# Advances in Analysis: The Legacy of Elias M. Stein

Charles Fefferman
Alexandru D. Ionescu
D. H. Phong
Stephen Wainger
Pages: 560
https://www.jstor.org/stable/j.ctt4cg99n

1. Front Matter
(pp. i-vi)
(pp. vii-viii)
3. Preface
(pp. ix-xii)
C. Fefferman, A. D. Ionescu, D. H. Phong and S. Wainger
4. Chapter One Selected Theorems by Eli Stein
(pp. 1-34)
Charles Fefferman

The purpose of this chapter is to give the general reader some idea of the scope and originality of Eli Stein’s contributions to analysis*. His work deals with representation theory, classical Fourier analysis, and partial differential equations. He was the first to appreciate the interplay among these subjects, and to preceive the fundamental insights in each field arising from that interplay. No one else really understands all three fields; therefore, no one else could have done the work I am about to describe. However, deep understanding of three fields of mathematics is by no means sufficient to lead to Stein’s...

5. Chapter Two Eli’s Impact: A Case Study
(pp. 35-46)
Charles Fefferman

In [7], we discussed some of Eli Stein’s contributions to analysis. Here, by picking out a single striking example, we illustrate the continuing powerful influence of Eli’s ideas.

We start by recalling Eli’s ideas on Littlewood-Paley theory, as well as several major developments in pure and applied mathematics, to which those ideas gave rise. We then discuss the remarkable recent work of Gressman and Strain [9–11] on the Boltzmann equation, and explain in particular its connection to Eli’s work.

Before Eli, Littlewood-Paley theory was one of the deepest parts of the classical study of Fourier series in one variable....

6. Chapter Three On Oscillatory Integral Operators in Higher Dimensions
(pp. 47-62)
Jean Bourgain

What follows is a discussion of progress towards problems originating from E. Stein’s seminal paper [St1]. It is mainly based on a conference talk on joint work of L. Guth and the author [BG]. By no means does it aim at a complete overview of all the recent advances in this field. Thus various significant contributions may be only briefly mentioned, or not at all.

It is by now well-known that the mapping properties of Fourier restriction operators to hypersurfaces in${{\mathbb{R}}^{n}}$and their variable coefficient generalizations are intimately related to questions of a combinatorial nature, such as the Kakeya...

7. Chapter Four Hölder Regularity for Generalized Master Equations with Rough Kernels
(pp. 63-83)
Luis Caffarelli and Luis Silvestre

We study evolution problems that are related to continuous time random walks (CTRW), having a discontinuous path for which both the jumps and the time elapsed in between them are random. These processes are governed by a generalizedmaster equationwhich is nonlocal both in space and time.

We consider kernelsK(t, x, s, y) in${{\mathbb{R}}^{n}}\times {{\mathbb{R}}^{n}}\times (0,\infty )\times (0,\infty )$. From kernels, we define an integral operator which is nonlocal both in space and time$Lu(t,x)=\int_{{{\mathbb{R}}^{n}}}{{}}\int_{0}^{\infty }{(u(t,x)-u(t-s,x+y))}K(t,x,s,y)\ ds\ dy.\caption {(1.1)}$

We will study equations that may or may not include a time derivation. The first model we are interested in, is the equation which is purely...

8. Chapter Five Extremizers of a Radon Transform Inequality
(pp. 84-107)
Michael Christ

Let$d\ge2$. Denote by${{\mathfrak{G}}_{d}}$the Grassmann manifold of all affine hyperplanes in${{\mathbb{R}}^{d}}$. There is a natural two-to-one mapping, with the exception of a null set, from$\mathbb{R}\times {{S}^{d-1}}$to${{\mathfrak{G}}_{d}}$given by$(r,\theta )\mapsto \pi =\left\{ x\in {{\mathbb{R}}^{d}}:x\cdot \theta =r \right\}.\caption {(1.1)}$

We equip${{\mathfrak{G}}_{d}}$with the measurewhich pulls back todr dθunder this two-to-one identification.

The Radon transform$\cal R$maps functions defined on${{\mathbb{R}}^{d}}$to functions defined on${{\mathfrak{G}}_{d}}$, by${\cal R}f(r,\theta )=\int_{x\cdot \theta =r}{f(x)d{{\sigma }_{r,\theta }}(x)}\caption {(1.2)}$where${{\sigma }_{r,\theta }}$is surface measure on the affine hyperplane$\{x:x\cdot \theta =r\}$. Since${\cal R}f(-r,-\theta )\equiv {\cal R}f(r,\theta )$,$\parallel {\cal R}f\parallel _{{{L}^{q}}({{\mathfrak G}_{d}},\mu )}^{q}=\frac{1}{2}\int_{\mathbb{R}}{{}}\int_{{{S}^{d-1}}}{|{\cal R}f(r,\theta ){{|}^{q}}\ dr\ d\theta .}\caption {(1.3)}$

The measureµon${{\mathfrak{G}}_{d}}$is natural in this context. It has certain invariance properties...

9. Chapter Six Should We Solve Plateau’s Problem Again?
(pp. 108-145)
Guy David

The main goal of this chapter is to give a partial account of the situation of Plateau’s problem on the existence and regularity of soap films with a given boundary. We intend to convince the reader that there are many reasonable ways to state a Plateau problem, most of which give interesting questions that are still wide open. This is even more true when we want our models to stay close to Plateau’s original motivation, which was to describe physical phenomena such as soap films.

Plateau problems led to lots of beautiful results; we shall start the chapter with a...

10. Chapter Seven Averages along Polynomial Sequences in Discrete Nilpotent Lie Groups: Singular Radon Transforms
(pp. 146-188)
Alexandru D. Ionescu, Akos Magyar and Stephen Wainger

A class of interesting problems arises in studying averages of functions along polynomial sequences in discrete nilpotent groups. More precisely, assume$\mathbb{G}$is a discrete nilpotent group of step$d\ge 1$and$A:\mathbb{Z}\to \mathbb{G}$is a polynomial sequence (see Definition 1.1 below), and consider the following problems:¹

Problem 1.(L²boundedness of maximal Radon transforms) Assume$f:\mathbb{G}\to \mathbb{C}$is a function and let${\mathcal M}f(g)=\underset{N\ge 0}{\mathop{\sup }}\,\frac{1}{2N+1}\sum\limits_{|n|\le N}^{{}}{|f({{A}^{-1}}(n)\cdot g)|,\quad g\in \mathbb{G}.}$

Then$\parallel\mathcal Mf{{\parallel }_{{{L}^{2}}(\mathbb{G})}}\lesssim \parallel f{{\parallel }_{{{L}^{2}}(\mathbb{G})}}.$

Problem 2.(L²pointwise ergodic theorems) Assume$\mathbb{G}$acts by measure-preserving transformations on a probability space X,$f\in {{L}^{2}}(X)$, and let${{A}_{N}}f(x)=\frac{1}{2N+1}\sum\limits_{|n|\le N}^{{}}{f({{A}^{-1}}(n)\cdot x),\quad x\in X.}$

Then the sequence ANf converges almost everywhere in X as$N\to \infty$.

Problem 3.(L²boundedness of singular...

11. Chapter Eight Internal DLA for Cylinders
(pp. 189-214)
David Jerison, Lionel Levine and Scott Sheffield

Internal Diffusion-Limited Aggregation (internal DLA) is a random lattice growth model. Consider the two-dimensional lattice, Z × Z. In the case of a single source at the origin, the random occupied setA(T) ofTlattice sites is defined inductively as follows. LetA(1) be the singleton set containing the origin. GivenA(T− 1), start a random walk in Z × Z at the origin. Then$A(T):=\{n\}\bigcup A(T-1)$where$n\in \mathbf Z\times \mathbf Z$is the first site reached by the random walk that is not inA(T− 1).

In this chapter, we will discuss the continuum limit of internal DLA, which is...

12. Chapter Nine The Energy Critical Wave Equation in 3D
(pp. 215-223)
Carlos Kenig

In this chapter we will discuss the energy critical nonlinear wave equation in 3 space dimensions.

We start by a review of the linear wave equation$(LW)\left\{ \begin{array}{ll} \partial _{t}^{2}w-\Delta w=h \\ w{{|}_{t=0}}={{w}_{0}} \\ {{\partial }_{t}}w{{|}_{t=0}}={{w}_{1}}. \\ \end{array} \right.\caption {(1)}$

We write the solution:$w(t)=S(t)({{w}_{0}},{{w}_{1}})+D(t)(h),$WhereS(t) denotes the solution of the homogeneous problem (h= 0) andD(t) the solution of the inhomogeneous one ((w0, w1) = (0, 0)).

One of the main properties of the linear wave equation is the finite speed propagation:

If supp$({{w}_{0}},{{w}_{1}})\bigcap \overline{B({{x}_{0}},a)}=\rlap{/}0$, supp$h\cap ({{\cup }_{0\le t\le a}}B({{x}_{0}},a-t)\times \{t\})=\rlap{/}0$, then$w\equiv0$on${{\bigcup }_{0\le t\le a}}B({{x}_{0}},a-t)\times \{t\}$.

An important estimate (Strichartz estimate) is (see [16]):$\parallel w{{\parallel }_{{{L}^{8}}_{x,t}}}\le C\left\{ \parallel ({{w}_{0}},{{w}_{1}}){{\parallel }_{{{{\dot{H}}}^{1}}\times {{L}^{2}}}}+\parallel {{D}^{\frac{1}{2}}}h{{\parallel }_{{{L}^{\frac{4}{3}}}_{x,t}}} \right\}.$

The energy critical nonlinear wave equation in...

13. Chapter Ten On the Bounded L² Curvature Conjecture
(pp. 224-244)
Sergiu Klainerman

I am delighted to dedicate this review of my recent work in collaboration with Igor Rodnianski and Jeremie Szeftel to my old friend and source of inspiration E. Stein. Anybody slightly familiar with my work over the years would easily notice the change in direction and methodology which occurred almost immediately after I joined the Princeton mathematics department. Indeed my work at NYU was all based on traditional PDE methods, that is, energy methods, Sobolev estimates, commuting vectorfields, and simple interpolations. A far-reaching extension of these techniques has culminated with a proof of the global stability of the Minkowski space...

14. Chapter Eleven On Div-Curl for Higher Order
(pp. 245-272)
Loredana Lanzani and Andrew S. Raich

In 2004 Stein and the first named author discovered a connection [LS] between the celebrated Gagliardo-Nirenberg inequality [G]-[N] for functions$\parallel f{{\parallel }_{{{L}^{r}}({{\mathbb{R}}^{n}})}}\le C\parallel \nabla f{{\parallel }_{{{L}^{1}}({{\mathbb{R}}^{n}})}},\quad r=n/(n-1)\caption {(1)}$and a recent estimate of Bourgain and Brezis [BB2] for divergence-free vector fields as proved by Van Schaftingen [VS1]$\parallel Z{{\parallel }_{{{L}^{r}}({{\mathbb{R}}^{n}})}}\le C\parallel {\mathrm {Curl}}\ Z{{\parallel }_{{{L}^{1}}({{\mathbb{R}}^{n}})}},\quad r=n/(n-1),\quad div\ Z=0.\caption {(2)}$

Such connection is provided by the exterior derivative operator acting on differential forms on${{\mathbb{R}}^{n}}$with (say) smooth and compactly supported coefficients$d:{{\Lambda }_{q}}({{\mathbb{R}}^{n}})\to {{\Lambda }_{q+1}}({{\mathbb{R}}^{n}}),\quad 0\le q\le n.$

It was proved in [LS] that the inequality$\parallel u{{\parallel }_{{{L}^{r}}({{\mathbb{R}}^{n}})}}\le C(\parallel du{{\parallel }_{{{L}^{1}}({{\mathbb{R}}^{n}})}}+\parallel {{d}^{*}}u{{\parallel }_{{{L}^{1}}({{\mathbb{R}}^{n}})}}),\quad r=n/(n-1)\caption {(4)}$holds for any formuof degreeqother thanq= 1 (unlessd*u= 0) andq=n− 1 (unlessdu=...

15. Chapter Twelve Square Functions and Maximal Operators Associated with Radial Fourier Multipliers
(pp. 273-302)
Sanghyuk Lee, Keith M. Rogers and Andreas Seeger

We begin with an overview on square functions for spherical and Bochner–Riesz means which were introduced by Eli Stein, and discuss their implications for radial multipliers and associated maximal functions. We then prove new endpoint estimates for these square functions, for the maximal Bochner–Riesz operator, and for more general classes of radial Fourier multipliers.

Square functions. The classical Littlewood–Paley functions on${{\mathbb{R}}^{d}}$are defined by$g[f]={{\left( {{\int_{0}^{\infty }{\left| \frac{\partial }{\partial t}{{P}_{t}}f \right|}}^{2}}t\ dt \right)}^{1/2}}$where (Pt)t>0is an approximation of the identity defined by the dilates of a “nice” kernel (for example, (Pt) may be the Poisson or the heat semigroup). Their significance in harmonic...

16. Chapter Thirteen Problems of Harmonic Analysis Related to Finite-Type Hypersurfaces in ${{\mathbb{R}}^{3}}$, and Newton Polyhedra
(pp. 303-345)
Detlef Müller

LetSbe a smooth, finite type hypersurface in${{\mathbb{R}}^{3}}$with Riemannian surface measure, and consider the compactly supported measure$d\mu \ :=\ \rho d\sigma$onS, where$0\le \rho \in C_{0}^{\infty }(S)$.

The problems on which I shall essentially focus are the following ones:.

A. Find, if possible, optimal uniform decay estimates for the Fourier transform of the surface carried measure.

B. If we denote byAtthe averaging operator${{A}_{t}}f(x):={{\int }_{S}}f(x-ty)d\mu (y)$, determine for which exponentspthe associated maximal operator${\cal M}f(x):=\underset{t>0}{\mathop{\sup }}\,|{{A}_{t}}f(x)|$is bounded on${{L}^{p}}({{\mathbb{R}}^{3}})$. For instance, ifSis a Euclidean sphere centered at the origin, then${\cal M}$is the spherical...

17. Chapter Fourteen Multi-Linear Multipliers Associated to Simplexes of Arbitrary Length
(pp. 346-401)
Camil Muscalu, Terence Tao and Christoph Thiele

The present chapter is a natural continuation of our previous work in [12] and [13]. In those articles we studied theLpboundedness properties of a tri-linear operatorT3defined by the formula${{T}_{3}}({{f}_{1}},{{f}_{2}},{{f}_{3}})(x)=\int_{{{\xi }_{1}}<{{\xi }_{2}}<{{\xi }_{3}}}{{{{\hat{f}}}_{1}}({{\xi }_{1}}){{{\hat{f}}}_{2}}({{\xi }_{2}}){{{\hat{f}}}_{3}}({{\xi }_{3}}){{e}^{2\pi ix({{\xi }_{1}}+{{\xi }_{2}}+{{\xi }_{3}})}}}d{{\xi }_{1}}d{{\xi }_{2}}d{{\xi }_{3}}\caption {(1)}$forf1,f2,f3Schwartz functions on the real line. A particular case of our main theorem there is the following¹

Theorem 1.1.T3extends to a bounded tri-linear operator from L² ×L² ×L² intoL2/3.

Related toT3is the well-known bi-linear Hilbert transformT2essentially defined by${{T}_{2}}({{f}_{1}},{{f}_{2}})(x)=\int_{{{\xi }_{1}}<{{\xi }_{2}}}{{{{\hat{f}}}_{1}}({{\xi }_{1}}){{{\hat{f}}}_{2}}({{\xi }_{2}}){{e}^{2\pi ix({{\xi }_{1}}+{{\xi }_{2}})}}}d{{\xi }_{1}}d{{\xi }_{2}}.\caption {(2)}$

From the work of Lacey and Thiele [7], [8], we know in particular the following...

18. Chapter Fifteen Diagonal Estimates for Bergman Kernels in Monomial-Type Domains
(pp. 402-418)
Alexander Nagel and Malabika Pramanik

We begin by briefly recalling the definition and some of the elementary properties of the Bergman projection and Bergman kernel. Let$\Omega \subset {{\mathbb{C}}^{n}}$be a domain, that is, an open connected set. The space${{A}^{2}}(\Omega )=\left\{ f\in {{L}^{2}}(\Omega ):f\ \text{is}\ \text{holomorphic}\ \text{on}\ \Omega \right\}$is then a closed subspace of the Hilbert space${{L}^{2}}(\Omega)$where the inner product is given by$\left\langle f,g \right\rangle ={{\int }_{\Omega }}f(w)\overline{g(w)}dw$. TheBergman projectionfor Ω is the orthogonal projection$B={{B}_{\Omega }}:{{L}^{2}}(\Omega )\to {{A}^{2}}(\Omega ).$

TheBergman kernelis the Schwartz kernel of this operatorB, and is given by a function$K={{K}_{\Omega }}:\Omega \times \Omega \to \mathbb{C}$. In fact, if$z\in\Omega$, the linear functional${{A}^{2}}(\Omega )\ni f\to f(z)$is bounded, and by the Riesz representation theorem there...

19. Chapter Sixteen On the Singularities of the Pluricomplex Green’s Function
(pp. 419-435)
D. H. Phong and Jacob Sturm

The Green’s function plays a central role in the study of functions of one complex variable or of two real variables. But while its natural generalization to functions of more real variables is as the fundamental solution of the Laplacian, its natural generalization to functions of several complex variables is rather as a fundamental solution of the complex Monge-Ampère equation. For our purposes, we shall consider the following broad definition. LetMbe ann-dimensional compact Kähler manifold with smooth boundary ∂M, and letωbe a smooth non-negative closed (1,1)-form onM. LetPSH(M,ω) the space of plurisubharmonic...

20. Chapter Seventeen Smoothness of Spectral Multipliers and Convolution Kernels in Nilpotent Gelfand Pairs
(pp. 436-446)
Fulvio Ricci

Most of the problems of harmonic analysis on the Heisenberg groupHnhave to do, in one way or another, with the left-invariant sublaplacianL, introduced by G. Folland and E. M. Stein in 1974 [11]. OftenLappears in combination with the central derivativeT= ∂t, like in the Folland-Stein operatorsL+iαT, which include the Kohn Laplacian on the boundary on the Siegel domain, or in the Laplace-Beltrami operatorsL+aT2,a> 0, of the left-invariant Riemannian metrics onHnwhich are also invariant under the action of the unitary group Un.

A widely studied...

21. Chapter Eighteen On Eigenfunction Restriction Estimates and L⁴-Bounds for Compact Surfaces with Nonpositive Curvature
(pp. 447-462)
Christopher D. Sogge and Steve Zelditch

Let (M,g) be a compact two-dimensional Riemannian manifold without boundary. We shall assume throughout that the curvature of (M,g) is everywhere nonpositive. If Δgis the Laplace-Beltrami operator associated with the metricg, then we are concerned with certain size estimates for the eigenfunctions$-{{\Delta }_{g}}{{e}_{\lambda }}(x)={{\lambda }^{2}}{{e}_{\lambda }}(x),\quad x\in M.$

Thus we are normalizing things so thateλis an eigenfunction of the first order operator$\sqrt{-{{\Delta }_{g}}}$with eigenvalue λ. Ifeλis also normalized to haveL²-norm one, we are interested in various size estimates for theeλwhich are related to how concentrated they may be along geodesics. If Π denotes...

22. List of Contributors
(pp. 463-464)
23. Index
(pp. 465-466)