiffiffiffiffiffiffi��p �ðxÞ, ð7:2Þ

then a general such uðt, xÞ can be written

uðt, xÞ ¼ pðt, x,DxÞRðt, xÞ þ qðt,x,DxÞ@tRðt, xÞ, ð7:3Þ

where pðt, x,DxÞ and qðt, x,DxÞ are smooth families of pseudodifferentialoperators on R

n.Let us first consider the case where pðt, x,DxÞ and qðt, x,DxÞ are inde-

pendent of t. Let us assume � > 0. We have

v1ðx, �Þ ¼

Zeit�pðx,DxÞRðt, xÞ dt

¼

ZZeit�pðx, �Þ

sin tj�j

j�jeix�� d� dt

¼

Zpðx, �Þ

�ð�� j�jÞ

j�jeix�� d�

¼ �n�2ZSn�1

pðx, �!Þei�x�! dSð!Þ: ð7:4Þ

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If we assume pðx, �Þ 2 Sm, with asymptotic expansion

pðx, �!Þ �Xj�0

�m�jpjðx,!Þ, ð7:5Þ

we have

v1ðx, �Þ � �mþn�2

Xj�0

��j�jðx, �xÞ, ð7:6Þ

where

�jðx, zÞ ¼

ZSn�1

pjðx,!Þeiz�! dSð!Þ: ð7:7Þ

Note that �j is smooth and satisfies the estimate

j�jðx, zÞj � Chzi�ðn�1Þ=2: ð7:8Þ

More precise asymptotics for �jðx, zÞ as jzj ! 1 follow via the stationaryphase method:

�jðx, zÞ �X�¼�

jzj�ðn�1Þ=2 e�ijzjX‘�0

�‘ ðx, z=jzjÞ jzj�‘, ð7:9Þ

with �‘ smooth on Rn� Sn�1. A paradigm is given by p0ðx,!Þ ¼ 1, n ¼ 3,

in which case

�0ðx, �xÞ ¼sin �jxj

�jxj: ð7:10Þ

Similarly, given qðx, �Þ 2 Sm�1, with expansion

qðx, �!Þ �Xj�0

�m�1�jqjðx,!Þ, ð7:11Þ

we have (for � > 0)

v2ðx, �Þ ¼

Zeit�qðx,DxÞ@tRðt, xÞ dt � �

mþn�2Xj�0

��j jðx, �xÞ, ð7:12Þ

where

jðx, zÞ ¼

ZSn�1

qjðx,!Þeiz�! dSð!Þ, ð7:13Þ

which also satisfies an estimate of the form (7.8).


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Next we treat the case pðt, x,DxÞ ¼ ’ðtÞpðx,DxÞ, with ’ 2 C10 ðRÞ. The

analysis of this case will suffice for the treatments of the Pinsky focus on Rn

and Tn in §8. In this case, (7.4) is replaced by

v3ðx, �Þ ¼

Z’ðtÞeit�pðx,DxÞRðt, xÞ dt ¼ ’’ � v1ðx, �Þ, ð7:14Þ

where v1ðx, �Þ is given by (7.4) and the convolution is with respect to �.Consequently

v3ðx, �Þ �Xj�0

ZSn�1

pjðx,!Þ ’’ ��mþn�2�j ei�x�!

�dSð!Þ, ð7:15Þ

where we can cut off �mþn�2�j for � � 1. It is elementary to produce theasymptotic expansion

’’ ��mþn�2�j ei�x�!

�� mþn�2�jei�x�!

Xk�0

’kðx � !Þ��k, ð7:16Þ

where

’0ðx � !Þ ¼ ’ðx � !Þ: ð7:17Þ

Hence we have

v3ðx, �Þ �Xj, k�0

�mþn�2�j�k jkðx, �xÞ, ð7:18Þ

with

jkðx, zÞ ¼

ZSn�1

pjðx,!Þ’kðx � !Þ eiz�! dSð!Þ, ð7:19Þ

functions having estimates and asymptotic expansions similar to (7.8)–(7.9).One has similar results when ’ðtÞ is inserted into the integrand in (7.12).

Now we are ready to consider

v4ðx, �Þ ¼

Zeit� pðt, x,DxÞRðt, xÞ dt, ð7:20Þ

given pðt, x, �Þ 2 Sm, compactly supported in t. We also assume pðt, x, �Þ ¼ 0for j�j � 1. We have

v4ðx, �Þ ¼

Zeit� pðt, x, �Þ

sin tj�j

j�jeix�� d� dt

¼ c

Zppð�� j�j, x, �Þ

eix��

j�jd�, ð7:21Þ

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mod Oð��1Þ, as �!þ1. Here ppð�, x, �Þ is an element of Smcl , rapidlydecreasing as j�j ! 1. We can write

j��j�1ppð�, x, ��Þ �Xj�0

�m�j�1 qjð�, x, �Þ, ð7:22Þ

and then


Xj�0

��jZ�qj

��ð1� j�jÞ, x, �

�ei�x�� d�: ð7:23Þ

Each integral here is changed only by Oð��1Þ if we insert a smooth cut-off’ð�Þ, supported on 1=2 � j�j � 2. We can then switch to spherical polarcoordinates and apply the following result.

Lemma 7.1. Given q, ’ 2 SðRÞ, there exists an asymptotic expansionZ�qð�rÞ’ðrÞ dr �

Xk�0

ak��k, � ! þ1: ð7:24Þ

Proof. Set t ¼ 1=� and write the left side of (7.24) as

AðtÞ ¼

ZqðsÞ’ðtsÞ ds, t > 0:

This is easily seen to extend continuously to t 2 ½0,1Þ, with Að0Þ given as’ð0Þ

RqðsÞ ds. Similarly we extend

A0ðtÞ ¼

ZsqðsÞ’0ðtsÞ ds,

and so on. Hence A 2 C1ð½0,1ÞÞ. Then the expansion (7.24) follows fromthe formal power series expansion of AðtÞ about t ¼ 0. &

An analogue of (7.24) also works with extra parameters attached to qand ’. In (7.23), we set � ¼ j�j!, r ¼ 1� j�j, and hence ei�x�� ¼ ei�x�!e�i�rx�!,and obtain


Xj, k�0

��j�k jkðx, �xÞ, ð7:25Þ

with

jkðx, zÞ ¼

ZSn�1

qjkðx,!Þeiz�! dSð!Þ, ð7:26Þ

having the same qualitative properties as (7.18)–(7.19).


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Finally, a similar analysis works onZeit� qðt, x,DxÞ @tRðt, xÞ dt: ð7:27Þ

8. FOURIER INVERSION NEAR A PINSKY FOCUS

Here we analyze the behavior of SR f ðxÞ uniformly for x in a neighbor-hood of p, when f 2 I�ðM, @BÞ and B is a ball centered at p. We begin in theclassical context of Fourier inversion on R

n.Thus consider a ball B � R

n, n � 2. We may as well assume its centeris p ¼ 0. We take f 2 I�ðRn, @BÞ, compactly supported, and analyze

SR f ðxÞ ¼1

�

Z 1�1

sinRt

tuðt, xÞ dt, ð8:1Þ

for x in a neighborhood of 0. As usual, uðt, xÞ ¼ cos tffiffiffiffiffiffiffiffi��p

f ðxÞ. Let a be theradius of B. Then uðt, xÞ has two perfect focus caustics, at x ¼ 0, t ¼ �a.Fix b 2 ð0, aÞ. We desire to analyze the behavior of (1) as R!1, uniformlyfor x 2 Bb ¼ fx: jxj � bg. To get started on this, fix c 2 ð0, a� bÞ, andintroduce a smooth partition of unity �jðtÞ, � 2 � j � 2, satisfying

supp �0 � ð�c, cÞ, supp��1 � �ðc=2, aþ 2Þ,

supp ��2 � � ðaþ 1,1Þ: ð8:2Þ

We can arrange that ��jðtÞ ¼ �jð�tÞ. With the �jðtÞ inserted into the inte-grand in (8.1), denote the resulting term by SjR f ðxÞ, � 2 � j � 2.

Since �0ðtÞuðt, xÞ is smooth on R� Bb, it is easy to see that

S0R f ðxÞ ¼ f ðxÞ þOðR

�1Þ, uniformly on x 2 Bb: ð8:3Þ

Also the functions ��2ðtÞuðt, xÞ are smooth on R� Bb. If n is odd, they arecompactly supported in t. If n is even they have asymptotic expansions ast!�1 of the form

��2ðtÞuðt, xÞ �Xj�0

vjðxÞjtj�ðn�1Þ�j , ð8:4Þ

so we have

S�2R f ðxÞ ¼ OðR�1Þ, uniformly on x 2 Bb: ð8:5Þ

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It remains to examine S�1R f ðxÞ. Note that, for t 2 supp�1, we have,mod C1,

uðt, xÞ ¼ cosðt� aÞffiffiffiffiffiffiffiffi��p

pðx,DÞ�ðxÞ þsinðt� aÞ

ffiffiffiffiffiffiffiffi��pffiffiffiffiffiffiffiffi

��p qðx,DÞ�ðxÞ,

ð8:6Þ

with

pðx,DÞ 2 OPS��ðnþ1Þ=2, qðx,DÞ 2 OPS��ðn�1Þ=2: ð8:7Þ

In fact here we can take pðx,DÞ ¼ pðDÞ and qðx,DÞ ¼ qðDÞ. A similaranalysis holds for t 2 supp ��1. Hence the results of §7 give the following:

Proposition 8.1. Given the ball B � Rn and f 2 I�ðRn, @BÞ, compactly sup-

ported, we have, for

S1R f ðxÞ ¼ S

�1R f ðxÞ ¼

1

�

ZðsinRtÞ

�1ðtÞ

tuðt, xÞ dt ð8:8Þ

an asymptotic expansion of the form

S�1R f ðxÞ �X�¼�

e�iaRXj�0

R�þðn�3Þ=2�j��j ðx,RxÞ, ð8:9Þ

uniformly on Bb, where the functions ��j ðx, zÞ are smooth and satisfy theestimates

j��j ðx, zÞj � Cjhzi�ðn�1Þ=2, ð8:10Þ

and indeed have the asymptotic expansion

��j ðx, zÞ �

X ¼�

jzj�ðn�1Þ=2 e ijzjX‘�0

j‘�ðx, z=jzjÞ jzj�‘, ð8:11Þ

as jzj ! 1, with �j‘� smooth on Bb � Sn�1. (Here � ¼ �.)

Specializing to the critical case � ¼ �ðn� 3Þ=2, we have:

Corollary 8.2. If f 2 I�ðn�3Þ=2ðRn, @BÞ, B a ball of radius a centered at 0, then,uniformly on Bb � B, we have

SR f ðxÞ ¼ f ðxÞ þX�¼�

e�iaR��0 ðx,RxÞ þOðR

�1Þ: ð8:12Þ


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Note that matching this with (1.2) we have for f 2 I0ðR3, @BÞ, piece-wise smooth with simple jump across @B,

X�¼�

e�iaR��0ð0, 0Þ ¼ �

2

�

�Avg@B½ f

�sin aR, ð8:13Þ

where ½ f denotes the jump of f across @B (from R3nB to B). In particular

we have the result of (5) that SR f ð0Þ ! f ð0Þ for such a piecewise smooth f ,if and only if the average value of the jump is zero. The uniform analysisprovides a counterpoint to this result on the behavior of SR f ð0Þ. Namely,suppose f is piecewise continuous with a simple jump across @B and that theaverage of the jump is zero, so SR f ð0Þ ! f ð0Þ. If there is a nontrivial jump,then the functions ��0 ð0, zÞ are not identically zero. As a consequence, (8.12)shows that SRf does not converge to f uniformly on a neighborhood ofzero, even though SRf ðxÞ ! f ðxÞ for each x 2 Bb. In fact, (8.12) records theprecise way in which uniform convergence fails. See (8.26) below for anexplicit example. This phenomenon is evidently somewhat parallel to theGibbs phenomenon.

We record some more explicit computations for specific examples offunctions singular on the boundary of the unit ball B ¼ B1ð0Þ � R

3. Forbrevity of notation we use O0ðR�kÞ to denote a remainder with asymptoticexpansion of the formX

�¼�

Xj�k

e�iR��j ðx,RxÞR

�j,

where the asymptotic behaviors as in (8.11) hold. Being O0ðR�kÞ impliesbeing OðR�kÞ, but it is much more precise. Note that in (8.12) we couldreplace OðR�1Þ by O0ðR�1Þ.

Our first example is

f1ðxÞ ¼ �BðxÞ: ð8:14Þ

By rotational symmetry, in the analysis of the perfect focus causticsproduced by cos t


f1, one has amplitudes pjðx,!Þ and qjðx,!Þ in (7.3)and (7.13) that are independent of !, at x ¼ 0, with a similar observationapplying to (7.19). Hence in this case (8.12) can be given the more explicitform

SR�BðxÞ ¼ 1�2

�ðsinRÞ s

ðx,RxÞ þ ðcosRÞ cðx,RxÞ þ O0ðR�1Þ,

ð8:15Þ

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uniformly for jxj � b < 1, where sðx, zÞ and c

ðx, zÞ are smooth andbehave like (8.11), and

sð0, zÞ ¼

sin jzj

jzj, c

ð0, zÞ ¼ 0: ð8:16Þ

The factor �ð2=�ÞðsinRÞ arises from (8.13). Next, take

f2ðxÞ ¼ jxj2�B: ð8:17Þ

Since this differs from f1 by an element of I�1ðR3, @BÞ, we have

SRf2ðxÞ ¼ jxj2�

2

�ðsinRÞ s

ðx,RxÞ þ ðcosRÞ cðx,RxÞ þ O0ðR�1Þ,

ð8:18Þ

uniformly for jxj � b < 1, with the same s and c as in (8.15). We use thisas a tool in the analysis of SR f3ðxÞ, where

f3ðxÞ ¼ ð1� jxj2Þ�B, ð8:19Þ

an element of I�1ðR3, @BÞ.To do this, we use the identity

SRD� f ðxÞ ¼ D�SR=� f ðxÞ, D� f ðxÞ ¼ f ð �xÞ,

which gives

SRLf ðxÞ ¼ LSRf ðxÞ � R@

@RSR f ðxÞ, ð8:20Þ

where

LgðxÞ ¼@

@�D�gðxÞ

��¼1¼ x � rgðxÞ: ð8:21Þ

Now Lf3 ¼ �2f2, so

R@

@RSR f3ð0Þ ¼ �

4

�sinRþO0ðR�1Þ, ð8:22Þ

and hence

SRf3ð0Þ ¼4

�

cosR

RþO

0ðR�2Þ: ð8:23Þ


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At this point, using the radial symmetry of f3ðxÞ and the same reasoning asapplied to get (8.15), we obtain

SR f3ðxÞ ¼ 1� jxj2 þ4

�

cosR

R c

3ðx,RxÞ þsinR

R s

3ðx,RxÞ þ O0ðR�2Þ,

c3ð0, zÞ ¼

sin jzj

jzj, s

3ð0, zÞ ¼ 0, ð8:24Þ

valid uniformly on jxj � b < 1. For our fourth example, we take

f4ðxÞ ¼ x1�B: ð8:25Þ

Note that f4 ¼ �ð1=2Þ@f3=@x1. Since SR commutes with @=@x1, we obtainfrom (8.24) that

SRf4ðxÞ ¼ x1 �2

�ðcosRÞ c

4ðx,RxÞ þ ðsinRÞ s4ðx,RxÞ þ O

0ðR�1Þ,

c4ð0, zÞ ¼

@

@z1

sin jzj

jzj, s

4ð0, zÞ ¼ 0, ð8:26Þ

uniformly on jxj � b < 1.We return to more general considerations. If f 2 I�ðRn, @BÞ and

� > �ðn� 3Þ=2, then S�1R f is not bounded on Bb. We can estimate Lp-norms, readily obtaining from (8.9) thatZ

Bb

jS�1R f ðxÞjp dx � CRp�þpðn�3Þ=2�n: ð8:27Þ

As long as the functions ��0 ð0,RxÞ in (8.9) are not identically zero, we alsohave a lower bound of the same form. Note that the right side of (8.27) isbounded for R 2 ½1,1Þ if and only if

� �n

p�n� 3

2: ð8:28Þ

We could replace Rn by a variety of noncompact Riemannian mani-

folds satisfying the strong scattering condition, but rather than dwelling onthat we now consider the case of a compact, n-dimensional, RiemannianmanifoldM. We assume B is a ball inM, of radius a, centered at p, and wewill make the standing assumption that the exponential map takes a ball inTpM diffeomorphically onto B. We will also assume that M and @B satisfythe hypotheses of Proposition 5.3. In particular we assume the geodesicflow starting at a fiber S�xM has no perfect focus. This hypothesis holds ifM is a flat torus T

n, or more generally has nonpositive sectional curvature

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everywhere, and in fact it holds generically. It does not hold for the standardspheres Sn, which will be discussed in further detail in §9.

Now take f 2 I�ðM, @BÞ. The analyses of S0Rf ðxÞ and S

�1R f ðxÞ proceed

as before, but we need to replace the analysis of S�2R f ðxÞ. Now, with� ¼ ��1 þ �0 þ �1, we have

S2R f ðxÞ þ S

�2R f ðxÞ ¼ T�R f ðxÞ, ð8:29Þ

and, by Proposition 5.3,

f 2 I�ðM, @BÞ ¼) limR!1

R��ðn�3Þ=2 kT�R f kL1 ¼ 0: ð8:30Þ

As a result we have the following. As before, we take Bb to be centered at pand have radius b < a.

Proposition 8.3. Given the ball B �M, and given f 2 I�ðM, @BÞ, we have (inexponential coordinates)

SR f ðxÞ ¼ f ðxÞ þX�¼�

e�iaR��0ðx,RxÞR

�þðn�3Þ=2þ oðR�þðn�3Þ=2Þ,

ð8:31Þ

uniformly for x 2 Bb.

This result holds for all � 2 R, but it is most useful when � ��ðn� 3Þ=2. It is useful to record Lp-estimates on the remainder in (8.31).Since (8.30) holds, we can apply Proposition 5.2, to obtain the following.

Proposition 8.4. Given the ball B �M and given f 2 I�ðM, @BÞ, we have, forpn < p � 1,

SR f � f �X�¼�


�þðn�3Þ=2

��Lp

¼ oðR�þðn�3Þ=2�n=pÞ:

ð8:32Þ

In particular, if n ¼ 3, then for p > 4 the Lp-norm of the remainder isoðR��3=pÞ. If n ¼ 2, then for p > 6 the Lp-norm of the remainder isoðR��1=2�2=pÞ.

As noted in (8.27) and the remark following it, the Lp-norm ofX�¼�


�þðn�3Þ=2ð8:33Þ


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is on the order of R�þðn�3Þ=2�n=p, so this dominates the quantity estimated in(8.32). Hence, we see that for p > pn,

kSR f � S�0R f kLp � CR

�þðn�3Þ=2�n=p, ð8:34Þ

for typical f 2 I�ðM, @BÞ. This implies that the estimate (5.7) is sharp(for � ¼ �0), which in turn implies that the Sogge estimate of kPR f kLp

for pn < p <1, arising via interpolating (5.3) with the estimatekPR f kL1 � CR

ðn�1Þ=2k f kL2 , is sharp. This was proven in (9), by a different

method.By the same token, this argument shows that Proposition 6.1 is sharp,

in that if n ¼ dimM � 4 and 0 < � < ðn� 3Þ=ð2n� 2Þ, then there existsf 2 I�ðM, @BÞ such that SRf is not bounded in L1=�

w ðMÞ, for R � 1, i.e.,such that (6.1) fails.

9. SPHERES, ZOLL SURFACES, AND VARIANTS

As indicated in the introduction, there are sharp results on Gibbsphenomena, Pinsky phenomena, and related issues involving eigenfunctionexpansions on spheres, even though the results of §5 fail in this case. Thebasic mechanism behind this analysis was explained in (2). Here we givefurther details and extend these considerations to Zoll surfaces and otherrelated contexts.

The key to the analysis on Sn is that the operator

A ¼ ��þn� 1

2

� 2 !1=2

ð9:1Þ

has the property

SpecA ¼n� 1

2þ k: k ¼ 0, 1, 2, . . .g:

�ð9:2Þ

Hence cos tA is periodic in t, of period 2� for n odd and of period 4� for neven. Thus, for n odd, we can set

SN f ðxÞ ¼1

2�

ZS1

sinðN þ ð1=2ÞÞt

sin ð1=2Þtuðt, xÞ dt, ð9:3Þ

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with uðt, xÞ ¼ ðcos tAÞ f ðxÞ, and then �RðAÞ ¼ SN for N < R < N þ 1. For neven we can set

SN f ðxÞ ¼1

4�

Z~SS


sin ð1=4Þtuðt, xÞ dt, ð9:4Þ

and then �RðAÞ ¼ SN for N þ 1=2 < R < N þ 3=2. Here S1¼ R=ð2�ZÞ and

~SS ¼ R=ð4�ZÞ, and N is taken in Zþ.

We briefly describe how the Gibbs effect shapes up in this case. Sayf is a function on Sn that is smooth away from a smooth surface �and has the form (2.9) near �. For convenience, assume n is odd. Let usagain bring in the cut-off �ðtÞ. This time one has the following variant of(2.6)–(2.7):

S�N f ðxÞ ¼XKj¼0

AjðxÞIjðx,NÞ þ JK ðx,NÞ, ð9:5Þ

with

Ijðx,NÞ ¼1

�

ZS1


sin ð1=2Þt�ðtÞ

� ðxÞ � t

��aþjþ

dt: ð9:6Þ

If we write

~��ðtÞ ¼t

sinð1=2Þt�ðtÞ, ð9:7Þ

and

~��ðtÞ ¼ ~��ð ðxÞÞ þh~��ðtÞ � ~��ð ðxÞÞ

i¼ ~��ð ðxÞÞ þ ðt� ðxÞÞ ðt, xÞ, ð9:8Þ

we can inductively trade �ðtÞ in (9.6) for factors that are just smooth func-tions of x. Then we obtain the following variant of (4.3):

S�N f ðxÞ ¼XKj¼0

eAAjðxÞ ðN þ ð1=2ÞÞa�jFa�jððN þ 1=2Þ ðxÞÞ

þ JK ðx,NÞ þOðN�1Þ, ð9:9Þ


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where eAAjðxÞ are obtained from the functions AjðxÞ of (2.5) in an explicitfashion, which we write down just for j ¼ 0:

eAA0ðxÞ ¼ ðxÞ

2 sin ðxÞ=2A0ðxÞ: ð9:10Þ

This treats the case of odd n. The analysis for even n is similar.The cut-off �ðtÞ might be supplemented by other cut-offs, to isolate

various focusing effects. However, in place of an analysis of large t effects onthe remainder SN f � S

�N f , here the following simple result suffices.

Proposition 9.1. Let K � Sn be compact and take A ¼ ft 2 S1 : �ðtÞ 6¼ 1g. Ifuðt, xÞ is smooth on a neighborhood of A� K � S1

� Sn, then

SNf ðxÞ � S�N f ðxÞ ¼ OðN

�1Þ, uniformly on x 2 K : ð9:11Þ

The uniform analysis near a Pinsky focus done in §8 extends tospheres, as follows. Let B � Sn be a ball; say its center is p1, with antipodep2, so B ¼ Ba1ðp1Þ and SnnB ¼ Ba2 ð p2Þ, with a1 þ a2 ¼ �. Then if f 2I�ðSn, @BÞ we have, in exponential coordinates centered at p1,

SN f ðxÞ ¼ f ðxÞþX�¼�

e�ia1ðNþð1=2ÞÞ��0ðx, ðNþð1=2ÞÞxÞ ðNþð1=2ÞÞ

�þðn�3Þ=2

þ oðN�þðn�3Þ=2Þ, ð9:12Þ

uniformly on a neighborhood of p1 ¼ 0, and similarly near p2, if n is odd,with a similar formula if n is even. The functions ��0 ðx, zÞ have the samegeneral form as in §8. However, if for example � ¼ 0 and f ¼ �B, the valuesof ��0 ð0, 0Þ will differ from those that work in (8.13).

To illustrate this point, let’s take n ¼ 3 and f ¼ �B. The functionuðt, xÞ ¼ cos tA f ðxÞ on R� S3 is given by

uðt, xÞ ¼ @tðsin t f xðjtjÞÞ

¼ ðcos tÞf xðjtjÞ þ ðsin tÞ@t f xðjtjÞ, ð9:13Þ

where f xðrÞ denotes the spherical mean of f , over a 2-sphere of radius rcentered at x 2 S3. For f ¼ �B we have

@t f p1ðjtjÞ ¼ �ðtþ a1Þ � �ðt� a1Þ,

@t f p2ðjtjÞ ¼ �ðt� a2Þ � �ðtþ a2Þ: ð9:14Þ

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In concert with (9.3) this yields

SN�Bð p1Þ ¼ 1�1

�

sin a1sinða1=2Þ

sinðN þ 1=2Þa1 þ oð1Þ,

SN�Bð p2Þ ¼1

�

sin a2sinða2=2Þ

sinðN þ 1=2Þa2 þ oð1Þ,

ð9:15Þ

as N !1, which can be compared with (1.2).The behaviors of (9.12) and (9.15) contrast with the behaviors of (1.2)

and (8.31), valid for eigenfunction expansions on Riemannian manifoldsMsatisfying the hypotheses of Proposition 5.3, in the following significant way.In the context of Corollary 8.2, for example, the values of SR f ðxÞ at aPinsky focus x ¼ p have only small jumps as R crosses points inSpec


(for large R). However, typically, for f 2 I�ðn�3Þ=2ðSn, @BÞ, thejumps from SNf ðpÞ to SNþ1ð pÞ at a Pinsky focus p are not small. Thisillustrates the breakdown of the results of §5 when M ¼ Sn, and of courseit is connected with the clustering of the spectrum of the Laplace operatoron Sn.

Phenomena such as described above also hold for other compact rank-one symmetric spaces, such as CP

n and HPn; cf. (2) for the analogues of

(9.1)–(9.4).There are also extensions of the results on spheres given above to Zoll

surfaces, smooth Riemannian manifolds with periodic geodesic flow, whichwe can assume to be of period 2�. Consider the Laplace operator � onsuch a manifold. More generally than


, we consider any self-adjointoperator A of the form

A ¼ffiffiffiffiffiffiffiffi��p

þ B, B 2 OPS�1ðMÞ: ð9:16Þ

The following result provides a replacement for (9.1)–(9.2). We assume thatM is connected and dim M � 2.

Proposition 9.2. Assume all geodesics on M have the minimal period 2�. Thereexists a positive, self-adjoint A 2 OPS1

ðMÞ, commuting with A, such that

A ¼ Aþ �I þ S, S 2 OPS�1ðMÞ, ð9:17Þ

and

e2�iA ¼ I : ð9:18Þ

This result is due to (15); see also §29.2 of (13). We make some remarkson the proof; for more details see (13). To start, we can say that


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e2�iA ¼ P 2 OPS0ðMÞ, and the global calculus of Fourier integral operators

yields that the principal symbol of P is constant, so

P ¼ cI þ R, jcj ¼ 1, R 2 OPS�1ðMÞ: ð9:19Þ

Hence SpecP has only c as an accumulation point, so we can apply thelogarithm and write

P ¼ e2�iQ, Q 2 OPS0ðMÞ, ð9:20Þ

where Q is self-adjoint and commutes with A. Furthermore,

Q ¼ �I þ S, e2�i� ¼ c, S 2 OPS�1ðMÞ: ð9:21Þ

Note the following relation between the principal symbols of R and S:

�ðSÞ ¼1

2�ic�ðRÞ: ð9:22Þ

Now the desired operator A is given by A ¼ A�Q. If necessary we can adda negative integer to � to ensure that A is positive.

Since SpecA � Zþ and S 2 OPS�1ðMÞ commutes with A, it follows

that SpecA clusters around small neighborhoods of N þ �, for largeN 2 Z

þ. The clusters are contained in intervals

IN ¼ ðN þ �� "N ,N þ �þ "NÞ, "N � CN�1: ð9:23Þ

Here we can take any C > kASk. Hence, for large R,

N þ �þ "N < R < ðN þ 1Þ þ �� "Nþ1

¼)�RðAÞ ¼ �Nþ1=2ðAÞ ¼ SN , ð9:24Þ

where SN is given by (9.3), with

uðt, xÞ ¼ cos tA f ðxÞ: ð9:25Þ

This time the equation

utt þA2u ¼ 0, uð0,xÞ ¼ f ðxÞ, utð0, xÞ ¼ 0 ð9:26Þ

is not quite a partial differential equation, but the solution operator cos tAretains enough of the general features of cos t


that the results on theGibbs phenomena and Pinsky phenomena described above persist in thismore general situation. This provides a description of �RðAÞ f as R!1,avoiding the clusters in IN , given by (9.23). In §10 we will say more aboutwhat can happen when R runs through these clusters.

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The global calculus of Fourier integral operators is useful in identify-ing c and the principal symbol of R in (9.19), but in some cases of interestmore elementary tools suffice for the calculation. For example, consider

A ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi��þ Vp

¼ A0 þ B, ð9:27Þ

with

A0 ¼ ��þn� 1

2

� 2 !1=2

, B 2 OPS�1ðMÞ, M ¼ Sn: ð9:28Þ

This case was studied in (16) and (17). In this case, P in (9.19) is equal toPð2�Þ, where

PðtÞ ¼ eitAe�itA0 : ð9:29Þ

We have

P0ðtÞ ¼ i½A0,PðtÞ þ iBPðtÞ, Pð0Þ ¼ I : ð9:30Þ

On the symbol level, this gives

@p

@t�Hap ¼ ibpþ i

Xj�j�1

ij�j

�!bð�Þpð�Þ

þ iXj�j�2

ij�j

�!

�að�Þpð�Þ � p

ð�Það�Þ

�: ð9:31Þ

If p � p0 þ p�1 þ � � �, we obtain

@p0@t�Ha1p0 ¼ 0, p0ð0, x, �Þ ¼ 1: ð9:32Þ

Hence p0 ) 1, so c ¼ 1 in (9.18). Given that p0 ) 1, the equation for p�1simplifies to

@p�1@t�Ha1p�1 ¼ ib�1p0, p�1ð0, x, �Þ ¼ 0, ð9:33Þ

so

p�1ðt, x, �Þ ¼ i

Z t

0

b�1

�F sðx, �Þ

�ds, ð9:34Þ


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where F s is the Hamiltonian flow on T�M ¼ T�Sn generated by Ha1 , i.e.,the geodesic flow in this case. One verifies that

b�1ðx, �Þ ¼1

2j�jV1ðxÞ, V1ðxÞ ¼ VðxÞ �

n� 1

2

� 2

, ð9:35Þ

so we have for the principal symbol of R:

�Rðx, �Þ ¼i

2j�j

Z 2�

0

V1ð�ðF sðx, �ÞÞÞ ds, ð9:36Þ

with �:T�Sn! Sn the standard projection. See §29.2 of (13) for general cal-culations of c and �R. The calculation above is done much in the spirit of (18).

10. SPHERICAL HARMONIC EXPANSIONS:

ANOTHER DIVERGENCE EFFECT

In §9 we considered the Laplace operator on spheres (and Zollsurfaces), and also various self-adjoint perturbations of the Laplace opera-tor, of order zero. We obtained fairly precise results on convergence ofeigenfunction expansions of functions with conormal singularities, as longas all the contributions from a cluster were grouped together. Here we willconsider some examples showing that the partial sums within clusters canbehave somewhat more wildly.

We look at operators of the following form: pick b 2 ð0, 1Þ and take

A ¼ ��þn� 1

2

� 2

þibL12

!1=2

, on Sn, ð10:1Þ

where L12 generates a one-parameter group of rotations of period 2� in thex1x2-plane in R

n. This is a zero-order perturbation of

A0 ¼ ��þn� 1

2

� 2 !1=2

,

SpecA0 ¼ kþn� 1

2¼ �k: k ¼ 0, 1, 2, . . .

� �, ð10:2Þ

and the �k-eigenspaces Vk of A0 split into 2kþ 1 eigenspaces Vk‘ for A,with eigenvalues �k‘ ¼ ð�

2k þ b‘Þ

1=2, spread out around the point �k, over

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an interval

Jk ¼ ½��k , �

þk , ��k ¼ ð�

2k � bkÞ

1=2� �k � b=2: ð10:3Þ

The example (10.1) does not quite fit into the class of operators studiedin §9, but the following does:

A1 ¼ ��þn� 1

2

� 2

þ ibL12A�10

!1=2

, ð10:4Þ

with A0 as in (10.2). This has the same eigenspaces as A, but eigenvalues�k‘ ¼ ð�

2k þ b‘=�kÞ

1=2, which are clustered about �k. Clearly the two opera-tors A and A1 give rise to the same eigenfunction expansions. We will phrasethe results in terms of the operator A, given by (10.1).

Note that, for " ¼ "ðkÞ sufficiently small,

��þkþ"ðAÞ ¼ ��kþ"ðA0Þ, ��

k�"ðAÞ ¼ ��k�"ðA0Þ, ð10:5Þ

and the analysis of ��k�"ðA0Þ has been described in §9. We are now interestedin analyzing �RðAÞ f as R crosses the various eigenvalues of A in the intervalJk. Note that, for R 2 Jk, R =2 SpecA,

�RðAÞ f ¼ ��k�"ðA0Þ f þX�k‘<R

Pk‘gk, ð10:6Þ

where Pk‘ is the orthogonal projection of Vk onto Vk‘ and

gk ¼ ��kþ"ðA0Þ f � ��k�"ðA0Þ f : ð10:7Þ

Regarding the sum in (10.6), we have the following observation. For q 2 Sn,let Cq denote the orbit of q under the group of rotations generated by L12.Then Cq is either a circle or the single point fqg. If Cq is a circle, then

Xk‘¼�k

Pk‘gkjCq ð10:8Þ

is precisely the Fourier series of gkjCq . If Cq ¼ fqg, then the only nonzeroterm in (10.8) is Pk0gkðqÞ ¼ gkðqÞ.

Here T1¼ R=ð2�ZÞ and Cq are identified via �$ Eqð�Þ ¼

Expð�L12ÞðqÞ. In particular, suppose

0 � j < k, �k,�j�1 < R1 < �k,�j, �kj < R2 < �k, jþ1: ð10:9Þ


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Then, with gkqð�Þ ¼ gk + Eqð�Þ, we have

h�R2ðAÞ f � �R1

ðAÞ fi+ Eqð�Þ ¼

Xj‘¼�j

ggkqð‘Þ ei‘�¼ Sjgkqð�Þ, ð10:10Þ

where the last identity defines Sjgkqð�Þ.Let us take f ¼ �B, with B ¼ BaðqÞ. We want to study the series

(10.10), with gk given by (10.7). We first note that the behavior of Sjgkqdepends mainly on gkqð�Þ only near � ¼ 0 (and � ¼ � if the antipode q0 iscontained in Cq). This is established as follows.

Lemma 10.1. If f ¼ �B and gk is given by (10.7), then, for any compactX � Snnfq, q0g,

supXjgkj � CX k

�1, supXjrgkj � CX : ð10:11Þ

Proof. We examine fgkg near @B. The behavior elsewhere (away from qand q0) is simpler. By (9.9), for x close to @B, gkðxÞ is given by

eAA0ðxÞF0

�ðkþ 1=2Þ ðxÞ

�� eAA0ðxÞF0

�ðk� 1=2Þ ðxÞ

�, ð10:12Þ

plus lower order terms. This is equal to eAA0ðxÞ times

ðxÞ

Z 1

0

F 00ððkþ sÞ ðxÞÞ ds, F 00ðsÞ ¼sin s

s, ð10:13Þ

and writing ðxÞ ¼ k�1ðk ðxÞÞ, we readily verify the estimates (10.11). &

Corollary 10.2. Let I be any compact subset of T1nf0g if q0=2Cq, or of

T1nf0,�g if q0 2 Cq. Then

supIjgkqj � CI k

�1, supIjg0kqj � CI : ð10:14Þ

Now the behavior of gkqð�Þ near � ¼ 0 is given by the behavior of gkðxÞnear x ¼ q, which in turn is given by (9.12). In exponential coordinatescentered at q, we have

gkðxÞ ¼X�¼�

e�iaðk�1=2Þ kðn�3Þ=2

� e�ia��0

�x, ðkþ 1=2Þx

��

0

�x, ðk� 1=2Þx

�n oþOðkðn�5Þ=2Þ: ð10:15Þ

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Note that the factor e�ia in front of the first ��0 destroys the cancellation onewould have without its presence. This is very important for the analysis thatfollows. It is also significant that the unwritten terms in the remainder in(10.15) have a form similar to the written terms, just with lower powers ofk attached.

Parallel to (8.15), we have

��0 ð0, zÞ ¼ c�n nðjzjÞ, nðrÞ ¼ r

1�n=2 Jn=2�1ðrÞ, ð10:16Þ

for certain nonzero constants c�n . Furthermore, using

��0 ðx, zÞ ��0 ð0, zÞ ¼ x

Z 1

0

ð@x��0 Þðtx, zÞ dt,

we have

j��0 ðx,RxÞ ��0 ð0,RxÞj � CjxjhRxi�ðn�1Þ=2

� CR�� jxj1�� hRxi��ðn�1Þ=2, ð10:17Þ

for 0 � � � 1. Note that, for n � 5,Rh yi��ðn�1Þ=2 dy is finite for all � < 1

(and finite for � ¼ 1 if n � 6). We also recall the classical formulaZ 10

J�ðrÞr�� dr ¼ � ¼ 2��

�ð1=2Þ

�ð�þ 1=2Þ, � >

1

2, ð10:18Þ

so � 6¼ 0 for � ¼ n=2� 1, n � 3.Putting the results above together, and keeping in mind that

Sjgkqð�Þ ¼ gkq �Djð�Þ, ð10:19Þ

where Djð�Þ is the Dirichlet kernel:

Djð�Þ ¼1

2�

sinð j þ 1=2Þ�

sinð1=2Þ�, ð10:20Þ

we have the following:

Proposition 10.3. Let f ¼ �B and gk be given by (10.7). Here B ¼ BaðqÞ. If theantipode q0=2Cq, then, for each fixed j 2 Z

þ, Sjgkq in (10.10) has the followingbehavior as k!1, provided n � 5:

Sjgkqð�Þ ¼ n=2�1 kðn�5Þ=2

X�¼�

c�ne�iaðk�1=2Þ

ðe�ia � 1Þ

!Djð�Þ

þ oðkðn�5Þ=2Þ: ð10:21Þ

There is an analogous result if q0 2 Cq.


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This in turn leads to the following:

Proposition 10.4. With f ¼ �B as in Proposition 10.3, if n � 5 we have thepointwise convergence

limR!1

�RðAÞ f ðxÞ ¼ f ðxÞ, ð10:22Þ

if and only if x =2Cq [ Cq0 . (As usual we define f ¼ 1=2 on @B.) If n ¼ 5 thedivergence on Cq [ Cq0 is bounded; it is unbounded if n � 6.

Proof. The convergence follows from Lemma 10.1 plus the previous materialon �RðA0Þ f . The divergence follows from (10.21). &

This result can be compared with a result of (19), treating B � Tn,

n � 5. There the divergence of the Fourier series of �B at points other thanthe center of B is derived from a number-theoretic analysis.

One can go further, replacing A in (10.1) by

A2 ¼ ��þn� 1

2

� 2

þ ib1L12þ ib2L34A�10 þ ib3L56A

�20

!1=2

, ð10:23Þ

and so on. The eigenspaces of A2 are formed by splitting the eigenspaces ofA, producing eigenvalues in clusters and then in further, tighter clusters.One can have further divergence phenomena for �RðA2Þ f , as R crosses thesevarious clusters, particularly in higher dimensions. We will not pursue thedetails.

ACKNOWLEDGMENT

The author’s research has been supported by the NSF grant. DMS-9877077.

REFERENCES

1. Colzani, L.; Vignati, M. The Gibbs Phenomenon for Multiple FourierIntegrals. J. Approximation Theory 1995, 80(1), 39–79.

2. Pinsky, M.; Taylor, M. Pointwise Fourier Inversion: A Wave EquationApproach. J. Fourier Anal. and Appl. 1997, 3(6), 647–703.

3. Brandolini, L.; Colzani, L. Localization and Convergence of Eigenfunc-tion Expansions. J. Fourier Anal. and Appl. 1999, 5(5), 431–447.

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4. Taylor, M. Pointwise Fourier Inversion on Tori and Other CompactManifolds. J. Fourier Anal. and Appl. 1999, 5(5), 449–463.

5. Pinsky, M. Pointwise Fourier Inversion and Related EigenfunctionExpansions. Comm. Pure Appl. Math. 1994, 47(5), 653–681.

6. Hormander, L. The Spectral Function of an Elliptic Operator. ActaMath. 1968, 121(3–4), 193–218.

7. Brandolini, L.; Colzani, L. Decay of Fourier Transforms andSummability of Eigenfunction Expansions. Preprint, 1999.

8. Taylor, M. Eigenfunction Expansions and the Pinsky Phenomenon onCompact Manifolds. J. Fourier Anal. and Appl. in press.

9. Sogge, C. Concerning the Lp Norm of Spectral Clusters for SecondOrder Elliptic Differential Operators on Compact Manifolds. J. Funct.Anal. 1988, 77(1), 123–134.

10. Taylor, M. Pseudodifferential Operators; Princeton Univ. Press:Princeton, New Jersey, 1981.

11. Taylor, M. Partial Differential Equations, Vols. 1–3; Springer-Verlag:New York, 1996.

12. Hormander, L. Pseudodifferential Operators and HypoellipticEquations. Proc. Symp. Pure Math. 1997, 10(1), 138–183.

13. Hormander, L. The Analysis of Linear Partial Differential Equations,Vol. 4; Springer-Verlag: New York, 1985.

14. Diustermaat, J.; Guillemin, V. The Spectrum of Positive EllipticOperators and Periodic Bicharacteristics. Invent. Math. 1975, 29(1),39–79.

15. Colin de Verdiere, Y. Sur le Spectre des Operateurs Elliptiques aBicharacteristiques Toutes Periodiques. Comment. Math. Helv. 1979,54(3), 508–522.

16. Weinstein, A. Asymptotics of Eigenvalue Clusters for the LaplacianPlus a Potential. Duke Math. J. 1977, 44(4), 883–892.

17. Guillemin, V. Some Spectral Results for the Laplace Operatorwith Potential on the n-Sphere. Adv. in Math. 1978, 27(3), 273–286.

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Received December 2000Revised May 2001


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