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#### Essays on Fourier Analysis in Honor of Elias M. Stein (PMS-42) ELIAS M. STEIN

**By Charles Fefferman, Robert Fefferman, Stephen Wainger**

**PRINCETON UNIVERSITY PRESS**

**Copyright © 1995 Princeton University Press**

All rights reserved.

ISBN: 978-0-691-08655-2

All rights reserved.

ISBN: 978-0-691-08655-2

CHAPTER 1

**Selected Theorems by Eli Stein**

*Charles Fefferman*

**INTRODUCTION**

The purpose of this survey article 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 perceive 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], [44], [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 article, 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

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

and the Hilbert transform

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

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

**COMPLEX INTERPOLATION**

Let us begin with Stein's work on interpolation of operators. As background, we state and prove a classical 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 [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] to [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], and from [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] to [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (Here, p0, p1, r0, r1 [member of] [1, ∞].) Then T is bounded for Lp (X) to Lr (Y) for 1/p = t/p1 + (1-t)/p0, 1/r = t/r1 + (1-t)/r0, 0 ≤ t ≤ 1.*

The Riesz Convexity Theorem says that the points (1/p , 1/r) for which *T* is bounded from Lp to Lr form 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 to L1 to L∞, and from L2 to L2. Therefore, *T* is bounded from Lp to the dual class for Lp' for ≤ p ≤ 2.

The idea of the proof of the Riesz Convexity Theorem is to estimate ∫Y (Tf) · g for f [member of] Lp and g [member of] Lr'. Say f = Feiφ and g = Geiψ with F, G ≥ 0 and φ, ψ real. Then we can define analytic families of functions fz, gz by setting fz = Faz + b+eiφ, gz = Gcz+deiψ, for real *a, b, c, d* to be picked in a moment.

Define

(1) Φ(z) = ∫Y (Tfz)gz.

Evidently, Φ is an analytic function of z.

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

(2) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] when Re z = 0;

(3) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] when Re z = 1;

(4) fz = f and gz = g when z = t.

From (2) we see that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]≤ C for Re z = 0. So the definition (1) and the assumption [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] show that

(5) |Φ (z)| ≤ C' for Re z = 0.

Similarly, (3) and the assumption [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] imply

(6) |Φ (z)| ≤ C' for Re z = 1.

Since Φ is analytic, (5) and (6) imply |Φ (z)| ≤ C' for 0 ≤ 1, 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 [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

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 operators on 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 [MATHEMATICAL NOT REPRODUCIBLE IN ASCII] when Re z = 0, and from [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] to [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] when Re z = 1. Then Tt is bounded from Lp to Lr, where 1/p = t/p1 + (1=t)/p0, 1/r = 1/r1 + (1-t)/r0 and 0 ≤ 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 fz)gz as in (1), we set Φ(z) = ∫Y (Tzfz)gz. 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

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Then TαR

(7)[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], if |1/p – ½| α/n – 1.

This follows immediately f r om 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 I**

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 = [+ or -] 1 and a real parameter *t*;

The *discrete series*, parametrized by a sign σ = [+ or -] and an integer k ≥ 0; and

The *complementary series*, parametrized by a real number t [element to] (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

[??](U) = ∫G f(g)Ugdg,

where *dg* denotes Haar measure on the group. Thus, [??] 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) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

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

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

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 Uστ of G, parametrized by a sign σ = [+ or -] 1 and a complex number τ in a strip Ω, with the following properties.*

**(A)***Uστ all act on the same Hilbert space H.*

(B) *For fixed σ = [+ or -]1, g [member of] G, and [xi], η [member of] H, the matrix element <(Uσ,τ)g[xi], η> is an analytic function of τ [member of] Ω.*

**(C)** The Uσ,τ for Re τ = ½ are equivalent to the representations of the principal series.

**(D)***The U+ 1,r 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 [xi] - exp(i[xi] · x) is a bounded analytic function on a strip |Im [xi]| ≤ 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 [??] (σ, τ) = ∫G f(g)(Uσ,τ)gdg for σ = [+ or -]1, τ [member of] Ω.

Thus, f [member of] L1(G) implies [??](σ,·) analytic and bounded on Ω. So we have continued analytically the restriction of [??] to the principal series. It is as if the Fourier transform of an L1 function on (-∞, ∞) were automatically analytic in a strip. If f [member of] L2(G), then [??] (σ, τ) is still defined on the line {Re τ = ½}, 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 [member of] Lp(G) (1 ≤ p 2) implies [??] (σ,·) analytic and satisfying an Lp' -inequality on a strip Ωp. As p increases from 1 to 2, the strip Ωp shrinks from Ω to the line {Re τ = ½}. Thus we obtain the following results.

**Corollary 1.***If [member of] Lp (G) (1 ≤ p 2), then [??] is bounded almost everywhere with respect to the Plancherel measure.*

**Corollary 2.**For 1 ≤ p 2 we have the convolution inequality [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

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 - [??](σ, τ) on the strip Ωp. Since an Lp'-function analytic on a strip Ωp is clearly bounded on an interior line {Re τ = ½}, it follows at once that [??] is bounded on the principal series. Regarding the discrete series Uσ,k we note that

**(9)** [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

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 [parallel][??](Uσ,k)[paralle] is immediate from (9). Thus the Fourier transform [??] is bounded on both the principal series and the discrete series, for f [member of] Lp(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 [absolute value of f] and [absolute value of 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 article. Now, however, we turn our attention to classical Fourier analysis.

*(Continues...)*

Excerpted fromEssays on Fourier Analysis in Honor of Elias M. Stein (PMS-42) ELIAS M. STEINbyCharles Fefferman, Robert Fefferman, Stephen Wainger. Copyright © 1995 Princeton University Press. Excerpted by permission of PRINCETON UNIVERSITY PRESS.

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