Fourier Restriction for Hypersurfaces in Three Dimensions and Newton Polyhedra - Rilegato

Ikromov, Isroil A.; Mller, Detlef

Rilegato

ISBN 10: 0691170541 ISBN 13: 9780691170541

Casa editrice: Princeton Univ Pr, 2016

Riassunto

This is the first book to present a complete characterization of Stein-Tomas type Fourier restriction estimates for large classes of smooth hypersurfaces in three dimensions, including all real-analytic hypersurfaces. The range of Lebesgue spaces for which these estimates are valid is described in terms of Newton polyhedra associated to the given surface.

Isroil Ikromov and Detlef Müller begin with Elias M. Stein's concept of Fourier restriction and some relations between the decay of the Fourier transform of the surface measure and Stein-Tomas type restriction estimates. Varchenko's ideas relating Fourier decay to associated Newton polyhedra are briefly explained, particularly the concept of adapted coordinates and the notion of height. It turns out that these classical tools essentially suffice already to treat the case where there exist linear adapted coordinates, and thus Ikromov and Müller concentrate on the remaining case. Here the notion of r-height is introduced, which proves to be the right new concept. They then describe decomposition techniques and related stopping time algorithms that allow to partition the given surface into various pieces, which can eventually be handled by means of oscillatory integral estimates. Different interpolation techniques are presented and used, from complex to more recent real methods by Bak and Seeger.

Fourier restriction plays an important role in several fields, in particular in real and harmonic analysis, number theory, and PDEs. This book will interest graduate students and researchers working in such fields.

Su questo libro

Informazioni sull?autore

Isroil A. Ikromov is professor of mathematics at Samarkand State University in Uzbekistan. Detlef Müller is professor of mathematics at the University of Kiel in Germany.

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Fourier Restriction for Hypersurfaces in Three Dimensions and Newton Polyhedra

By Isroil A. Ikromov, Detlef Müller

PRINCETON UNIVERSITY PRESS

Copyright © 2016 Princeton University Press
All rights reserved.

ISBN: 978-0-691-17054-1

Contents

Chapter 1 Introduction, 1, 
Chapter 2 Auxiliary Results, 29, 
Chapter 3 Reduction to Restriction Estimates near the Principal Root Jet, 50, 
Chapter 4 Restriction for Surfaces with Linear Height below 2, 57, 
Chapter 5 Improved Estimates by Means of Airy-Type Analysis, 75, 
Chapter 6 The Case When hlin([??]) = 2: Preparatory Results, 105, 
Chapter 7 How to Go beyond the Case hlin ([??])= 5, 131, 
Chapter 8 The Remaining Cases Where m = 2 and B = 3 or B = 4, 181, 
Chapter 9 Proofs of Propositions 1.7 and 1.17, 244, 
Bibliography, 251, 
Index, 257, 

CHAPTER 1

Introduction

Let S be a smooth hypersurface in R3 with Riemannian surface measure ds. We shall assume that S is of finite type, that is, that every tangent plane has finite order of contact with S. Consider the compactly supported measure dµ := ?ds on S, where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. The central problem that we shall investigate in this monograph is the determination of the range of exponents p for which a Fourier restriction estimate

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.1)

holds true.

This problem is a special case of the more general Fourier restriction problem, which asks for the exact range of exponents p and q for which an Lp-Lq Fourier restriction estimate

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.2)

holds true and which can be formulated for much wider classes of subvarieties S in arbitrary dimension n and suitable measures dµ supported on S. In fact, as observed by G. Mockenhaupt [M00] (see also the more recent work by I. Laba and M. Pramanik [LB09]), it makes sense in much wider settings, even for measures dµ supported on "thin" subsets S of Rn, such as Salem subsets of the real line.

The Fourier restriction problem presents one important instance of a wide circle of related problems, such as the boundedness properties of Bochner Riesz means, dimensional properties of Kakeya type sets, smoothing effects of averaging over time intervals for solutions to the wave equation (or more general dispersive equations), or the study of maximal averages along hypersurfaces. The common question underlying all these problems asks for the understanding of the interplay between the Fourier transform and properties of thin sets in Euclidean space, for instance geometric properties of subvarieties. Some of these aspects have been outlined in the survey article [M14], from which parts of this introduction have been taken.

The idea of Fourier restriction goes back to E. M. Stein, and a first instance of this concept is the determination of the sharp range of Lp-Lq Fourier restriction estimates for the circle in the plane through work by C. Fefferman and E. M. Stein [F70] and A. Zygmund [Z74], who obtained the endpoint estimates (see also L. Hörmander [H73] and L. Carleson and P. Sjölin [CS72] for estimates on more general related oscillatory integral operators). For subvarieties of higher dimension, the first fundamental result was obtained (in various steps) for Euclidean spheres Sn-1 by E. M. Stein and P. A. Tomas [To75], who proved that an Lp-L2 Fourier restriction estimate holds true for Snn-1, n = 3, if and only if p' = 2(2/(n - 1) + 1), where p' denotes the exponent conjugate to p, that is, 1/p + 1/p' = 1 (cf. [S93] for the history of this result). A crucial property of Euclidean spheres which is essential for this result is the non-vanishing of the Gaussian curvature on these spheres, and indeed an analogous result holds true for every smooth hypersurface Swith nonvanishing Gaussian curvature (see [Gl81]).

Fourier restriction estimates have turned out to have numerous applications to other fields. For instance, their great importance to the study of dispersive partial differential equations became evident through R. Strichartz' article [Str77], and in the PDE-literature dual versions which invoke also Plancherel's theorem are often called Strichartz estimates.

The question as to which Lp-Lq Fourier restriction estimates hold true for Euclidean spheres is still widely open. It is conjectured that estimate (1.2) holds true for S = Sn-1 if and only if p' > 2n/(n - 1) and p' = q(2/(n - 1) + 1), and there has been a lot of deep work on this and related problems by numerous mathematicians, including J. Bourgain, T. Wolff, A. Moyua, A. Vargas, L. Vega, and T. Tao (see, e.g., [Bou91], [Bou95], [W95], [MVV96], [TVV98], [W00], [TV00], [T03], and [T04] for a few of the relevant articles, but this list is far from being complete). There has been a lot of work also on conic hypersurfaces and some on even more general classes of hypersurfaces with vanishing Gaussian curvature, for instance in Barcelo [Ba85], [Ba86], in Tao, Vargas, and Vega [TVV98], in Wolff [W01], and in Tao and Vargas [TV00], and more recently by A. Vargas and S. Lee [LV10] and S. Buschenhenke [Bu12]. Again, these citations give only a sample of what has been published on this subject.

Recent work by J. Bourgain and L. Guth [BG11], making use also of multilinear estimates from work by J. Bennett, A. Carbery, and T. Tao [BCT06], has led to further important progress. Nevertheless, this and related problems continue to represent one of the major challenges in Euclidean harmonic analysis, bearing various deep connections with other important open problems, such as the Bochner-Riesz conjecture, the Kakeya conjecture and C. Sogge's local smoothing conjecture for solutions to the wave equation. We refer to Stein's book [S93] for more information on and additional references to these topics and their history until 1993, and to more recent related essays by Tao, for instance in [T04].

As explained before, we shall restrict ourselves to the study of the Stein-Tomastype estimates (1.1). For convex hypersurfaces of finite line type, a good understanding of this type of restriction estimates is available, even in arbitrary dimension (we refer to the article [I99] by A. Iosevich, which is based on work by J. Bruna, A. Nagel and S. Wainger [BNW88], providing sharp estimates for the Fourier transform of the surface measure on convex hypersurfaces). However, our emphasis will be to allow for very general classes of hypersurfaces S [subset]R3, not necessarily convex, whose Gaussian curvature may vanish on small, or even large subsets.

Given such a hypersurface S, one may ask in terms of which quantities one should describe the range of ps for which (1.1) holds true. It turns out that an extremely useful concept to answer this question is the notion of Newton polyhedron. The importance of this concept to various problems in analysis and related fields has been revealed by V.I. Arnol'd and his school, in particular through groundbreaking work by A. N. Varchenko [V76] and subsequent work by V. N. Karpushkin [K84] on estimates for oscillatory integrals, and came up again in the seminal article [PS97] by D. H. Phong and E. M. Stein on oscillatory integral operators.

Indeed, there is a close connection between estimates for oscillatory integrals and Lp-L Fourier restriction estimates, which had become evident already through the aforementioned work by Stein and Tomas. The underlying principles had been formalized in a subsequent article by A. Greenleaf [Gl81]. For the case of hypersurfaces, Greenleaf's classical restriction estimate reads as follows:

Theorem 1.1 (Greenleaf).Assume that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Then the restriction estimate (1.1) holds true for every p = 1 such that p' = 2(h + 1).

Observe next that in order to establish the restriction estimate (1.1), we may localize the estimate to a sufficiently small neighborhood of a given point x0 on S. Notice also that if estimate (1.1) holds for the hypersurface S, then it is valid also for every affine-linear image of S, possibly with a different constant if the Jacobian of this map is not one. By applying a suitable Euclidean motion of R3 may and shall therefore assume in the sequel that x0 = (0, 0, 0) and that S is the graph

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

of a smooth function [??] defined on a sufficiently small neighborhood [ohm] of the origin, such that [??] (0, 0) = 0 and [nabla][??] (0, 0) = 0.

Then we may write [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] as an oscillatory integral,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where ? [member of] C80 (O). Since [nabla] [??] (0, 0) = 0, the complete phase in this oscillatory integral will have no critical point on the support of ? unless |[xi]1| + |[xi]2|« | [xi]3|, provided [ohm] is chosen sufficiently small. Integrations by parts then show that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], for every N [member of] N, unless |[xi]1| + |[xi]2| « | [xi]3|.

We may thus focus on the latter case. In this case, by writing ? = [xi]3 and [xi]j = -sj, ?, j = 1, 2, we are reduced to estimating two-dimensional oscillatory integrals of the form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where we may assume without loss of generality that ? » 1 and that s = (s1, s2) [member of] R2 is sufficiently small, provided that ? is supported in a sufficiently small neighborhood of the origin. The complete phase function is thus a small, linear perturbation of the function [??].

If s = 0, then the function I(?; 0) is given by an oscillatory integral of the form [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], and it is well known ([BG69], [At70]) that for any analytic phase function [??] defined on a neighborhood of the origin in Rn satisfying [??] (0) = 0, such an integral admits an asymptotic expansion as ? -> 8 of the form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.3)

provided the support of ? is sufficiently small. Here, the rk form an increasing sequence of rational numbers consisting of a finite number of arithmetic progressions, which depends only on the zero set of [??] and the aj,k(?) are distributions with respect to the cutoff function ?. The proof is based on Hironaka's theorem on the resolution of singularities.

We are interested in the case n = 2. If we denote the leading exponent r0 in (1.3) by r0 = 1/h, then we find that the following estimate holds true:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.4)

where v may be 0, or 1. Assuming that this estimate is stable under sufficiently small analytic perturbations of [??] then we find in particular that I(? s) satisfies the same estimate (1.4) for |s| sufficiently small, so that we obtain the following uniform estimate for [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.5)

provided the support of ? is sufficiently small. Greenleaf's theorem then shows that the Fourier restriction estimate (1.1) holds true for p' = 2(h + 1), if v = 0, and at least for p' > 2(h + 1), if v = 1, where 1/h denotes the decay rate of the oscillatory integral I(?; 0), hence ultimately of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Moreover, for instance for hypersurfaces with nonvanishing Gaussian curvature, this yields the sharp restriction result mentioned before.

However, as we shall see, for large classes of hypersurfaces, the relation between the decay rate of the Fourier transform of dµ and the range of p's for which (1.1) holds true will not be so close anymore.

Nevertheless, uniform decay estimates of the form (1.5) will still play an important role.

The first major question that arises is thus the following one: given a smooth phase function [??] how can one determine the sharp decay rate 1/h and the exponent v in the estimate (1.4) for the oscillatory integral [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] This question has been answered by Varchenko for analytic [??] in [V76], where he identified h as the so-called height of the Newton polyhedron associated to [??] in "adapted" coordinates and also gave a corresponding interpretation of the exponent v. Subsequently, Karpushkin [K84] showed that the estimates given by Varchenko are stable under small analytic perturbations of the phase function [??], which in particular leads to uniform estimates of the form (1.5). More recently, in [IM11b], we proved, by a quite different method, that Karpushkin's result remains valid even for smooth, finite-type functions [??], at least for linear perturbations, which is sufficient in order to establish uniform estimates of the form (1.5).

In order to present these results in more detail, let us review some basic notations and results concerning Newton polyhedra (see [V76], [IM11a]).

1.1 NEWTON POLYHEDRA ASSOCIATED WITH [??], ADAPTED COORDINATES, AND UNIFORM ESTIMATES FOR OSCILLATORY INTEGRALS WITH PHASE [??]

We shall build on the results and technics developed in [IM11a] and [IKM10], which will be our main sources, also for references to earlier and related work. Let us first recall some basic notions from [IM11a], which essentially go back to Arnol'd (cf. [Arn73], [AGV88]) and his school, most notably Varchenko [V76].

If [??] is given as before, consider the associated Taylor series

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

of [??] centered at the origin. The set

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

will be called the Taylor support of [??] at (0, 0). We shall always assume that the function [??] is of finite type at every point, that is, that the associated graph S of [??] is of finite type. Since we are also assuming that [??](0, 0) = 0 and [nabla][??](0, 0) = 0, the finite-type assumption at the origin just means that

T (phi) ? [??].

The Newton polyhedron N([??]) of [??] at the origin is defined to be the convex hull of the union of all the quadrants (a1, a2) + R2+ in R2, with (a1, a2) [member of] T([??]). The associated Newton diagram Nd([??]) of [??] in the sense of Varchenko [V76] is the union of all compact faces of the Newton polyhedron; here, by a face, we shall mean an edge or a vertex.

We shall use coordinates (t1, t2) for points in the plane containing the Newton polyhedron, in order to distinguish this plane from the (x1, x2)-plane.

The Newton distance in the sense of Varchenko, or shorter distance, d = d ([??]) between the Newton polyhedron and the origin is given by the coordinate d of the point (d, d) at which the bisectrix t1 = t2 intersects the boundary of the Newton polyhedron. (See Figure 1.1.)

The principal face p ([??]) of the Newton polyhedron of [??] is the face of minimal dimension containing the point (d, d). Deviating from the notation in [V76], we shall call the series

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

the principal part of [??] In case that p ([??]) is compact, [??]pr is a mixed homogeneous polynomial; otherwise, we shall consider [??]pr as a formal power series.

Note that the distance between the Newton polyhedron and the origin depends on the chosen local coordinate system in which ? is expressed. By a local coordinate system (at the origin) we shall mean a smooth coordinate system defined near the origin which preserves 0. The height of the smooth function [??] is defined by

h(phi):= sup {dy},

where the supremum is taken over all local coordinate systems y = (y1, y2) at the origin and where dy is the distance between the Newton polyhedron and the origin in the coordinates y.

A given coordinate system x is said to be adapted to [??] if h([??]) = dx. In [IM11a] we proved that one can always find an adapted local coordinate system in two dimensions, thus generalizing the fundamental work by Varchenko [V76] who worked in the setting of real-analytic functions [??] (see also [PSS99]).

Notice that if the principal face of the Newton polyhedron N([??]) is a compact edge, then it lies on a unique principal line

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

with k1, k2 > 0. By permuting the coordinates x1 and x2, if necessary, we shall always assume that k1 = k2. The weight k = (k1, k2) will be called the principal weight associated with [??]. It induces dilations [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], and we find that on R2, so that the principal part [??]pr of [??] is k-homogeneous of degree one with respect to these dilations, that is, [??]pr (dr(x1, x2)) = r [??]pr (x1, x2) for every r > 0, and we find that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

It can then easily be shown (cf. Proposition 2.2 in [IM11a]) that [??]pr can be factored as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.6)

with M = 1, distinct nontrivial "roots" ?i [member of] C \ {0} of multiplicities nli [member of] N \ {0}, and trivial roots of multiplicities V1, v2 [member of] N at the coordinate axes. Here, p and q are positive integers without common divisor, and k2/k1 = p/q.

(Continues...)

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Dettagli dell?edizione

• Editore: Princeton Univ Pr
• Data di pubblicazione: 2016
• Lingua: Inglese
• ISBN 10: 0691170541
• ISBN 13: 9780691170541
• Rilegatura: Copertina rigida
• Numero di pagine: 258
• Contatto del produttore: non disponibile
• Persona responsabile: non disponibile