Jump to ContentJump to Main Navigation

## Charles Fefferman, Alexandru D. Ionescu, D.H. Phong, and Stephen Wainger

Print publication date: 2014

Print ISBN-13: 9780691159416

Published to Princeton Scholarship Online: October 2017

DOI: 10.23943/princeton/9780691159416.001.0001

Show Summary Details
Page of

PRINTED FROM PRINCETON SCHOLARSHIP ONLINE (www.princeton.universitypressscholarship.com). (c) Copyright Princeton University Press, 2017. All Rights Reserved. Under the terms of the licence agreement, an individual user may print out a PDF of a single chapter of a monograph in HSO for personal use (for details see http://www.universitypressscholarship.com/page/privacy-policy).date: 17 March 2018

# Selected Theorems by Eli Stein

Chapter:
(p.1) Chapter One Selected Theorems by Eli Stein
Source:
Advances in Analysis
Publisher:
Princeton University Press
DOI:10.23943/princeton/9780691159416.003.0001

# Abstract and Keywords

This chapter is an overview 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 perceive the fundamental insights in each field arising from that interplay. No one else really understands all three fields, hence the need for this chapter. However, deep understanding of three fields of mathematics is by no means sufficient to lead to Stein's main ideas. Rather, at crucial points, Stein has shown extraordinary originality, without which no amount of work or knowledge could have succeeded.

# Introduction

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 main ideas. Rather, at crucial points, Stein has shown extraordinary originality, without which no amount of work or knowledge could have succeeded. Also, large parts of Stein’s work (e.g., the fundamental papers [26, 38, 41, 59] on complex analysis in tube domains) don’t fit any simple one-paragraph description such as the one above.

It follows that no single mathematician is competent to present an adequate survey of Stein’s work. As I attempt the task, I am keenly aware that many of Stein’s papers are incomprehensible to me, while others were of critical importance to my own work. Inevitably, therefore, my survey is biased, as any reader will see. Fortunately, S. Gindikin provided me with a layman’s explanation of Stein’s contributions to representation theory, thus keeping the bias (I hope) within reason. I am grateful to Gindikin for his help, and also to Y. Sagher for a valuable suggestion.

For purposes of this chapter, representation theory deals with the construction and classification of the irreducible unitary representations of a semisimple Lie group. Classical Fourier analysis starts with the Lp-boundedness of two fundamental operators, the maximal function.

$Display mathematics$

and the Hilbert transform

$Display mathematics$

Finally, we shall be concerned with those problems in partial differential equations that come from several complex variables.

# (p.2) Complex Interpolation

Let us begin with Stein’s work on interpolation of operators. As background, we state and prove a clasical result, namely the

M. Riesz Convexity Theorem. Suppose X, Y are measure spaces, and suppose T is an operator that carries functions on X to functions on Y. Assume T is bounded from$L p 0 (X)to L r 0 (Y),$ and from $L p 1 (X)to L r 1 (Y)(Here, p 0 , p 1 , r 0 , r 1 ∈[1,∞].)$ Then T is bounded from $L p (X)​to L r (Y)for 1 p = t p 1 + (1−t) p 0 , 1 r = t r 1 + (1−t) r 0 ,0≤t≤1.$

The Riesz Convexity Theorem says that the points $( 1 p , 1 r )$ forwhich T is bounded from Lp to Lr from a convex region in the plane. A standard application is the Hausdorff-Young inequality: We take T to be the Fourier transform on Rn, and note that T is obviously bounded from L1 to L, and from L2 to L2. Therefore, T is bounded from Lp to the dual class $L p ′ for1≤p≤2.$

The idea of the proof of the Riesz Convexity Theorem is to estimate $∫ Y (Tf)⋅g$for$f∈ L p andg∈ L r ′$ Say $f=F e iϕ andg=G e iψ$ with F,G ≥ 0 and $ϕ,Ψ$ real. Then we can define analytic families of functions fz, gz by setting $f z = F az+b e iϕ ,$ $g z = G cz+d e iψ ,$ for real a, b, c, d to be picked in a moment.

Define

(1)
$Display mathematics$

Evidently, is an analytic function of z.

For the correct choice of a, b, c, d we have

(2)
$Display mathematics$

(3)
$Display mathematics$

(4)
$Display mathematics$

From (2) we see that $∥ f z ∥ L p 0 ,∥ g z ∥ L 0 r ′ ≤C$for Re z = 0. So the definition (1) and Lr the assumption $T: L p 0 → L r 0$ show that

(5)
$Display mathematics$

Similarly, (3) and the assumption $T: L p 1 → L r 1$ imply

(6)
$Display mathematics$

Since is analytic, (5) and (6) imply by the maximum principle for a strip. In particular, $|Φ(t)|≤ C ′$ In view of (4), this means that $| ∫ Y (Tf)g|≤ C ′$ with C determined by $∥f ∥ L p and∥g ∥ L r ′$

Thus, T is bounded from Lp to Lr , and the proof of the Riesz Convexity Theorem is complete.

This proof had been well-known for over a decade, when Stein discovered an amazingly simple way to extend its usefulness by an order of magnitude. He realized that an ingenious argument by Hirschman [H] on certain multiplier operatorson (p.3) Lp(Rn) could be viewed as a Riesz Convexity Theorem for analytic families of operators. Here is the result.

Stein Interpolation Theorem. Assume Tz is an operator depending analytically on z in the strip 0 ≤ Re z ≤ 1. Suppose Tz is bounded from Lp0 to Lr0 when Re z = 0, and from Lp1toLr1 when Re z = 1. Then Tt is bounded from Lp to Lr , 1= t + (1−t) 1= t + (1−t) ≤ ≤ where , $1 p = t p 1 + (1−t) p o , 1 r = t r 1 + (1−t) r 0 and0≤t≤1$

Remarkably, the proof of the theorem comes from that of the Riesz Convexity Theorem by adding a single letter of the alphabet. Instead of taking $Φ(z)= ∫ y (T f z ) g z$as in (1), we set (z) $Φ(z)= ∫ Y ( T z f z ) g z$ The proof of the Riesz Convexity Theorem then applies with no further changes.

Stein’s Interpolation Theorem is an essential tool that permeates modern Fourier analysis. Let me just give a single application here, to illustrate what it can do. The example concerns Cesaro summability of multiple Fourier integrals.

We define an operator Tα‎R on functions on Rn by setting

$Display mathematics$

Then

(7)
$Display mathematics$

This follows immediately from the Stein Interpolation Theorem. We let α‎ play the role of the complex parameter z, and we interpolate between the elementary cases p = 1 and p = 2. Inequality (7), due to Stein, was the first non-trivial progress on spherical summation of multiple Fourier series.

# Representation Theory 1

Our next topic is the Kunze-Stein phenomenon, which links the Stein Interpolation Theorem to representations of Lie groups. For simplicity we restrict attention to G = SL(2,R), and begin by reviewing elementary Fourier analysis on G. The irreducible unitary representations of G are as follows:

The principal series, parametrized by a sign σ‎ = ±1 and a real parameter t ; The discrete series, parametrized by a sign σ‎ = ±1 and an integer k ≥ 0; and The complementary series, parametrized by a real number t ∈ (0, 1).

We don’t need the full description of these representations here.

The irreducible representations of G give rise to a Fourier transform. If f is a function on G, and U is an irreducible unitary representation of G, then we define

$Display mathematics$

(p.4) where dg denotes Haar measure on the group. Thus, f ˆ is an operator-valued function defined on the set of irreducible unitary representations of G. As in the Euclidean case, we can analyze convolutions in terms of the Fourier transform. In fact,

(8)
$Display mathematics$

as operators. Moreover, there is a Plancherel formula for G, which asserts that

$Display mathematics$

for a measure μ‎ (the Plancherel measure). The Plancherel measure for G is known, but we don’t need it here. However, we note that the complementary series has measure zero for the Plancherel measure.

These are, of course, the analogues of familiar results in the elementary Fourier analysis of Rn. Kunze and Stein discovered a fundamental new phenomenon in Fourier analysis on G that has no analogue on Rn. Their result is as follows.

Theorem (Kunze-Stein Phenomenon). There exists a uniformly bounded representation Uσ‎,τ‎ of G, parametrized by a sign σ‎ = ±1 and a complex number τ‎ in a strip , with the following properties.

1. (A) The Uσ‎,τ‎ all act on the same Hilbert space H.

2. (B) For fixed σ‎ = ±1, g ∈ G, and ξ‎ , η‎ ∈ H, the matrix element $〈 ( U σ,τ ) g ξ,η〉$ is an analytic function of τ‎ ∈ .

3. (C) The Uσ‎,τ‎ for Re τ‎ = are equivalent to the representations of the principal series.

4. (D) The U+1,τ‎ for suitable τ‎ are equivalent to the representations of the complementary series.

(See [14] for the precise statement and proof, as well as Ehrenpreis-Mautner [EM] for related results.)

The Kunze-Stein Theorem suggests that analysis on G resembles a fictional version of classical Fourier analysis in which the basic exponential $ξ↦exp(iξ⋅x)$ is a bounded analytic function on strip $|Imξ|≤C$ uniformly for all x.

As an immediate consequence of the Kunze-Stein Theorem, we can give an analytic continuation of the Fourier transform for G. In fact, we set $f ^ (σ,τ)= ∫ G f(g)( U σ,τ ) g dgforσ=±1,τ∈Ω.$

Thus, $f∈ L 1 (G)$ implies $f ^ (σ,⋅)$ analytic and bounded on . So we have continued analytically the restriction of f ˆ to the principal series. It is as if the Fourier transform of an L1 function on (−∞,∞) were automatically analytic in a strip. If $f∈ L 2 (G)then f ^ (σ,τ)$is still defined on the line ${Reτ= 1 2 }$ by virtue of the Plancherel formula and part (C) of the Kunze-Stein Theorem. Interpolating between L1(G) and L2(G) using the Stein Interpolation Theorem, we see that $f∈ L p (G)(1≤p<2)$ implies $f ^ (σ,⋅)$ analytic and satisfying an Lp -inequality on a strip p. As p increases from 1 to 2, the strip p shrinks from to the line ${Reτ= 1 2 }$ Thus we obtain the following results.

Corollary 1. $f∈ L p (G)(1≤p<2)$ then f ˆis bounded almost everywhere with respect to the Plancherel measure.

(p.5)

Corollary 2. For 1 ≤ p < 2 we have the convolution inequality $∥f∗g ∥ L 2 (G) ≤ C p ∥f ∥ L p (G) ∥g ∥ L 2 (G)$

To check Corollary 1, we look separately at the principal series, the discrete series, and the complementary series. For the principal series, we use the Lp - inequality established above for the analytic function $τ↦ f ^ (σ,τ)$ on the strip p. Since an Lp-function analytic on a strip p is clearly bounded on an interior line ${Reτ= 1 2 }$ it follows at once that f ˆ is bounded on the principal series. Regarding the discrete series Uσ‎,k we note that

(9)
$Display mathematics$

for suitable weights μ‎σ‎,k and for 1 ≤ p ≤ 2. The weights μ‎σ‎,k amount to the Plancherel measure on the discrete series, and (9) is proved by a trivial interpolation, just like the standard Hausdorff-Young inequality. The boundedness of the $∥ f ^ ( U σ,k )∥$is immediate from (9). Thus the Fourier transform f ˆ is bounded on both the principal series and the discrete series, for $f∈ L p (G)(1≤p<2)$ The complementary series has measure zero with respect to the Plancherel measure, so the proof of Corollary 1 is complete. Corollary 2 follows trivially from Corollary 1, the Plancherel formula, and the elementary formula (8).

This proof of Corollary 2 poses a significant challenge. Presumably, the corollary holds because the geometry of G at infinity is so different from that of Euclidean space. For example, the volume of the ball of radius R in G grows exponentially as R→∞. This must have a profound impact on the way mass piles up when we take convolutions on G. On the other hand, the statement of Corollary 2 clearly has nothing to do with cancellation; proving the corollary for two arbitrary functions f , g is the same as proving it for |f | and |g|. When we go back over the proof of Corollary 2, we see cancellation used crucially, e.g., in the Plancherel formula for G; but there is no explicit mention of the geometry of G at infinity. Clearly there is still much that we do not understand regarding convolutions on G.

The Kunze-Stein phenomenon carries over to other semisimple groups, with profound consequences for representation theory. We will continue this discussion later in the chapter. Now, however, we turn our attention to classical Fourier analysis.

# Curvature and the Fourier Transform

One of the most fascinating themes in Fourier analysis in the last two decades has been the connection between the Fourier transform and curvature. Stein has been the most important contributor to this set of ideas. To illustrate, I will pick out two of his results. The first is a “restriction theorem,” i.e., a result on the restriction f ˆ | of the Fourier transform of a function $f∈ L p ( ℝ n )$ to a set of measure zero. If p > 1, then the standard inequality $f ^ ∈ L p ′ ( ℝ n )$ suggests that f ˆ should not even bewell-defined (p.6) on , since has measure zero. Indeed, if is (say) the x-axis in the plane R2, then we can easily find functions $f( x 1 , x 2 )=φ( x 1 )ψ( x 2 )∈ L p ( ℝ 2 )$ for $f ^ | Γ$ which f | is infinite everywhere. Fourier transforms of $f∈ L p ( ℝ 2 )$ clearly cannot be restricted to straight lines. Stein proved that the situation changes drastically when is curved. His result is as follows.

Stein’s Restriction Theorem. Suppose is the unit circle, $1≤p< 8 7$ and $f∈ C 0 ∞ ( ℝ 2 )$ Then we have the a priori inequality $∥ f ^ | Γ ∥ L 2 ≤ C p ∥f ∥ L p ( ℝ 2 )$ with Cp depending only on p.

Using this a priori inequality, we can trivially pass from the dense subspace C0 to define the operator $f↦ f ^ | Γ forallf∈ L p ( ℝ 2 )$ Thus, the Fourier transform of $f∈ L p (p< 8 7 )$ may be restricted to the unit circle. Improvements and generalizations were soon proven by other analysts, but it was Stein who first demonstrated the phenomenon of restriction of Fourier transforms. Stein’s proof of his restriction theorem is amazingly simple. If μ‎ denotes uniform measure on the circle $Γ⊂ ℝ 2 c$ then for $f∈ C 0 ∞ ( ℝ 2 )$we have

(10)
$Display mathematics$

The Fourier transform μ‎ˆ (ξ‎ ) is a Bessel function. It decays like |ξ‎ |1/2 at infinity, a fact intimately connected with the curvature of the circle. In particular, μ‎ˆ∈ Lq for 4 < q ≤ ∞, and therefore $∥f∗ μ ^ ∥ L p ′ ≤ C p ∥f ∥ L p for1≤p< 8 7$ by the usual elementary estimates for convolutions. Putting this estimate back into (10), we see that $∫ Γ | f ^ | 2 ≤ C p ∥f ∥ L p 2$ which proves Stein’s Restriction Theorem. The Stein Restriction Theorem means a lot to me personally, and has strongly influenced my own work in Fourier analysis.

The second result of Stein’s relating the Fourier transform to curvature concerns the differentiation of integrals on Rn.

Theorem. Suppose $f∈ L p ( ℝ n )$ with $n≥3​andp> n n−1$ For $x∈ ℝ n$ and r > 0, let F(x, r) denote the average of f on the sphere of radius r centered at x. Then limr0 F(x, r) = f (x) almost everywhere.

The point is that unlike the standard Lebesgue Theorem, we are averaging f over a small sphere instead of a small ball. As in the restriction theorem, we are seemingly in trouble because the sphere has measure zero in Rn, but the curvature of the sphere saves the day. This theorem is obviously closely connected to the smoothness of solutions of the wave equation.

The proof of the above differentiation theorem relies on an

Elementary Tauberian Theorem. Suppose that $lim R→0 1 R ∫ 0 R F(r)dr$exists and $∫ 0 ∞ r| dF dr | 2 dr<∞$ Then $lim R→0 F(R)$exists, and equals $lim R→0 1 R ∫ 0 R F(r)dr$

This result had long been used, e.g., to pass from Cesaro averages of Fourier series to partial sums. (See Zygmund [Z].) On more than one occasion, Stein has shown the surprising power hidden in the elementary Tauberian Theorem. Here weapply (p.7) it to F(x, r) for a fixed x. In fact, we have $F(x,r)= ∫ f(x+ry)dμ(y)$ with μ‎ equal to normalized surface measure on the unit sphere, so that the Fourier transforms of F and f are related by $F ^ (ξ,r)= f ^ (ξ) μ ^ (rξ)$for each fixed r. Therefore, assuming f ∈ L2 for simplicity, we obtain

$Display mathematics$

(Here we make crucial use of curvature, which causes μ‎ˆ to ∞ decay at infinity, so that the integral in curly brackets converges.) It follows that $∫ 0 ∞ r| ∂ ∂r F(x,r )| 2 dr<∞$for almost every x ∈ Rn. On the other hand, $1 R ∫ 0 R F(x,r)dr$ is easily seen to be the convolution of f with a standard approximate identity. Hence the usual Lebesgue differentiation theorem shows that lim$lim R→0 1 R ∫ 0 R F(x,r)dr=f(x)$ for almost every x.

So for almost all x ∈ Rn, the function F(x, r) satisfies the hypotheses of the elementary Tauberian theorem. Consequently,

$Display mathematics$

almost everywhere, proving Stein’s differentiation theorem for f ∈ L2(Rn).

To prove the full result for $f∈ L p ( ℝ n ),p> n n−1 ,$ we repeat the above argument with surface measure μ‎ replaced by an even more singular distribution on Rn. Thus we obtain a stronger conclusion than asserted, when f ∈ L2. On the other hand, for f ∈ L1+ε‎ we have a weaker result than that of Stein, namely Lebesgue’s differentiation theorem. Interpolating between L2 and L1+ε‎, one obtains the Stein differentiation theorem.

The two results we picked out here are only a sample of the work of Stein and others on curvature and the Fourier transform. For instance, J. Bourgain has dramatic results on both the restriction problem and spherical averages.

We refer the reader to Stein’s address at the Berkeley congress [128] for a survey of the field.

# HP -Spaces

Another essential part of Fourier analysis is the theory of Hp-spaces. Stein transformed the subject twice, once in a joint paper with Guido Weiss, and again in a joint paper with me. Let us start by recalling how the subject looked before Stein’s work. The classical theory deals with analytic functions F(z) on the unitdisc. (p.8) Recall that F belongs to $H p (0 if the norm $∥F ∥ H p ≡ lim r→1− ( ∫ 0 2π |F(r e iθ )| p dθ) 1/p$ is finite.

The classical Hp-spaces serve two main purposes. First, they provide growth conditions under which an analytic function tends to boundary values on the unit circle. Secondly, Hp serves as a substitute for Lp to allow basic theorems on Fourier series to extend from 1 < p < ∞ to all p > 0. To prove theorems about F ∈ Hp, the main tool is the Blaschke product

(11)
$Display mathematics$

where {zv} are the zeroes of the analytic function F in the disc, and θ‎v are suitable phases. The point is that B(z) has the same zeroes as F, yet it has absolute value 1 on the unit circle. We illustrate the role of the Blaschke product by sketching the proof of the Hardy-Littlewood maximal theorem for Hp. The maximal theorem says that $∥ F ∗ ∥ L p ≤ C p ∥F ∥ H p$ where $F ∗ (θ)= sup z∈Γ(θ) |F(z)|$ and (θ‎) is the convex hull of eiθ‎ and the circle of radius about the origin.

This basic result is closely connected to the pointwise convergence of F(z) as z ∈ (θ‎) tends to eiθ‎ . To prove the maximal theorem, we argue as follows.

First suppose p > 1. Then we don’t need analyticity of F. We can merely assume that F is harmonic, and deduce the maximal theorem from real variables. In fact, it is easy to show that F arises as the Poisson integral of an Lp function f on the unit circle. The maximal theorem for f , a standard theorem of real variables, says that $∥Mf ∥ L p ≤ C p ∥f ∥ L p$ where $Mf(θ)= sup h>0 ( 1 2h ∫ θ−h θ+h |f(t)|dt)$ It is quite simple to show that $F ∗ (θ)Therefore $∥ F ∗ ∥ L p ≤C∥Mf ∥ L p < C ′ ∥F ∥ H p$ and the maximal theorem is proven for Hp (p > 1).

If p ≤ 1, then the problem is more subtle, and we need to use analyticity of F(z). Assume for a moment that F has no zeroes in the unit disc. Then for 0 < q < p, we can define a single-valued branch of (F (z))q , which will belong to Hp/q since F ∈ Hp. Since p˜≡ pq > 1, the maximal theorem for H is already known. Hence, maxz∈(θ‎) $max z∈Γ(θ) |(F(z )) q |∈ L p/q$ with norm

$Display mathematics$

That is, $∥ F ∗ ∥ L p ≤ C p,q ∥F ∥ H p$ proving the maximal theorem for functions without zeroes.

To finish the proof, we must deal with the zeroes of an F ∈ Hp (p ≤ 1). We bring in the Blaschke product B(z), as in (11). Since B(z) and F(z) have the same zeroes and since |B(z)| = 1 on the unit circle, we can write F(z) = G(z)B(z) with G analytic, and |G(z)| = |F(z)| on the unit circle. Thus, GHp = FHp. Inside the circle, G has no zeroes and |B(z)| ≤ 1. Hence |F| ≤ |G|, so

$Display mathematics$

by the maximal theorem for functions withour zeroes. The proof of the maximal theorem is complete. (We have glossed over difficulties that should not enter an expository paper.)

(p.9) Classically, Hp theory works only in one complex variable, so it is useful only for Fourier analysis in one real variable. Attempts to generalize Hp to several complex variables ran into a lot of trouble, because the zeroes of an analytic function F(z1, … zn) ∈ Hp form a variety V with growth conditions. Certainly V is much more complicated than the discrete set of zeroes {zv} in the disc. There is no satisfactory substitute for the Blaschke product. For a long time, this blocked all attempts to extend the deeper properties of Hp to several variables.

Stein and Weiss [13] realized that several complex variables was the wrong generalization of Hp for purposes of Fourier analysis. They kept clearly in mind what Hp-spaces are supposed to do, and they kept an unprejudiced view of how to achieve it. They found a version of Hp theory that works in several variables.

The idea of Stein and Weiss was very simple. They viewed the real and imaginary parts of an analytic function on the disc as the gradient of a harmonic function. In several variables, the gradient of a harmonic function is a system u= (u1, u2, … , un) of functions on Rn that satisfies the Stein-Weiss Cauchy-Riemann equations

(12)
$Display mathematics$

In place of the Blaschke product, Stein and Weiss used the following simple observation. If u= (u1, … , un) satisfies (12), then $| u → | p =( u 1 2 + u 2 2 +⋯+ u n 2 ) p/2$ is n−uun1222 subharmonic for p > n−2 1 . We sketch the simple proof of this fact, then explain how an Hp theory can be founded on it.

To see that |u|p is subharmonic, we first suppose |u| = 0 and calculate (|u|p) in coordinates that diagonalize the symmetric matrix ( ∂∂uj ) at a given point. The xk result is

(13)
$Display mathematics$

with wk = ∂uk and vk = ukwk. ∂xk

Since kn=1wk = 0 by the Cauchy-Riemann equations, we have

$Display mathematics$

||2 ≤ n−1 | w|2. |v|2 ≤ ||2)|u|2 ≤ n−1 2|

i.e., wkn Hence $| v → | 2 ≤( max k | w k | 2 )| u → | 2 ≤( n−1 n )| w → | 2 | u → | 2$ so the n expression in curly brackets in (13) is non-negative for p ≥ nn−−2 1, and |u|p is subharmonic.

So far, we know that |u|p is subharmonic where it isn’t equal to zero. Hence for 0 < r < r(x) we have

(14)
$Display mathematics$

provided |u(x)| = 0. However, (14) is obvious when |u(x)| = 0, so it holds for any x. That is, |u|p is a subharmonic function for p ≥ nn−−2 1 , as asserted.

Now let us see how to build an Hp theory for Cauchy-Riemann systems, based on subharmonicity of |u|p. To study functions on Rn−1(n ≥ 2), we regard Rn−1 asthe (p.10) boundary of $ℝ + n ={( x 1 ,…, x n )| x n >0}$ and we define Hp(Rn+) as the space of all Cauchy-Riemann systems (u1, u2, … , un) for which the norm

$Display mathematics$

is finite. For n = 2 this definition agrees with the usual Hp-spaces for the upper half-plane.

Next we show how the Hardy-Littlewood maximal theorem extends from the disc to Rn+.

Define the maximal function $M( u → )(x)= sup |y−x|.

Then for $u → ∈ H p ( ℝ + n ), n−2 n−1 we have $M( u → )∈ L p ( ℝ n−1 )$ with norm $∫ ℝ n−1 (M( u → )) p dx≤C∥ u → ∥ H p p$

As in the classical case, the proof proceeds by reducing the problem to the maximal theorem for Lp(p > 1). For small h > 0, the function Fh(x, t) = |u(x, t + h)| nn−−2 1 (x ∈ Rn−1, t ≥ 0) is subharmonic on Rn+ and continuous up to the boundary. Therefore,

(15)
$Display mathematics$

where P.I. is the Poisson integral and fh(x) = Fh(x, 0) = |u(x, h)| nn−−2 1 . By definition of the Hp-norm, we have

(16)
$Display mathematics$

On the other hand, since the Poisson integral arises by convoling with an approximate identity, one shows easily that

(17)
$Display mathematics$

with

$Display mathematics$

The standard maximal theorem of real variables gives

$Display mathematics$

since p˜ > 1. Hence (15), (16), and (17) show that

(18)
$Display mathematics$

The constant Cp is independent of h, so we can take the limit of (18) as h → 0 to obtain the maximal theorem for Hp. The point is that subharmonicity of |u| nn−−2 1 substitutes for the Blaschke product in this argument.

(p.11) Stein and Weiss go on in [13] to obtain n-dimensional analogues of the classical theorems on existence of boundary values of Hp functions. They also extend to Rn+ the classical F. and M. Riesz theorem on absolute continuity of H1 boundary values. They begin the program of using Hp(Rn+) in place of Lp(Rn−1), to extend the basic results of Fourier analysis to p = 1 and below. We have seen how they deal with the maximal function. They prove also an Hp-version of the Sobolev theorem.

It is natural to try to get below p = nn−−2 1 , and this can be done by studying higher gradients of harmonic functions in place of (12). See Calderón-Zygmund [CZ].

A joint paper [60] by Stein and me completed the task of developing basic Fourier analysis in the setting of the Hp-spaces. In particular, we showed in [60] that singular integral operators are bounded on Hp(Rn+) for 0 < p < ∞. We proved this by finding a good viewpoint, and we found our viewpoint by repeatedly changing the definition of Hp. With each new definition, the function space Hp remained the same, but it became clearer to us what was going on. Finally we arrived at a definition of Hp with the following excellent properties. First of all, it was easy to prove that the new definition of Hp was equivalent to the Stein-Weiss definition and its extensions below $p= n−2 n−1$ Secondly, the basic theorems of Fourier analysis, which seemed very hard to prove from the original definition of Hp(Rn+), became nearly obvious in terms of the new definition. Let me retrace the steps in [60].

Burkholder-Gundy-Silverstein [BGS] had shown that an analytic function F = u + iv on the disc belongs to Hp (0 < p < ∞) if and only if the maximal function u(θ‎) = supz∈(θ‎) |u(z)| belongs to Lp (Unit Circle). Thus, Hp can be defined purely in terms of harmonic functions u, without recourse to the harmonic conjugate υ‎. Stein and I showed in [60] that the same thing happens in n dimensions. That is, a Cauchy-Riemann system (u1, u2, … , un) on Rn+ belongs to the Stein-Weiss Hp-space (p > nn−−2 1 ) if and only if the maximal function u(x) = sup|y−x|<t |un(y, t)| belongs to Lp(Rn−1). (Here, the nth function un plays a special role because R+ nis defined by {xn > 0}.) Hence, Hp may be viewed as a space of harmonic functions u(x, t) on R+n. The result extends below p = nn−−2 1 if we pass to higher gradients of harmonic functions.

The next step is to view Hp as a space of distributions f on the boundary Rn−1. Any reasonable harmonic function u(x, t) arises as the Poisson integral of a distribution f , so that u(x, t) = ϕ‎t ∗ f (x), ϕ‎t = Poisson kernel. Thus, it is natural to say that f ∈ Hp(Rn−1) if the maximal function

(19)
$Display mathematics$

belongs to Lp. Stein and I found in [60] that this definition is independent of the choice of the approximate identity ϕ‎t , and that the “grand maximal function”

(20)
$Display mathematics$

belongs to Lp, provided f ∈ Hp. Here A is a neighborhood of the origin in a suitable space of approximate identities. Thus, f ∈ Hp if and only if f ∈ Lp for some reasonable approximate identity. Equivalently, f ∈ Hp ifand (p.12) only if the grand maximal function Mf belongs to Lp. The proofs of these various equivalencies are not hard at all.

We have arrived at the good definition of Hp mentioned above.

To transplant basic Fourier analysis from Lp(1 < p < ∞) to Hp(0 < p < ∞), there is a simple algorithm. Take Calderón-Zygmund theory, and replace every application of the standard maximal theorem by an appeal to the grand maximal function. Only small changes are needed, and we omit the details here. Our paper [60] also contains the duality of H1 and BMO. Before leaving [60], let me mention an application of Hp theory to Lp-estimates. If σ‎ denotes uniform surface measure on the unit sphere in Rn, then f −→ ( ∂∂ x )α‎σ‎ ∗ f is bounded on Lp(Rn), provided n ≥ 3 and | 1p − | ≤ − nα‎ −1 . Clearly, this result gives information on solutions to the wave equation.

The proof uses complex interpolation involving the analytic family of operators

$Display mathematics$

as is clear to anyone familiar with the Stein interpolation theorem. The trouble here is that (−)α‎/2 fails to be bounded on L1 when α‎ is imaginary. This makes it impossible to prove the sharp result $(| 1 p − 1 2 |= 1 2 − α n−1 )$ using Lp alone. To overcome the difficulty, we use H1 in place of L1 in the interpolation argument. Imaginary powers of the Laplacian are singular integrals, which we know to be bounded on H1. To show that complex interpolation works on H1, we combined the duality of H1 and BMO with the auxiliary function $f # (x)= sup Q∍x 1 |Q| ∫ Q |f(y)−(mea n Q f)|dy$ We refer the reader to [60] for an explanation of how to use f #, and for other applications.

Since [60], Stein has done a lot more on Hp, both in “higher rank” settings, and in contexts related to partial differential equations.

# Representation Theory II

Next we return to representation theory. We explain briefly how the Kunze-Stein construction extends from SL(2,R) to more general semisimple Lie groups, with profound consequences for representation theory. The results we discuss are contained in the series of papers by Kunze-Stein [20, 22, 33, 63], Stein [35, 48, 70], and Knapp-Stein [43, 46, 50, 53, 58, 66, 73, 93, 97]. Let G be a semisimple Lie group, and let Uπ‎ be the unitary principal series representations of G, or one of its degenerate variants. The Uπ‎ all act on a common Hilbert space, whose inner product we denote by ξ‎, η‎. We needn’t write down Uπ‎ here, nor even specify the parameters on which it depends. A finite group W, the Weyl group, acts on the parameters π‎ in such a way that the representations Uπ‎ and Uwπ‎ are unitarily equivalent for w ∈ W. Thus there is an intertwining operator A(w, π‎) so that

A(w, π‎)Uwπ‎ g = Uπ‎ g A(w, π‎) for g ∈ G, w ∈ W,and for all π‎. (21)

If Uπ‎ is irreducible (which happens for most π‎), then A(w, π‎) is uniquely determined by (21) up to multiplication by an arbitrary scalar a(w, π‎). The crucialidea (p.13) is as follows. If the A(w, π‎) are correctly normalized (by the correct choice of a(w, π‎)), then A(w, π‎) continues analytically to complex parameter values π‎. Moreover, for certain complex (w, π‎), the quadratic form

(21)
$Display mathematics$

is positive definite.

In addition, the representation Uπ‎ (defined for complex π‎ by a trivial analytic continuation) is unitary with respect to the inner product (22). Thus, starting with the principal series, we have constructed a new series of unitary representations of G. These new representations generalize the complementary series for SL(2,R). Applications of this basic construction are as follows.

1. (1) Starting with the unitary principal series, one obtains understanding of the previously discovered complementary series, and construction of new ones, e.g., on Sp(4,C). Thus, Stein exposed a gap in a supposedly complete list of complementary series representations of Sp(4,C) [GN]. See [33].

2. (2) Starting with a degenerate unitary principal series, Stein constructed new irreducible unitary representations of SL(n,C), in startling contradiction to the standard, supposedly complete list [GN] of irreducible unitary representations of that group. Much later, when the complete list of representations of SL(n,C) was given correctly, the representations constructed by Stein played an important role.

3. (3) The analysis of intertwining operators required to carry out analytic continuation also determines which exceptional values of π‎ lead to reducible principal series representations. For example, such reducible principal series representations exist already for SL(n,R), again contradicting what was “known.” See Knapp-Stein [53].

A very recent result of Sahi and Stein [139] also fits into the same philosophy. In fact, Speh’s representation can also be constructed by a more complicated variant of the analytic continuation defining the complementary series. Speh’s representation plays an important role in the classification of the irreducible unitary representations of SL(2n,R).

The main point of Stein’s work in representation theory is thus to analyze the interwining operators A(w, π‎). In the simplest non-trivial case, A(w, π‎) is a singular integral operator on a nilpotent group N. That is, A(w, π‎) has the form

(22)
$Display mathematics$

with K(y) smooth away from the identity and homogeneous of the critical degree with respect to “dilations”

(23)
$Display mathematics$

In (23), dy denotes Haar measure on N. We know from the classical case N = R1 that (23) is a bounded operator only when the convolution kernel K(y) satisfies a (p.14) cancellation condition. Hence we assume $∫ B 1 \ B 0 K(y)dy=0$ where the Bi are = B1B0 dilates (Bi δ‎ti(B)) of a fixed neighborhood of the identity in N.

It is crucial to show that such singular integrals are bounded on L2(N), generalizing the elementary L2-boundedness of the Hilbert transform.

# Cotlar-Stein Lemma

In principle, L2-boundedness of the translation-invariant operator (23) should be read off from the representation theory of N. In practice, representation theory provides a necessary and sufficient condition for L2-boundedness that no one knows how to check. This fundamental analytic difficulty might have proved fatal to the study of intertwining operators. Fortunately, Stein was working simultaneously on a seemingly unrelated question, and made a discovery that saved the day. Originally motivated by desire to get a simple proof of Calderón’s theorem on commutator integrals [Ca], Stein proved a simple, powerful lemma in functional analysis. His contribution was to generalize to the critically important non-commutative case the remarkable lemma of Cotlar [Co]. The Cotlar-Stein lemma turned out to be the perfect tool to prove L2-boundedness of singular integrals on nilpotent groups. In fact, it quickly became a basic, standard tool in analysis. We will now explain the Cotlar-Stein lemma, and give its amazingly simple proof. Then we will return to its application to singular integrals on nilpotent groups.

The Cotlar-Stein lemma deals with a sum $T= Σ v T v$of operators on a Hilbert space. The idea is that if the Tv are almost orthogonal, like projections onto the various coordinate axes, then the sum T will have norm no larger than maxv Tv. The precise statement is as follows.

Cotlar-Stein Lemma. Suppose $T= Σ k=1 M T k$is a sum of operators onHilbert space.Assume $∥ T j ∗ T k ∥≤a(j−k)$and$∥ T j T k ∗ ∥≤a(j−k)$Then$∥T∥≤ Σ −M M a(j)$

Proof. T = (T T )1/2, so

$Display mathematics$

We can estimate the summand in two different ways.

Writing Tj1T j2 · · · Tj2s−1T j2s = (Tj1T j∗ 2)(Tj3T ∗ j4 ) · · · (Tj2s−1T ∗ j2s ), we get

(24)
$Display mathematics$

On the other hand, writing

(25)
$Display mathematics$

(p.15) we see that

$Display mathematics$

Taking the geometric mean of (25), (26) and putting the result into (24), we conclude that

(26)
$Display mathematics$

Thus, $∥T∥≤ ( max j ∥ T j ∥⋅M) 1/2s ⋅ ( Σ l a(l) ) 2s−1 2s$ Letting s →∞, we obtain the conclusion of the Cotlar-Stein Lemma.

To apply the Cotlar-Stein lemma to singular integral operators, take a partition of unity $1= Σ v φ v (x)$(x) on N, so that each ϕ‎v is a dilate of a fixed C0 function that vanishes in a neighborhood of the origin. Then T : f → K ∗ f may be decomposed into a sum T = vTv, with $T v :f→( φ v K)∗f$ The hypotheses of the Cotlar-Stein lemma are verified trivially, and the boundedness of singular integral operators follows. The L2-boundedness of singular integrals on nilpotent groups is the Knapp-Stein Theorem.

Almost immediately after this work, the Cotlar-Stein lemma became the standard method to prove L2-boundedness of operators. Today one knows more, e.g., the T (1) theorem of David and Journé. Still it is fair to say that the Cotlar-Stein lemma remains the most important tool for L2-boundedness.

Singular integrals on nilpotent groups were soon applied by Stein in a context seemingly far from representation theory.

# -Problems

We prepare to discuss Stein’s work on the ¯∂-problems of several complex variables and related questions. Let us begin with the state of the subject before Stein’s contributions. Suppose we are given a domain D ⊂ Cn with smooth boundary. If we try to construct analytic functions on D with given singularities at the boundary, then we are led naturally to the following problems.

1. I. Given a (0, 1) form $α= Σ k=1 n f k d z k ¯$ D, find a function u on D that solves$∂ ¯ u=α$ where $∂ ¯ u= Σ k=1 n ∂ ¯ u ∂ z ¯ k d z k ¯$ Naturally, this is possible only ifα‎ satisfies the consistency condition$∂ ¯ α=0,i.e., ∂ ∂ z ¯ k f j = ∂ ∂ z ¯ j f k$ Moreover,uis determined only modulo addition of an arbitrary analytic function on D. To make u unique, we demand that u be orthogonal to analytic functions in L2(D).

2. (p.16) II. There is a simple analogue of the ¯∂-operator for functions defined only on the boundary ∂D. In local coordinates, we can easily find (n − 1) linearly independent complex vector fields $L 1 ⋯ L n−1$for type(0,1)(i.e,$L j = a j1 ∂ ∂ z ¯ 1 + a j2 ∂ ∂ z ¯ 2 +⋯+ a jn ∂ ∂ z ¯ n$for smooth, complex-valued ajk ) whose real z1 ∂z2 ∂zn and imaginary parts are all tangent to ∂D. The restriction u of an analytic function to ∂D clearly satisfies ¯∂bu = 0, where in local coordinates$∂ ¯ b u=( L 1 u, L 2 u,…, L n−1 u)$ The boundary analogue of the ¯∂-problem (I) Ln− is the inhomogeneous ∂-equation ¯∂= α‎. Again, this is possible only if α‎ ¯ b¯bu satisfies a consistency condition ∂bα‎ = 0, and we impose the side condition that u be orthogonal to analytic functions in L2(∂D).

Just as analytic functions of one variable are related to harmonic functions, so the first-order systems (I) and (II) are related to second-order equations and b, the -Neumann and Kohn Laplacians. Both fall outside the scope of standard elliptic theory. Even for the simplest domains D, they posed a fundamental challenge to workers in partial differential equations. More specifically, is simply the Laplacian in the interior of D, but it is subject to non-elliptic boundary conditions. On the othe hand, b is a non-elliptic system of partial differential operators on ∂D, with no boundary conditions (since ∂D has no boundary). Modulo lower-order terms (which, however, are important), b is the scalar operator $L= Σ k=1 n−1 ( X k 2 + Y k 2 )$ where Xk and Yk are the real and imaginary parts of the basic complex vector fields Lk. At a given point in ∂D, the Xk and Yk are linearly independent, but they don’t span the tangent space of ∂D. This poses the danger that L will behave like a partial Laplacian such as $Δ ′ = ∂ 2 ∂ x 2 + ∂ 2 ∂ y 2$ acting on function u(x, y, z). The equation ∂x2 ∂y2 u = f is very bad. For instance, we can take u(x, y, z) to depend on z alone, so that u = 0 with u arbitrarily rough. Fortunately, L is more like the full Laplacian than like , because the Xk and Yk together with their commutators [Xk, Yk] span the tangent space of ∂D for suitable D. Thus, L is a well-behaved operator, thanks to the intervention of commutators of vector fields.

It was Kohn in the 1960s who proved the basic C regularity theorems for ,b, ¯∂and ¯∂b on strongly pseudoconvex domains (the simplest case). His proofs were based on subelliptic estimates such as $〈 □ b w,w〉≥c∥w ∥ (ε) 2 −C∥w ∥ 2 ,$ and brought to light the importance of commutators. Hörmander proved a celebrated theorem on C regularity of operators,

$Display mathematics$

where X0,X1, … , XN are smooth, real vector fields which, together with their repeated commutators, span the tangent space at every point.

If we allow X0 to be a complex vector field, then we get a very hard problem that is not adequately understood to this day, except in very special cases.

Stein made a fundamental change in the study of the ¯∂-problems by bringing in constructive methods. Today, thanks to the work of Stein with several collaborators, we know how to write down explicit solutions to the ¯∂-problems modulo negligible errors on strongly pseudoconvex domains. Starting from these explicitsolutions, (p.17) it is then possible to prove sharp regularity theorems. Thus, the ∂¯equations on strongly pseudoconvex domains are understood completely. It is a major open problem to achieve comparable understanding of weakly pseudoconvex domains.

Now let us see how Stein and his co-workers were able to crack the strongly pseudoconvex case. We begin with the work of Folland and Stein [67]. The simplest example of a strongly pseudoconvex domain is the unit ball. Just as the disc is equivalent to the half-plane, the ball is equivalent to the Siegel domain $D Siegel ={(z,w)∈ ℂ n−1 ×ℂ|Imw>|z | 2 }$ Its boundary $H=∂ D Siegel$ has an important symmetry group, including the following.

1. (a) Translations $(z,w)↦(z,w)⋅( z ′ , w ′ )≡(z+ z ′ ,w+ w ′ +2iz⋅ z ¯ ')$ for $( z ′ , w ′ )∈H$

2. (b) Dilations $δ t :(z,w)↦(tz, t 2 w)fort>0;$

3. (c) Rotations $(z,w)↦(Uz,w)$for unitary (n − 1) × (n − 1) matrices U.

The multiplication law in (a) makes H into a nilpotent Lie group, the Heisenberg group. Translation-invariance of the Seigel domain allows us to pick the basic complex vector-fields L1 · · ·Ln−1 to be translation-invariant on H. After we make a suitable choice of metric, the operators L and b become translation- and rotation-invariant, and homogeneous with respect to the dilations δ‎t . Therefore, the solution1 of bw = α‎ should have the form of a convolution w = K ∗ α‎ on the Heisenberg group. The convolution kernel K is homogeneous with respect to the dilations δ‎t and invariant under rotations. Also, since K is a fundamental solution, it satisfies bK = 0 away from the origin. This reduces to an elementary ODE after we take the dilation- and rotation-invariance into account. Hence one can easily find K explicitly and thus solve the b-equation for the Siegel domain. To derive sharp regularity theorems for b, we combine the explicit fundamental solution with the Knapp-Stein theorem on singular integrals on the Heisenberg group. For instance, if bw ∈ L2, then $L j L k w, L ¯ j L k w,and L ¯ j L ¯ k w$ all belong to L2. To see this, we write

$Display mathematics$

and note that LjLkK has the critical homogeneity and integral 0. Thus LjLkK is a singular integral kernel in the sense of Knapp and Stein, and it follows that $∥ L j L k w∥≤C∥α∥$ For the first time, nilpotent Lie groups have entered into the LjLkw¯ study of ∂-problems.

Folland and Stein viewed their results on the Heisenberg group not as ends in themselves, but rather as a tool to understand general strongly pseudoconvex CR-manifolds. A CR-manifold M is a generalization of the boundary of a smooth domain D ⊂ Cn. For simplicity we will take M = ∂D here. The key idea is that near any point w in a strongly pseudoconvex M, the CR-structure for M is very nearly equivalent to that of the Heisenberg group H via a change of coordinates w : M → H. More precisely, w carries w to the origin, and it carries the CR-structure on M to a CR-structure on H that agrees with the usual one at the origin.

(p.18) Therefore, if w = K ∗ α‎ is our known solution of bw = α‎ on the Heisenberg group, then it is natural to try.

$Display mathematics$

as an approximate solution of bw = α‎ on M. (Since w and α‎ are sections of bundles, one has to explain carefully what (27) really means.) If we apply b to the w defined by (27), then we find that

(27)
$Display mathematics$

where E is a sort of Heisenberg version of (−)−1/2. In particular, E gains smoothness, so that (I − E)−1 can be constructed modulo infinitely smoothing operators by means of a Neumann series. Therefore (27) and (28) show that the full solution of bw = α‎ is given (modulo infinitely smoothing errors) by

(28)
$Display mathematics$

from which one can deduce sharp estimates to understand completely b 1 on M.

The process is analogous to the standard method of “freezing coefficients” to solve variable-coefficient elliptic differential equations. Let us see how the sharp results are stated. As on the Heisenberg group, there are smooth, complex vector fields Lk that span the tangent vectors of type (0, 1) locally. Let Xj be the real and imaginary parts of the Lk. In terms of the Xj we define “non-Euclidean” versions of standard geometric and analytic concepts. Thus, the non-Euclidean ball B(z, ρ‎) may be defined as an ellipsoid with principal axes of length ∼ ρ‎ in the codimension 1 hyperplane spanned by the Xj , and length ∼ρ‎2 perpendicular to that hyperplane. In terms of B(z, ρ‎), the non-Euclidean Lipschitz spaces α‎(M) are defined as the set of functions u for which |u(z) − u(w)| < Cρ‎α‎ for w ∈ B(z, ρ‎). (Here, 0 < α‎ < 1. There is a natural extension to all α‎ > 0.) The non-Euclidean Sobolev spaces Sm,p(M) consist of all distributions u for which all $X j 1 X j 2 ⋯ X j s u∈ L p (M)$ for 0 ≤ s ≤ m.

Then the sharp results on b are as follows. If bw = α‎ and α‎ ∈ Sm,p(M), then w ∈ Sm+2,p(M) for m ≥ 0, 1 < p < ∞. If bw = α‎ and α‎ ∈ α‎(M), then$X j X k w∈ Γ α (M)for<α<1$ (say). For additional sharp estimates, and for comparisons between the non-Euclidean and standard function spaces, we refer the reader to [67].

To prove their sharp results, Folland and Stein developed the theory of singular integral operators in a non-Euclidean context. The Cotlar-Stein lemma proves the crucial results on L2-boundedness of singular integrals. Additional difficulties arise from the non-commutativity of the Heisenberg group. In particular, standard singular integrals or pseudodifferential operators commute modulo lower-order errors, but non-Euclidean operators are far from commuting. This makes more difficult the passage from Lp estimates to the Sobolev spaces M). ¯Sm,p(

Before we continue with Stein’s work on ∂, let me explain the remarkable paper of Rothschild-Stein [72]. It extends the Folland-Stein results and viewpoint to general Hörmander operators $L= Σ j=1 N X j 2 + X 0$ Actually, [72] deals with systemswhose (p.19) second-order part is $Σ j X j 2 ,$ but for simplicity we restrict attention here to L. In explaining the proofs, we simplify even further by supposing X0 = 0. The goal of the Rothschild-Stein paper is to use nilpotent groups to write down an explicit parametrix for L and prove sharp estimates for solutions of Lu = f. This ambitious hope is seemingly dashed at once by elementary examples. For instance, take $L= X 1 2 + X 2 2$with

(29)
$Display mathematics$

Then X1 and [X1,X2] span the tangent space, yet L clearly cannot be approximated by translation-invariant operators on a nilpotent Lie group in the sense of Folland-Stein. The trouble is that L changes character completely from one point to another. Away from the y-axis {x = 0},L is elliptic, so the only natural nilpotent group we can reasonably use is R2. On the y-axis, L degenerates, and evidently cannot be approximated by a translation-invariant operator on R2. The problem is so obviously fatal, and its solution by Rothschild and Stein so simple and natural, that [72] must be regarded as a gem. Here is the idea:

Suppose we add an extra variable t and “lift” X1 and X2 in (30) to vector fields

(30)
$Display mathematics$

Then the Hörmander operator L˜= X˜21+ X˜22 looks the same at every point of R3, and may be readily understood in terms of nilpotent groups as in Folland-Stein [67]. In particular, one can essentially write down a fundamental solution and prove sharp estimates for L˜−1. On the other hand, L˜reduces to L when ˜acting on functions u(x, y, t) that do not depend on t . Hence, sharp results on Lu = f imply sharp results on Lu = f.

Thus we have the Rothschild-Stein program: First, add new variables and lift the given vector fields · · · ,XN to new vector fields X˜· · · X˜whose underlying X1, 1 N ˜structure does not vary from point to point. Next, approximate by$L ˜ = Σ 1 N X ˜ 2$ a translation-invariant operator $L ^ = Σ 1 N Y j 2$ on a nilpotent Lie group N. Then analyze the fundamental solution of Lˆ, and use it to write down an approximate fundamental solution for L˜. From the approximate solution, derive sharp estimates for solutions of L˜u = f . Finally, descend to the original equation Lu = f by restricting attention to functions u, f that do not depend on the extra variables.

To carry out the first part of their program, Rothschild and Stein prove the following result.

Theorem A. Let X1 · · ·XN be smooth vector fields on a neighborhood of the origin in Rn. Assume that the Xj and their commutators [[[Xj1,Xj2 ],Xj3] · · ·Xjs ] of order up to r span the tangent space at the origin. Then we can find smooth vector fields1 · · · X˜N on a neighborhoodof the origin in Rn+m with the following properties.

1. (a) Thej and their commutators up to oder r are linearly independent at each point of U˜, except for the linear relations that follow formally from the antisymmetry of the bracket and the Jacobi identity.

2. (p.20) (b) Thej and their commutators up to order r span the tangent space of U˜.

3. (c) Acting on functions on Rn+m that do not depend on the last m coordinates, thej reduce to the given Xj .

Next we need a nilpotent Lie group N appropriate to the vector fields X˜1 · · · X˜N. The natural one is the free nilpotent group NNr of step r on N generators. Its Lie algebra is generated by Y1 · · · YN whose Lie brackets of order higher than r vanish, but whose brackets of order ≤ r are linearly independent, except for relations forced by antisymmetry of brackets and the Jacobi identity. We regard the Yj as translation-invariant vector fields on NNr. It is convenient to pick a basis {yα‎}α‎∈A for the Lie algebra of N, consisting of Y1 · · · and some of their commutators. OnNNr we form the Hörmander operator $L ^ = Σ 1 N Y j 2$ Then Lis translation-invariant and homogeneous under the natural dilations on NNr. Hence Lˆ−1 is given by convolution on Nwith a homogeneous kernel K(·) having a weak singularity Nr at the origin. Hypoellipticity of Lˆshows that K is smooth away from the origin. Thus we understand the equation Lˆu = f very well.

We want to use Lˆto approximate L˜at each point y ∈ U˜. To do so, we have to identify a neighborhood of y in U˜with a neighborhood of N ˆthe ˜origin in Nr. This has to be done just right, or else Lwill fail to approximate L. The idea is to use exponential coordinates on both U ˜and N. Thus, if $x=exp( Σ α∈A t α Y α )(identity)∈ N Nr$then we use$( t α ) α∈A$ (as coordinates for x. Similarly, let ( Xbe the com-Nrtα‎)α‎∈A α‎)α‎∈A ˜˜˜ mutators of X· · · Xanalogous to the , and let y ∈ Ube given. Then given a nearby point$x=exp( Σ α∈A t α X ˜ α )y∈ U ˜$ we use$( t α ) α∈A$as coordinates for x.

Now we can identify Uwith a neighborhood of the identity in NNr, simply by identifying points with the same coordinates. Denote the identification by $Θ y : U ˜ → N Nr$ and note that y(y) = identity.

In view of the identification Θ‎ y , the operators Lˆand L˜live on the same space. The next step is to see that they are approximately equal. To formulate this, we need some bookkeeping on the nilpotent group $N Nr Let { δ t } t>0$ be the natural dilations ∞Nrt> on NNr. If $φ∈ C 0 ∞ ( N Nr )$then write ϕ‎t for the function $x↦φ( δ t x).$ When ϕ‎ is fixed and t is large, then ϕ‎t is supported in a tiny neighborhood of the identity. Let D be a differential operator acting on functions on N. We say that D has Nr “degree” at most k if for each $φ∈ C 0 ∞ ( N Nr )$ we have $|D( φ t )|=O( t k )$ for large, positive t . According to this definitions, Y1, … YN have degree 1 while [Yj, Yk] has degree 2, and the degree of a(x)[Yj, Yk] depends on the behavior of a(x) near the identify. Now we can say in what sense L˜and Lˆare approximately equal. The crucial result is as follows.

Theorem B. Under the map y 1 , the vector fieldj pulls back to Yj + Zy,j , where Zy,j is a vector field on NNr of “degree” ≤ 0.

Using Theorem B and the map y , we can produce a parametrix for L˜and prove that it works. In fact, we take

(31)
$Display mathematics$

(p.21) where K is the fundamental solution of $L ^$. For fixed y, we want to know that

(32)
$Display mathematics$

where δ‎y(·) is the Dirac delta-function and E(x, y) has only a weak singularity at x = y. To prove this, we use $Θ y$ to pull back to NNr. Recall that $L ˜ = Σ 1 N X ˜ j 2$ while $L ^ = Σ 1 N Y j 2$ . Hence by Theorem B, $L ˜$ pulls back to an operator of the form $L ^ + D y$ , j with Dy having “degree” at most 1. Therefore (33) reduces to proving that

(33)
$Display mathematics$

where $E ^$ has only a weak singularity at the identity. Since $L ^ K(x)= δ id. (x)$, (34) means simply that DyK(x) has only a weak singularity at the identity. However, this is obvious from the smoothness and homogeneity if K(x), and from the fact that Dy has degree ≤ 1. Thus, $K ˜ (x,y)$ is an approximate fundamental solution for $L ˜$.

From the explicit fundamental solution for the lifted operator $L ˜$, one can “descend” to deal with the original Hörmander operator L in two different ways.

1. a. Prove sharp estimates for the lifted problems, then specialize to the case of functions that don’t depend on the extra variables.

2. b. Integrate out the extra variables from the fundamental solution for $L ˜$, to obtain a fundamental solution for L.

Rothschild and Stein used the first approach. They succeeded in proving the estimate

(34)
$Display mathematics$

This is the most natural and the sharpest estimate for Hörmander operators. It was new even for p = 2. Rothschild and Stein also proved sharp estimates in spaces analogous to the α‎ and Sm,p of Folland-Stein [67], as well as in standard Lipschitz and Sobolev spaces. We omit the details, but we point out that commuting derivatives past a general Hörmander operator here requires additional ideas.

Later, Nagel, Stein, and Wainger [119] returned to the second approach (“b” above) and were able to estimate the fundamental solution of a general Hörmander operator. This work overcomes substantial problems.

In fact, once we descend from the lifted problem to the original equation, we again face the difficulty that Hörmander operators cannot be modelled directly on nilpotent Lie groups. So it isn’t even clear how to state a theorem on the fundamental solution of a Hörmander operator. Nagel, Stein, and Wainger [119], realized that a family of non-Euclidean “balls” BL(x, ρ‎) associated to the Hörmander operator L plays the basic role. They defined the BL(x, ρ‎) and proved their essential properties. In particular, they saw that the family of balls survives the projection from the lifted problem back to the original equation, even though the nilpotent Liegroup (p.22) structure is destroyed. Non-Euclidean balls had already played an important part in Folland-Stein [67]. However, it was simple in [67] to guess the correct family of balls. For general Hörmander operators L the problem of defining and controlling non-Euclidean balls is much more subtle. Closely related results appear also in [FKP], [FS].

Let us look first at a nilpotent group such as NNr, with its family of dilations ${ δ t } t>0$. Then the correct family of non-Euclidean balls $B N Nr (x,ρ)$ is essentially Nr dictated by translation and dilation-invariance, starting with a more or less arbitrary harmless “unit ball” BNNr (identity, 1). Recall that the fundamental solution for $L ^ = ∑ 1 N Y j 2$ is given by a kernel K(x) homogeneous with respect to the δ‎t . Estimates that capture the size and smoothness of K(x) may be phrased entirely in terms of the non-Euclidean balls $B N Nr (x,ρ)$. In fact, the basic estimate is as follows.

(35)
$Display mathematics$

Next we associate non-Euclidean balls to a general Hörmander operator. For simplicity, take $L= ∑ 1 N X j 2$ as in our discussion of Rothschild-Stein [72]. One definition of the balls BL(x, ρ‎) involves a moving particle that starts at x and travels along the integral curve of Xj1 for time t1. From its new position x the particle then travels along the integral curve of Xj2 for time t2. Repeating the process finitely many times, we can move the particle from its initial position x to a final position y in a total time $t= t 1 +⋯+ t m$. The ball BL(x, ρ‎) consists of all y that can be reached in this way in time t < ρ‎. For instance, if L is elliptic, then BL(x, ρ‎) is essentially the ordinary (Euclidean) ball about x of radius ρ‎. If we take $L ^ = ∑ 1 N Y j 2$ on NN,r , then the balls $B L ^ (x,ρ)$ behave naturally under translations and dilations; hence they are essentially the same as the $B N N,r (x,ρ)$ appearing in (36). Nagel-Wainger-Stein analyzed the relations between $B L ^ (x,ρ), B L ˜ (x,ρ)and B L (x,ρ)$ for an arbitrary Hörmander operator L. (Here $L ˜$ and $L ^$ are as in our previous discussion of Rothschild-Stein.) This allowed them to integrate out the extra variables in the fundamental solution of $L ˜$, to derive the following sharp estimates from (36).

Theorem. Suppose X1 · · ·XN and their repeated commutators span the tangent space. Also, suppose we are in dimension greater than 2. Then the solution of (Xj 2)u = f is given by u(x) = K(x, y)f (y)dy with

(36)
$Display mathematics$

Here the Xji act either in the x- or the y- variable.

Let us return from Hörmander operators to the $∂ ¯$-problems on strongly pseudo-convex domains D ⊂ Cn. Greiner and Stein derived sharp estimates for the Neumann Laplacian w = α‎ in their book [78]. This problem is hard, because twodifferent (p.23) families of balls play an important role. On the one hand, the standard (Euclidean) balls arise here, because is simply the Laplacian in the interior of D. On the other hand, non-Euclidean balls (as in Folland-Stein [67]) arise on ∂D, because they are adapted to the non-elliptic boundary Conditions for . Thus, any understanding of requires notions that are natural with respect to either family of balls. A key notion is that of an allowable vector field on $D ¯$. We say that a smooth vector field X is allowable if its restriction to the boundry ∂D lies in the span of the complex vector fields $L 1 ,…, L n−1 , L ¯ 1 … L ¯ n−1$. Here we have retained the Ln¯−1 n− notation of our earlier discussion of $∂ ¯$-problems. At an interior point, an allowable vector field may point in any direction, but at a boundary point it must be in the natural codimension-one subspace of the tangent space of ∂D. Allowable vector fields are well-suited both to the Euclidean and the Heisenberg balls that control . The sharp estimates of Greiner-Stein are as follows.

Theorem. Suppose w = α‎ on a strictly pseudoconvex domain D ⊂ Cn. If α‎ belongs to the Sobolev space $L k p$, then w belongs to $L k−1 p (1. Moreover, k if X and Y are allowable vector fields, then XYw belongs to $L k p$. Also, $L ¯ w$ belongs k to L$L k+1 p$ if $L ¯$is a smooth complex vector field of type (0, 1). Similarly, if α‎ belongs to the Lipschitz space Lip(β‎) (0 < β‎ < 1), then the gradient of w belongs to Lip(β‎) as well. Also the gradient of $L ¯ w$ belongs to Lip(β‎) if L is a smooth complex vector field of type (0, 1); and XYw belongs to Lip(β‎) for X and Y allowable vector fields.

These results for allowable vector fields were new even for L2. We sketch the proof.

Suppose w = α‎. Ignoring the boundary conditions for a moment, we have w = α‎ in D, so

$Display mathematics$

where $w ˜$ is defined on ∂D, and G, P.I. denote the standard Green’s operator and Poisson integral, respectively. The trouble with (37) is that we know nothing about so far. The next step is to bring in the boundary condition for w = α‎. According to Calderón’s work on general boundary-value problems, (37) satisfies the $∂ ¯$-Neumann boundary conditions if and only if

(37)
$Display mathematics$

for a certain differential operater B on D, and a certain pseudodifferential operator A on ∂D. Both A and B can be determined explicitly from routine computation.

Greiner and Stein [78] derive sharp regularity theorems for the pseudodifferential equation $A w ˜ =g$, and then apply those results to (38) in order to understand w˜in terms of α‎. Once they know sharp regularity theorems for w˜, formula (37) gives the behavior of w.

Let us sketch how Greiner-Stein analyzed $A w ˜ =g$. This is really a system of n pseudodifferential equations for n unknown functions (n = dimCn). In a suitable frame, one component of the system decouples from the rest of the problem (modulo negligible errors) and leads to a trivial (elliptic) pseudodifferential equation. (p.24) The non-trivial part of the problem is a first-order system of (n − 1) pseudodifferential operators for (n − 1) unknowns, which we write as

(38)
$Display mathematics$

Here α‎# consists of the non-trivial components of {B(Gα‎)}|∂D,w# is the unknown, and + may be computed explicitly.

Greiner and Stein reduce (39) to the study of the Kohn-Laplacian b. In fact, they produce a matrix of first-order pseudodifferential operators similar to +, and then show that −+ = b modulo negligible errors.2 Applying to (39) yields

(39)
$Display mathematics$

From Folland-Stein [67] one knows an explicit integral operator K that inverts b modulo negligible errors. Therefore,

(40)
$Display mathematics$

Equations (37) and (41) express w in terms of α‎ as a composition of various explicit operators, including: the Poisson integral; restriction to the boundary; ; K; G. Because the basic notion of allowable vector fields is well-behaved with respect to both the natural families of balls for w = α‎, one can follow the effect of each of these very different operators on the relevant function spaces without losing information. To carry this out is a big job. We refer the reader to [78] for the rest of the story.

There have been important recent developments in the Stein program for several complex variables. In particular, we refer the reader to Phong’s paper in Essays in Fourier Analysis in Honor of Elias M. Stein (Princeton University Press, 1995) for a discussion of singular Radon transforms; and to Nagel-Rosay-Stein-Wainger [131], D.-C. Chang-Nagel-Stein [132], and [McN], [Chr], [FK] for the solution of the $∂ ¯$-problems on weakly pseudoconvex domains of finite type in C2.

Particularly in several complex variables we are able to see in retrospect the fundamental interconnections among classical analysis, representation theory, and partial differential equations, which Stein was the first to perceive.

I hope this chapter has conveyed to the reader the order of magnitude of Stein’s work. However, let me stress that it is only a selection, picking out results which I could understand and easily explain. Stein has made deep contributions to many other topics, e.g.,

• Limits of sequences of operators

• Extension of Littlewood-Paley Theory from the disc to Rn

• Differentiability of functions on sets of positive measure

• Fourier analysis on RN when N →∞

• Function theory on tube domains

• Analysis of diffusion semigroups

• Pseudodifferential calculus for subelliptic problems.

• The list continues to grow.

(p.25) Bibliography of E.M. Stein

Bibliography references:

[1] “Interpolation of linear operators.” Trans. Amer. Math. Soc. 83 (1956), 482–492.

[2] “Functions of exponential type.” Ann. of Math. 65 (1957), 582–592.

[3] “Interpolation in polynomial classes and Markoff’s inequality.” Duke Math. J. 24 (1957), 467–476.

[4] “Note on singular integrals.” Proc. Amer. Math. Soc. 8 (1957), 250–254.

[5] (with G.Weiss) “On the interpolation of analytic families of operators acting on Hp spaces.” Tohoku Math. J. 9 (1957), 318–339.

[6] (with E. H. Ortrow) “A generalization of lemmas of Marcinkiewicz and Fine with applications to singular integrals.” Annali Scuola Normale Superiore Pisa 11 (1957), 117–135.

[7] “A maximal function with applications to Fourier series.” Ann. of Math. 68 (1958), 584–603.

[8] (with G. Weiss) “Fractional integrals on n-dimensional Euclidean space.” J. Math. Mech. 77 (1958), 503–514.

[9] (with G.Weiss) “Interpolation of operators with change of measures.” Trans. Amer. Math. Soc. 87 (1958), 159–172.

[10] “Localization and summability of multiple Fourier series.” Acta Math. 100 (1958), 93–147.

[11] “On the functions of Littlewood-Paley, Lusin, Marcinkiewicz.” Trans. Amer. Math. Soc. 88 (1958), 430–466.

[12] (with G. Weiss) “An extension of a theorem of Marcinkiewicz and some of its applications.” J. Math. Mech. 8 (1959), 263–284.

[13] (with G. Weiss) “On the theory of harmonic functions of several variables I, The theory of Hp spaces.” Acta Math. 103 (1960), 25–62.

[14] (with R. A. Kunze) “Uniformly bounded representations and harmonic analysis of the 2 × 2 real unimodular group.” Amer. J. Math. 82 (1960), 1–62.

[15] “The characterization of functions arising as potentials.” Bull. Amer. Math. Soc. 67 (1961), 102–104; II, 68 (1962), 577–582.

[16] “On some functions of Littlewood-Paley and Zygmund.” Bull. Amer. Math. Soc. 67 (1961), 99–101.

[17] “On limits of sequences of operators.” Ann. of Math. 74 (1961), 140–170.

(p.26) [18] “On the theory of harmonic functions of several variables II. Behavior near the boundary.” Acta Math. 106 (1961), 137–174.

[19] “On certain exponential sums arising in multiple Fourier series.” Ann. of Math. 73 (1961), 87–109.

[20] (with R. A. Kunze) “Analytic continuation of the principal series.” Bull. Amer. Math. Soc. 67 (1961), 543–546.

[21] “On the maximal ergodic theorem.” Proc. Nat. Acad. Sci. 47 (1961), 1894– 1897.

[22] (with R. A. Kunze) “Uniformly bounded representations II. Analytic continuation of the principal series of representations of the n × n complex unimodular groups.” Amer J. Math. 83 (1961), 723–786.

[23] (with A. Zygmund) “Smoothness and differentiability of functions.” Ann. Sci. Univ. Budapest, Sectio Math., III-IV (1960–61), 295–307.

[24] “Conjugate harmonic functions in several variables.” Proceedings of the International Congress of Mathematicians, Djursholm-Linden, Institut Mittag-Leffler (1963), 414–420.

[25] (with A. Zygmund) “On the differentiability of functions.” Studia Math. 23 (1964), 248–283.

[26] (with G. and M. Weiss) “Hp-classes of holomorphic functions in tube domains.” Proc. Nat. Acad. Sci. 52 (1964), 1035–1039.

[27] (with B. Muckenhoupt) “Classical expansions and their relations to conjugate functions.” Trans. Amer. Math. Soc. 118 (1965), 17–92.

[28] “Note on the boundary values of holomorphic functions.” Ann. of Math. 82 (1965), 351–353.

[29] (with S. Wainger) “Analytic properties of expansions and some variants of Parseval-Plancherel formulas.” Arkiv. Math., Band 5 37 (1965), 553–567.

[30] (with A. Zygmund) “On the fractional differentiability of functions.” London Math. Soc. Proc. 34A (1965), 249–264.

[31] “Classes H2, multiplicateurs, et fonctions de Littlewood-Paley.” Comptes Rendues Acad. Sci. Paris 263 (1966), 716–719; 780–781; also 264 (1967), 107–108.

[32] (with R. Kunze) “Uniformly bounded representations III. Intertwining operators.” Amer. J. Math. 89 (1967), 385–442.

[33] “Singular integrals, harmonic functions and differentiability properties of functions of several variables.” Proc. Symp. Pure Math. 10 (1967), 316–335.

(p.27) [34] “Analysis in matrix spaces and some new representations of SL(N,C).” Ann. of Math. 86 (1967), 461–490.

[35] (with A. Zygmund) “Boundedness of translation invariant operators in Hölder spaces and Lp spaces.” Ann. of Math. 85 (1967), 337–349.

[36] “Harmonic functions and Fatou’s theorem.” In Proceeding of the C.I.M.E. Summer Course on Homogeneous Bounded Domains, Cremonese, 1968.

[37] (with A. Koranyi) “Fatou’s theorem for generalized halfplanes.” Annali di Pisa 22 (1968), 107–112.

[38] (with G. Weiss) “Generalizations of the Cauchy-Riemann equations and representations of the rotation group.” Amer. J. Math. 90 (1968), 163–196.

[39] (with A. Grossman and G. Loupias) “An algebra of pseudodifferential operators and quantum mechanics in phase space.” Ann. Inst. Fourier, Grenoble 18 (1968), 343–368.

[40] (with N. J. Weiss) “Convergence of Poisson integrals for bounded symmetric domains.” Proc. Nat. Acad. Sci. 60 (1968), 1160–1162.

[41] “Note on the class LlogL.” Studia Math. 32 (1969), 305–310.

[42] (with A. W. Knapp) “Singular integrals and the principal series.” Proc. Nat. Acad. Sci. 63 (1969), 281–284.

[43] (with N. J. Weiss) “On the convergence of Poisson integrals.” Trans. Amer. Math. Soc. 140 (1969), 35–54.

[44] Singular integrals and differentiability properties of functions. Princeton Mathematical Series, 30. Princeton University Press, 1970.

[45] (with A. W. Knapp) “The existence of complementary series.” In Problems in Analysis. Princeton University Press, 1970.

[46] Topics in harmonic analysis related to the Littlewood-Paley theory. Annals of Mathematics Studies, 103. Princeton University Press, 1970.

[47] “Analytic continuation of group representations.” Adv. Math. 4 (1970), 172–207.

[48] “Boundary values of holomorphic functions.” Bull. Amer. Math. Soc. 76 (1970), 1292–1296.

[49] (with A. W. Knapp) “Singular integrals and the principal series II.” Proc. Nat. Acad. Sci. 66 (1970), 13–17.

[50] (with S. Wainger) “The estimating of an integral arising in multipier transformations.” Studia Math. 35 (1970), 101–104.

(p.28) [51] (with G. Weiss) Introduction to Fourier analysis on Euclidean spaces. Princeton University press, 1971.

[52] (with A. Knapp) “Intertwining operators for semi-simple groups.” Ann. of Math. 93 (1971), 489–578.

[53] (with C. Fefferman) “Some maximal inequalities.” Amer. J. Math. 93 (1971), 107–115.

[54] “Lp boundedness of certain convolution operators.” Bull. Amer. Math. Soc. 77 (1971), 404–405.

[55] “Some problems in harmonic analysis suggested by symmetric spaces and semi-simple groups.” Proceedings of the International Congress of Mathematicians, Paris: Gauthier-Villers 1 (1971), 173–189.

[56] “Boundary behavior of holomorphic functions of several complex variables.” Princeton Mathematical Notes. Princeton University Press, 1972.

[57] (with A. Knapp) Irreducibility theorems for the principal series. (Conference on Harmonic Analysis, Maryland) Lecture Notes in Mathematics, No. 266. Springer Verlag, 1972.

[58] (with A. Koranyi) “H2 spaces of generalized half-planes.” Studia Math. XLIV (1972), 379–388.

[59] (with C. Fefferman) “Hp spaces of several variables.” Acta Math. 129 (1972), 137–193.

[60] “Singular integrals and estimates for the Cauchy-Riemann equations.” Bull. Amer. Math. Soc. 79 (1973), 440–445.

[61] “Singular integrals related to nilpotent groups and ¯estimates.” Proc. Symp. Pure Math. 26 (1973), 363–367.

[62] (with R. Kunze) “Uniformly bounded representations IV. Analytic continuation of the principal series for complex classical groups of types Bn, Cn, Dn.” Adv. Math. 11 (1973), 1–71.

[63] (with G. B. Folland) “Parametrices and estimates for the ¯∂b complex on strongly pseudoconvex boundaries.” Bull. Amer. Math. Soc. 80 (1974), 253–258.

[64] (with J. L. Clerc) “Lp multipliers for non-compact symmetric spaces.” Proc. Nat. Acad. Sci. 71 (1974), 3911–3912.

[65] (with A. Knapp) “Singular integrals and the principal series III.” Proc. Nat. Acad. Sci. 71 (1974), 4622–4624.

[66] (with G. B. Folland) “Estimates for the ¯∂b complex and analysis on the Heisenberg group.” Comm. Pure and Appl. Math. 27 (1974), 429–522.

(p.29) [67] “Singular integrals, old and new.” In Colloquium Lectures of the 79th Summer Meeting of the American Mathematical Society, August 18–22, 1975. American Mathematical Society, 1975.

[68] “Necessary and sufficient conditions for the solvability of the Lewy equation.” Proc. Nat. Acad. Sci. 72 (1975), 3287–3289.

[69] “Singular integrals and the principal series IV.” Proc. Nat. Acad. Sci. 72 (1975), 2459–2461.

[70] “Singular integral operators and nilpotent groups.” In Proceedings of the C.I.M.E., Differential Operators on Manifolds. Edizioni, Cremonese, 1975: 148–206.

[71] (with L. P. Rothschild) “Hypoelliptic differential operators and nilpotent groups.” Acta Math. 137 (1976), 247–320.

[72] (with A. W. Knapp) “Intertwining operators for SL(n, r).” Studies in Math. Physics. E. Lieb, B. Simon and A. Wightman, eds. Princeton University Press, 1976: 239–267.

[73] (with S. Wainger) “Maximal functions associated to smooth curves.” Proc. Nat. Acad. Sci. 73 (1976), 4295–4296.

[74] “Maximal functions: Homogeneous curves.” Proc. Nat. Acad. Sci. 73 (1976), 2176–2177.

[75] “Maximal functions: Poisson integrals on symmetric spaces.” Proc. Nat. Acad. Sci. 73 (1976), 2547–2549.

[76] “Maximal functions: Spherical means.” Proc. Nat. Acad. Sci. 73 (1976), 2174–2175.

[77] (with P. Greiner) “Estimates for the ¯∂-Neumann problem.” Mathematical Notes 19. Princeton University Press, 1977.

[78] (with D. H. Phong) “Estimates for the Bergman and Szegö projections.” Duke Math. J. 44 (1977), 695–704.

[79] (with N. Kerzman) “The Szegö kernels in terms of Cauchy-Fantappie kernels.” Duke Math. J. 45 (1978), 197–224.

[80] (with N. Kerzman) “The Cauchy kernels, the Szegö kernel and the Riemann mapping function.” Math. Ann. 236 (1978), 85–93.

[81] (with A. Nagel and S. Wainger) “Differentiation in lacunary directions.” Proc. Nat. Acad. Sci. 73 (1978), 1060–1062.

[82] (with A. Nagel) “A new class of pseudo-differential operators.” Proc. Nat. Acad. Sci. 73 (1978), 582–585.

(p.30) [83] (with N. Kerzman) “The Szegö kernel in terms of the Cauchy-Fantappie kernels.” In Proceeding of the Conference on Several Complex Variables, Cortona, 1977. 1978.

[84] (with P. Greiner) “On the solvability of some differential operators of the type b.” In Proceedings of the Conference on Several Complex Variables, Cortona, 1978.

[85] (with S. Wainger) “Problems in harmonic analysis related to curvature.” Bull. Amer. Math. Soc. 84 (1978), 1239–1295.

[86] (with R. Gundy) “Hp theory for the poly-disc.” Proc. Nat. Acad. Sci. 76 (1979), 1026–1029.

[87] “Some problems in harmonic analysis.” Proc. Symp. Pure and Appl. Math. 35 (1979), Part I, 3–20.

[88] (with A. Nagel and S. Wainger) “Hilbert transforms and maximal functions related to variable curves.” Proc. Symp. Pure and Appl. Math. 35 (1979), Part I, 95–98.

[89] (with A. Nagel) “Some new classes of pseudo-differential operators.” Proc. Symp. Pure and Appl. Math. 35 (1979), Part II, 159–170.

[90] “A variant of the area integral.” Bull. Sci. Math. 103 (1979), 446–461.

[91] (with A. Nagel) “Lectures on pseudo-differential operators: Regularity theorems and applications to non-elliptic problems.” Mathematical Notes 24. Princeton University Press, 1979.

[92] (with A. Knapp) “Intertwining operators for semi-simple groups II.” Invent. Math. 60 (1980), 9–84.

[93] “The differentiability of functions in Rn.” Ann. of Math. 113 (1981), 383– 385.

[94] “Compositions of pseudo-differential operators.” In Proceedings of Journées Equations aux derivées partielles, Saint-Jean de Monts, Juin 1981, Sociéte Math. de France, Conférence #5, 1–6.

[95] (with A. Nagel and S. Wainger) “Boundary behavior of functions holomorphic in domains of finite type.” Proc. Nat. Acad. Sci. 78 (1981), 6596–6599.

[96] (with A. Knapp) “Some new intertwining operators for semi-simple groups.” In Non-commutative harmonic analysis on Lie groups, Colloq. Marseille-Luminy, 1981. Lecture Notes in Mathematics, no. 880. Springer Verlag, 1981.

[97] (with M. H. Taibleson and G. Weiss) “Weak type estimates for maximal operators on certain Hp classes.” Rendiconti Circ. mat. Pelermo, Suppl. n. 1 (1981), 81–97.

(p.31) [98] (with D. H. Phong) “Some further classes of pseudo-differential and singular integral operators arising in boundary value problems, I, Composition of operators.” Amer. J. Math. 104 (1982), 141–172.

[99] (with D. Geller) “Singular convolution operators on the Heisenberg group.” Bull. Amer. Math. Soc. 6 (1982), 99–103.

[100] (with R. Fefferman) “Singular integrals in product spaces.” Adv. Math. 45 (1982), 117–143.

[101] (with G. B. Folland) “Hardy spaces on homogeneous groups.” Mathematical Notes 28. Princeton University Press, 1982.

[102] “The development of square functions in the work of A. Zygmund.” Bull. Amer. Math. Soc. 7 (1982).

[103] (with D. M. Oberlin) “Mapping properties of the Radon transform.” Indiana Univ. Math. J. 31 (1982), 641–650.

[104] “An example on the Heisenberg group related to the Lewy operator.” Invent. Math. 69 (1982), 209–216.

[105] (with R. Fefferman, R. Gundy, and M. Silverstein) “Inequalities for ratios of functionals of harmonic functions.” Proc. Nat. Acad. Sci. 79 (1982), 7958– 7960.

[106] (with D. H. Phong) “Singular integrals with kernels of mixed homogeneities.” (Conference in Harmonic Analysis in honor of Antoni Zygmund, Chicago, 1981), W. Beckner, A. Calderón, R. Fefferman, P. Jones, eds. Wadsworth, 1983.

[107] “Some results in harmonic analysis in Rn, for n → ∞.” Bull. Amer. Math. Soc. 9 (1983), 71–73.

[108] “An H1 function with non-summable Fourier expansion.” In Proceedings of the Conference in Harmonic Analysis, Cortona, Italy, 1982. Lecture Notes in Mathematics, no. 992. Springer Verlag, 1983.

[109] (with R. R. Coifman and Y. Meyer) “Un nouvel espace fonctionel adapté a l’étude des opérateurs définis pour des intégrales singulières.” In Proceedings of the Conference in Harmonic Analysis, Cortona, Italy, 1982. Lecture Notes in Mathematics, no. 992. Springer Verlag, 1983.

[110] “Boundary behaviour of harmonic functions on symmetric spaces: Maximal estimates for Poisson integrals,” Invent. Math. 74 (1983), 63–83.

[111] (with J. O. Stromberg) “Behavior of maximal functions in Rn for large n.” Arkiv f. math. 21 (1983), 259–269.

[112] (with D. H. Phong) “Singular integrals related to the Radon transform and boundary value problems.” Proc. Nat. Acad. Sci. 80 (1983), 7697–7701.

(p.32) [113] (with D. Geller) “Estimates for singular convolution operators on the Heisenberg group.” Math. Ann. 267 (1984), 1–15.

[114] (with A. Nagel) “On certain maximal functions and approach regions.” Adv. Math. 54 (1984), 83–106.

[115] (with R. R. Coifman and Y. Meyer) “Some new function spaces and their applications to harmonic analysis.” J. Funct. Anal. 62 (1985), 304–335.

[116] “Three variations on the theme of maximal functions.” (Proceedings of the Seminar on Fourier Analysis, El Escorial, 1983) Recent Progress in Fourier Analysis. I. Peral and J. L. Rubio de Francia, eds.

[117] Appendix to the paper “Unique continuation… .” Ann. of Math. 121 (1985), 489–494.

[118] (with A. Nagel and S. Wainger) “Balls and metrics defined by vector fields I: Basic properties.” Acta Math. 155 (1985), 103–147.

[119] (with C. Sogge) “Averages of functions over hypersurfaces.” Invent. Math. 82 (1985), 543–556.

[120] “Oscillatory integrals in Fourier analysis.” In Beijing lectures on Harmonic analysis. Annals of Mathematics Studies, 112. Princeton University Press, 1986.

[121] (with D. H. Phong) “Hilbert integrals, singular integrals and Radon transforms II.” Invent. Math. 86 (1986), 75–113.

[122] (with F. Ricci) “Oscillatory singular integrals and harmonic analysis on nilpotent groups.” Proc. Nat. Acad. Sci. 83 (1986), 1–3.

[123] (with F. Ricci) “Homogeneous distributions on spaces of Hermitian matricies.” Jour. Reine Angw. Math. 368 (1986), 142–164.

[124] (with D. H. Phong) “Hilbert integrals, singular integrals and Radon transforms I.” Acta Math. 157 (1986), 99–157.

[125] (with C. D. Sogge) “Averages over hypersurfaces: II.” Invent. math. 86 (1986), 233–242.

[126] (with M. Christ) “A remark on singular Calderón-Zygmund theory.” Proc. Amer. Math. Soc. 99, 1 (1987), 71–75.

[127] “Problems in harmonic analysis related to curvature and oscillatory integrals.” Proc. Int. Congress of Math., Berkeley 1 (1987), 196–221.

[128] (with F. Ricci) “Harmonic analysis on nilpotent groups and singular integrals I.” J. Funct. Anal. 73 (1987), 179–194.

[129] (with F. Ricci) “Harmonic analysis on nilpotent groups and singular integrals II.” J. Funct. Anal. 78 (1988), 56–84.

(p.33) [130] (with A. Nagel, J. P. Rosay, and S.Wainger) “Estimates for the Bergman and Szegö kernels in certain weakly pseudo-convex domains.” Bull. Amer. Math. Soc. 18 (1988), 55–59.

[131] (with A. Nagel and D. C. Chang) “Estimates for the ¯∂-Neumann problem for pseudoconvex domains in C2 of finite type.” Proc. Nat. Acad. Sci. 85 (1988), 8771–8774.

[132] (with A. Nagel, J. P. Rosay, and S. Wainger) “Estimates for the Bergman and Szegö kernels in C2.” Ann. of Math. 128 (1989), 113–149.

[133] (with D. H. Phong) “Singular Radon transforms and oscillatory integrals.” Duke Math. J. 58 (1989), 347–369.

[134] (with F. Ricci) “Harmonic analysis on nilpotent groups and singular integrals III.” J. Funct. Anal. 86 (1989), 360–389.

[135] (with A. Nagel and F. Ricci) “Fundamental solutions and harmonic analysis on nilpotent groups.” Bull. Amer. Math. Soc. 23 (1990), 139–143.

[136] (with A. Nagel and F. Ricci) “Harmonic analysis and fundamental solutions on nilpotent Lie groups in Analysis and P.D.E.” A collection of papers dedicated to Mischa Cotlar. Marcel Decker, 1990.

[137] (with C. D. Sogge) “Averages over hypersurfaces, smoothness of generalized Radon transforms.” J. d’ Anal. Math. 54 (1990), 165–188.

[138] (with S. Sahi) “Analysis in matrix space and Speh’s representations.” Invent. Math. 101 (1990), 373–393.

[139] (with S. Wainger) “Discrete analogues of singular Radon transforms.” Bull. Amer. Math. Soc. 23 (1990), 537–544.

[140] (with D. H. Phong) “Radon transforms and torsion.” Duke Math. J. (Int. Math. Res. Notices) #4 (1991), 44–60.

[141] (with A. Seeger and C. Sogge) “Regularity properties of Fourier integral operators.” Ann. of Math. 134 (1991), 231–251.

[142] (with J. Stein) “Stock price distributions with stochastic volatility: an analytic approach.” Rev. Fin. Stud. 4 (1991), 727–752.

[143] (with D. C. Chang and S. Krantz) “Hardy spaces and elliptic boundary value problems.” In the Madison Symposium on Complex Analysis, Contemp. Math. 137 (1992), 119–131.

(p.34) Other References

Bibliography references:

[BGS] D. Burkholder, R. Gundy, and M. Silverstein. “A maximal function characterization of the class Hp.” Trans. Amer. Math. Soc. 157 (1971): 137–153.

[Ca] A. P. Calderón. “Commutators of singular integral operators.” Proc. Nat. Acad. Sci. 53 (1965): 1092–1099.

[Chr] M. Christ. “On the ¯∂b-equation and Szegö projection on a CR manifold.” In Proceedings, El Escorial Conference on Harmonic Analysis 1987. Lecture Notes in Mathematics, no. 1384. Springer Verlag, 1987.

[Co] M. Cotlar. “A unified theory of Hilbert transforms and ergodic theory.” Rev. Mat. Cuyana I (1955): 105–167.

[CZ] A. P. Calderón and A. Zygmund. “On higher gradients of harmonic functions.” Studia Math. 26 (1964): 211–226.

[EM] L. Ehrenpreis and F. Mautner. “Uniformly bounded representations of groups.” Proc. Nat. Acad. Sci. 41 (1955): 231–233.

[FK] C. Fefferman and J. J. Kohn. “Estimates of kernels on three-dimensional CR manifolds.” Rev. Mat. Iber. 4, no. 3 (1988): 355–405.

[FKP] C. Fefferman, J. J. Kohn, and D. Phong. “Subelliptic eigenvalue problems.” In Proceedings, Conference in Honor of Antoni Zygmund.Wadsworth, 1981.

[FS] C. Fefferman and A. Sanchez-Calle. “Fundamental solutions for second order subelliptic operators.” Ann. of Math. 124 (1986): 247–272.

[GN] I. M. Gelfand and M. A. Neumark. Unitäre Darstellungen der Klassischen Gruppen. Akademie Verlag, 1957.

[H] I. I. Hirschman, Jr. “Multiplier transformations I.” Duke Math. J. 26 (1956): 222–242; “Multiplier transformations II.” Duke Math. J. 28 (1961): 45–56.

[McN] J. McNeal, “Boundary behaviour of the Bergman kernel function in C2.” Duke Math. J. 58 (1989): 499–512.

[Z] A. Zygmund. Trigonometric Series. Cambridge University Press, 1959.

## Notes:

(*) Reprinted from Essays on Fourier Analysis in Honor of Elias M. Stein, Princeton University Press, 1995.

(1) Kohn’s work showed that bw = α‎ has a solution if we are in complex dimension > 2. In two complex dimensions, bw = α‎ has no solution for most α‎. We assume dimension > 2 here.

(2) This procedure requires significant changes in two complex variables, since then b isn’t invertible.