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ESSAYS ON FOURffiR ANALYSIS IN HONOR OF
ELIAS M. STEIN

Princeton Mathematical Series
EDITORS: LUIS A. CAFFARELLI, JOHN N. MATHER, and ELIAS M. STEIN
1. The Classical Groups by Hermann Weyl
3. An Introduction to Differential Geometry by Luther Pfahler Eisenhart
4. Dimension Theory by W. Hurewicz and H. Wallman
6. The Laplace Transform by D.V. Widder
7. Integration by Edward J. McShane
8. Theory of Lie Groups: I by C. Chevalley
9. Mathematical Methods of Statistics by Harald Cramer
10. Several Complex Variables by S. Bochner and W. T. Martin
11. Introduction to Topology by S. Lefschetz
12. Alegebraic Geometry and Topology edited by R. H. Fox, D. C. Spencer, and A. W.
Tucker
14. The Topology of Fibre Bundles by Norman Steenrod
15. Foundations of Algebraic Topology by Samuel Eilenberg and Norman Steenrod
16. Functionals of Finite Riemann Surfaces by Menahem Schiffer and Donald C. Spencer
17. Introduction to Mathematical Logic, Vol I by Alonzo Church
19. Homological Algebra by H. Cartan and S. Eilenberg
20. The Convolution Transform by 1.1. Hirschman and D. V. Widder
21. Geometric Integration Theory by H. Whitney
22. Qualitiative Theory of Differential Equations by V. V. Nemytskii and V. V. Stepanov
23. Topological Analysis by Gordon T. Whyburn (revised 1964)
24. Analytic Functions by Ahlfors, Behnke, Bers, Grauert et al.
25. Continuous Geometry by John von Neumann
26. Riemann Surfaces by L. Ahlfors and L. Sario
27. Differential and Combinatorial Topology edited by S. S. Cairns
28. Convex Analysis by R. T. Rockafellar
29. Global Analysis edited by D. C. Spencer and S. Iyanaga
30. Singular Integrals and Differentiability Properties of Functions by Ε. M Stein
31. Problems in Analysis edited by R. C. Gunning
32. Introduction to Fourier Analysis on Euclidean Spaces by E. M. Stein and G. Weiss
33. Etale Cohomology by J. S. Milne
34. Pseudodifferential Operators by Michael E. Taylor
36. Representation Theory of Semisimple Groups: An Overview Based on Examples by
Anthony W. Knapp
37. Foundations of Algebraic Analysis by Masaki Kashiwara, Takahiro Kawai, and Tatsuo
Kimura. Translated by Goro Kato
38. Spin Geometry by H. Blaine Lawson, Jr., and Marie-Louise Michelsohn
39. Topology of 4-Manifolds by Michael H. Freedman and Frank Quinn
40. Hypo-Analytic Structures: Local Theory by Frangois Treves
41. The Global Nonlinear Stability of the Minkowksi Space by Demetrios Christodoulou
and Sergiu Klainerman
42. Essays on Fourier Analysis in Honor of Elias M. Stein edited by C. Fefferman, R.
Fefferman, and S. Wainger

ESSAYS ON FOURIER ANALYSIS IN HONOR OF
ELIAS M. STEIN
EDITED BY
Charles Fefferman, Robert Fefferman
and Stephen Wainger
PRINCETON UNIVERSITY PRESS
PRINCETON, NEW JERSEY

Copyright © 1995 by Princeton University Press
Published by
Princeton University Press, 41 Williara Street,
Princeton, New Jersey 08540
In the United Kingdom: Princeton University Press, Chichester, West Sussex
All Rights Reserved
Library of Congress Catalogfng-In-Publication Data
Essays on Fourier Analysis in honor of Hlias M. Stein / edited by Charles
Fefferman, Robert Fefferman, and Stephen Wainger.
p. cm.—(Princeton mathematical series ; 42)
Proceedings of the Princeton Conference in Harmonic
Analysis,
held May 13-17, 1991.
Includes bibliographical references.
ISBN 0-691-08655-9 (aik. paper)
J. Fouria- analysis—Congresses. I. Stein, Hlias M., 1931-.
Π. Fefferman, Charles, 1949-. ΙΠ. Fefferman, Robert, 1951-. IV.
Wainger,
Stephen, 1936-. V. Series.
QA403.5.P76 1993
515'.2433—dc20 92-43054
ClP
Princeton University Press books are printed on acid-free paper
and meet the guidelines for permanence and durability of the Committee
on Production Guidelines for Book Longevity of
the Council
on Library Resources
Printed in the United States of America
10 987654321

PRINCETON UNIVERSITY LIBRARY PAIR>
32 01 268 599
CONTENTS
INTRODUCTION
CHARLES F
EFFERMAN, ROBERT FEFFERMAN, AND STEPHEN
WAINGER
ONE
Selected Theorems by Eli Stein · C
HARLES FEFFERMAN
TWO
Geometric Inequalities in Fourier Analysis · WILLIAM BECKNER 36
THREE
Representing Measures for Holomorphic Functions on Type 2 Wedges 1
A
L BOGGESS AND ALEXANDER NAGEL 69
FOUR
Some New Estimates on Oscillatory Integrals · J
EAN BOURGAIN 83
FTVE
Dilations Associated to Flat Curves in R"
· ANTHONY CARBERY,
JAMES VANCE, S
TEPHEN WAINGER, AND DAVID WATSON 113
SIX
Nonexistence of Invariant Analytic Hypoelliptic Differential Operators
on
Nilpotent Groups of Step Greater than Two · MICHAEL
CHRIST 127
SEVEN
Operateurs Bilineaires et Renormalisation · R. R. COIFMAN,
S.
DOBYINSKY, AND Y. M
EYER 146
EIGHT
Numerical Harmonic Analysis · R. R. COIFMAN, Y. MEYER, AND
V.
WLCKERHAUSER 162
NINE
Some Topics from Harmonic Analysis and Partial Differential Equations
• R
OBERT A. FEFFERMAN

VL CONTENTS
TEN
Function Spaces on Spaces of Homogeneous Type · YONGSHENG
HAN
AND GUIDO WEISS 211
ELEVEN
The First Nodal Set of a Convex Domain · DAVID
JERISON 225
TWELVE
On Removable Sets for Sobolev Spaces in the Plane · PETER
W. JONES 250
THIRTEEN
Oscillatory Integrals and Non-Linear Dispersive Equations · CARLOS
E.
KENIG 268
FOURTEEN
Singular Integrals and Fourier Integral Operators · D. Η. PHONG 286
FIFTEEN
Counterexamples with Harmonic Gradients in E3 · THOMAS
Η. WOLFF 321

INTRODUCTION
This is the proceedings of the Princeton Conference in Harmonic Analysis, which
was held from May 13 through May 17, 1991. This conference was a very special
event because it celebrated Elias M. Stein's extraordinary impact on
mathematics,
on the occasion of his sixtieth birthday. Through his brilliant research, books,
lectures, and teaching, E. M. Stein has changed
the way central problems in analysis
are approached. The articles presented here are a clear reflection
of this and a tribute
to his stellar contribution.
We would like to take this opportunity to express our gratitude to those who
made the conference possible. We thank the National Science Foundation for
funding and the Princeton Mathematics Department for both its hospitality and its
financial support. It is a pleasure to acknowledge the contributions of M. Christ and
M. Machedon, who provided us with financial assistance through their PYI grants.
Thanks go to J. Ryff for his many helpful suggestions and his
encouragement, and
to Scott Kenney and the staff of the Princeton Mathematics
Department for the
great amount of work they did to make things run smoothly. Finally, we wish to
thank the speakers and the many participants in the conference for their energy
and enthusiasm, which made possible its success.
Charles Fefferman
Robert Fefferman
Stephen
Wainger

ESSAYS ON FOURIER ANALYSIS IN HONOR
OF
ELIAS M. STEIN

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 impor
tance 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 rea
son. 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 L^-boundedness of two fundamental

2 CHAPTER 1
operators, the maximal function
and the Hilbert transform
Finally
, we shall be concerned with those problems in partial differential equations
that come from several complex variables.
COMPLE
X INTERPOLATION
Let us begin with Stein's work on interpolation of operators. As background, we
stat
e 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 , and from
Then
T is bounded from
The Riesz Convexity Theorem says that the points for which T is
bounded from to form a convex region in the plane. A standard appli-
cation is the Hausdorff-Young inequality: We take T to be the Fourier transform
on K", and note that T is obviously bounded from to , and from L2 to L2.
Therefore
, T is bounded from Lp to the dual class for
Th
e idea of the proof of the Riesz Convexity Theorem is to estimate • g
for and Say . ' and with and
real
. Then we can define analytic families of functions fz, gz by setting
, for real a, b, c,d to be picked in a moment.
Defin
e
(1)
Evidently, $ is an analytic function of z.
Fo
r the correct choice of a, b, c, d we have
andwhe
n
and when
an
d when

SELECTED THEOREMS BY ELI STEIN 3
From (2) we see that r for ReSo the definition (1)
and the assumption show that
(5)
Similarly
, (3) and the assumption imply
(6) for
Since is analytic, (5) and (6) imply for by the
maximu
m principle for a strip. In particular, In view of (4), this
mean
s that with C" determined by
Thus, T is bounded from LptoLr, and the proof of the Riesz Convexity Theorem
is complete.
Thi
s 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 real-
ize
d that an ingenious argument by Hirschman [H] on certain multiplier operators
on LP(M") could be viewed as a Riesz Convexity Theorem for analytic families
of operators. Here is the result.
Stei
n Interpolation Theorem. Assume Tz is an operator depending analytically
on z in the strip . Suppose Tz is bounded from when
Re z — 0, and from LPi to Z/' when Re z = 1. Then T, is bounded from Lp to
where
and
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
as in (1), we sel . The proof of the Riesz Convexity
Theore
m 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.
Th
e example concerns Cesaro summability of multiple Fourier integrals.
We define an operator TaR on functions on R" by setting
The
n
(7)
This follows immediately from the Stein Interpolation Theorem. We let a play
the role of the complex parameter z, and we interpolate between the elementary

4 CHAPTER 1
cases ρ = 1 and ρ — 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 = ±1 and a real parameter f,
The discrete series, parametrized by
a sign σ = ±1 and an integer k > 0; and
The complementary series, parametrized by a real number ί e (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 / is a
function on G, and U is an irreducible unitary representation of G, then we define
f(U) = f f(g)U
gdg,
JG
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) /Tg
= / - g
as operators. Moreover, there is a Plancherel formula
for G, which asserts that
IIZIli
2(G) = J 11/(^) Il Hilbert-Schmidti^ (^)
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 M". Kunze and Stein discovered a fundamental new phenomenon in
Fourier analysis on G that has no analogue on Mn. Their result is as follows.
Theorem
(Kunze-Stein Phenomenon). There exists a uniformly bounded rep
resentation Urjill of G, parametrized by a sign σ = ±1 and a complex number τ
in a strip Ω, with the following properties.
(A) The Ua r all act on the same Hilbert space H.

SELECTED THEOREMS BY ELl STEIN 5
(B) For fixed σ = ±1, g 6 G, and ξ, η e Η, the matrix element ((UaiT)g%, η)
is an analytic function of τ G Ω.
(C) The U
σ τ for Ke τ = are equivalent to the representations of the principal
series.
(D) The U+ i,
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 ξ ι—> exp(/f ·
x) is
a bounded analytic function on a strip | Im ξ | < 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, r) =
/c f(g)(Ua,t)gdg for σ = ±1, τ € Ω.
Thus, / € 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 (—oo, oo) were automatically analytic in
a strip. If / e L2(G), then f{o, τ) is still defined on
the line {Re τ = \},
by virtue of the Plancherel formula and part (C) of the Kunze-Stein Theorem.
Interpolating between L1(G) and L
2(G) using the Stein Interpolation Theorem,
we see that / € Lp(G) (1 < ρ < 2) implies /(σ, ·) analytic and satisfying an
Lp -inequality on a strip Ω^,. As ρ increases from 1 to 2, the strip Ω^ shrinks from
Ω to the line {Re τ = |Thus we obtain the following results.
Corollary
1. If f e Lp(G) (1
< ρ < 2), then f is bounded almost everywhere
with respect to the Plancherel measure.
Corollary
2. For 1 < ρ < 2 we have the convolution inequality * g\L2{G) <
cP\LnG)MmG)·
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 ι—• /(σ, r) on
the strip
Ωρ. Since an Lp -function analytic on a strip Ω,, is clearly bounded on an interior
line
{Re τ = \}, it follows at once that / is bounded on the principal series.
Regarding the discrete series UaJi we note that
(9) ^ "/I' LO(G)

6 CHAPTER 1
for suitable weights μσ * and for 1 < ρ < 2. The weights μσ± amount to the
Plancherel measure on the discrete series, and (9) is proved by a trivial interpola
tion, just like the standard Hausdorff-Young inequality. The boundedness of the
Il f (UcfJi)Il is immediate
from (9). Thus the Fourier transform / is bounded on
both the principal series and the discrete series, for / G Lp(G) (1 < ρ < 2).
The complementary series has measure zero with respect to the Plancherel mea
sure, 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 Corol
lary 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 —>- oo. 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 /, g is the same as proving it for |/| 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 discus
sion later in the article. 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 /Ir
of the Fourier transform of a function / e Lp(Mn) to a set Γ of
measure zero. If ρ > 1, then the standard inequality / € LP(M") suggests that
/ should
not even be well-defined on Γ, since Γ has measure zero. Indeed, if Γ
is (say) the x-axis in the plane M2, then
we can easily find functions f(x\, xj) =
φ(χ)·ψ(χι) € Lp(Μ
2) for which /| r is infinite everywhere. Fouriertransforms
of / e Lp(R2) clearly cannot be restricted to straight lines. Stein
proved that the
situation changes drastically when Γ is curved. His result is as follows.
Stein's Restriction Theorem. Suppose Γ is the unit circle, 1 < ρ < |, and
f £ Cqc(M2). Then we have the a priori inequality Wf \r\L2 < Cpll/IL·^),
with
Cp depending only on p.

SELECTED THEOREMS BY ELI STEIN 7
Using this a priori inequality, we can trivially pass from the dense subspace
t
o define the operator for all Thus, the Fourier transform
of may be restricted to the unit circle.
Improvement
s and generalizations were soon proven by other analysts, but it was
Stei
n who first demonstrated the phenomenon of restriction of Fourier transforms.
Stein'
s proof of his restriction theorem is amazingly simple. If ¡i denotes
unifor
m measure on the circle then for > we have
(10
)
The Fourier transform /¿(£) is a Bessel function. It decays like at infinity,
a
fact intimately connected with the curvature of the circle. In particular,
for and therefore , by
the usual elementary estimates for convolutions. Putting this estimate back into
(10)
, we see that 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 M".
Theorem
. Suppose with and . For and
denote the average of f on the sphere of radius r centered at
x. Then almost everywhere.
The point is that unlike the standard Lebesgue Theorem, we are averaging /
over a small sphere instead of a small ball. As in the restriction theorem, we are
seemingl
y in trouble because the sphere has measure zero in M", but the curvature
of the sphere saves the day. This theorem is obviously closely connected to the
smoothnes
s of solutions of the wave equation.
The proof of the above differentiation theorem relies on an
Elementar
y Tauberian Theorem. Suppose that
ists and Thenexists, and equals
li
m
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
show
n the surprising power hidden in the elementary Tauberian Theorem. Here we
appl
y it to for a fixed x. In fact, we have
wit
h fi equal to normalized surface measure on the unit sphere, so that the Fourier
transform
s of F and / are related by for each fixed r.

8 CHAPTER 1
Therefore, assuming for simplicity, we obtain
(Here we make crucial use of curvature, which causes to decay at infinity, so that
the integral in curly brackets converges.) It follows that
for almost every . On the other hand, F(x, r)dr is easily seen
to be the convolution of / with a standard approximate identity. Hence the usual
Lebesgu
e differentiation theorem shows that
for almost every x.
So for almost all , the function F(x,r) satisfies the hypotheses of the
elementar
y Tauberian theorem. Consequently,
almost everywhere, proving Stein's differentiation theorem for
To prove the full result for , we repeat the above argu-
men
t with surface measure n replaced by an even more singular distribution on ¡R".
Thu
s we obtain a stronger conclusion than asserted, when On the other
hand
, for we have a weaker result than that of Stein, namely Lebesgue's
differentiation theorem. Interpolating between and , one obtains the Stein
differentiatio
n 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
dramati
c 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.
»'-SPACE
S
Another essential part of Fourier analysis is the theory of -spaces. Stein trans-
formed the subject twice, once in a joint paper with Guido Weiss, and again
i
n a joint paper with me. Let us start by recalling how the subject looked be-
fore Stein's work. The classical theory deals with analytic functions F(z) on

SELECTED THEOREMS BY ELI STEIN 9
the unit disc. Recall that F belongs to if th e norm
is finite.
The classical Hp-spaces serve two main purposes. First, they provide growth
condition
s under which an analytic function tends to boundary values on the unit
circle
. Secondly, serves as a substitute for to allow basic theorems on
Fourier series to extend from to all To prove theorems about
the main tool is the Blaschke product
(11
)
where are the zeroes of the analytic function F in the disc, and 6V are suitable
phases. The point is that B(z) has the same zeroes as F, yet it has absolute value 1
o
n the unit circle. We illustrate the role of the Blaschke product by sketching the
proo
f of the Hardy-Littlewood maximal theorem for Hp. The maximal theorem
says that for where
and T(0) is the convex hull of e'6 and the circle of radius about the origin.
This basic result is closely connected to the pointwise convergence of F(z) as
tend
s to To prove the maximal theorem, we argue as follows.
Firs
t suppose Then we don't need analyticity of F. We can merely
assum
e 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 / on
the unit circle. The maximal theorem for /, a standard theorem of real variables,
say
s that , where It is
quit
e simple to show that Therefore
an
d the maximal theorem is proven for
If , 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
,
we can define a single-valued branch of , which will belong
to H « since the maximal theorem for Hp is already
known
. Hence, with norm
Tha
t is, proving the maximal theorem for functions
withou
t zeroes.
To finish the proof, we must deal with the zeroes of an We
brin
g in the Blaschke product B(z), as in (11). Since Biz) and F(z) have the same
zeroe
s and since on the unit circle, we can write
wit
h G analytic, and on the unit circle. Thus,

10 CHAPTER I
Insid
e the circle, G has no zeroes and Hence , so
by the maximal theorem for functions without zeroes. The proof of the maximal
theore
m is complete. (We have glossed over difficulties that should not enter an
expositor
y paper.)
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
comple
x variables ran into a lot of trouble, because the zeroes of an analytic
functio
n form a variety V with growth conditions. Certainly
V is much more complicated than the discrete set of zeroes {zP} in the disc. There
is no satisfactory substitute for the Blaschke product. For a long time, this blocked
al
l attempts to extend the deeper properties of Hp to several variables.
Stei
n and Weiss [13] realized that several complex variables was the wrong
generalizatio
n of Hp for purposes of Fourier analysis. They kept clearly in mind
wha
t Hp spaces are supposed to do, and they kept an unprejudiced view of how
t
o 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
imaginar
y parts of an analytic function on the disc as the gradient of a har-
moni
c function. In several variables, the gradient of a harmonic function is a
syste
m of functions on that satisfies the Stein-Weiss
Cauchy-Rieman
n equations
(12)
In place of the Blaschke product, Stein and Weiss used the following simple ob-
servation
. If satisfies (12), then is
subharmoni
c for . We sketch the simple proof of this fact, then explain
how an theory can be founded on it.
To see that is subharmonic, we first suppose and calculate
i
n coordinates that diagonalize the symmetric matrix at a given point. The
resul
t is
(13)
wit
h
Since by the Cauchy-Riemann equations, we have

SELECTED THEOREMS BY ELI STEIN 11
i.e.
, . Hence , so
the expression in curly brackets in (13) is non-negative for, and is
subharmonic
.
So far, we know that is subharmonic where it isn't equal to zero. Hence for
we have
(14
)
provided However, (14) is obvious when so it holds for
any x. That is, is a subharmonic function for , as asserted.
No
w let us see how to build an Hp theory for Cauchy-Riemann systems, based
on subharmonicity of To study functions on we regard
as the boundary of and we define as the
spac
e of all Cauchy-Riemann systems for which the norm
is finite. For this definition agrees with the usual spaces for the upper
half-plane
.
Next we show how the Hardy-Littlewood maximal theorem extends from the
dis
c to
Define the maximal function for
The
n for we have with norm
As in the classical case, the proof proceeds by reducing the problem to the
maxima
l theorem for For small the function
is subharmonic on and continuous up to
th
e boundary. Therefore,
(15)
where RI. is the Poisson integral and By
definitio
n of the Hp-norm, we have
(16) with
On the other hand, since the Poisson integral arises by convolving with an
approximat
e identity, one shows easily that
(17)
with

12 CHAPTER 1
The standard maximal theorem of real variables gives
sinc
e . Hence (15), (16), and (17) show that
(18)
The constant Cp is independent of h, so we can take the limit of (18) as h —<• 0 to
obtai
n the maximal theorem for Hp. The point is that subharmonicity of
substitute
s for the Blaschke product in this argument.
Stein and Weiss go on in [13] to obtain «-dimensional analogues of the classical
theorem
s on existence
of boundary values of Hp functions. They also extend to
the classical F and M Riesz theorem on absolute continuity of H1 boundary
values
. They begin the program of using in place of , to extend
the basic results of Fourier analysis to and below. We have seen how they
dea
l with the maximal function. They prove also an Hp-\crsion of the Sobolev
theorem
.
It is natural to try to get below , and this can be done by studying
highe
r gradients of harmonic functions in place of (12). See Calderon-Zygmund
A
joint paper [60] by Stein and me completed the task of developing basic
Fourie
r analysis in the setting of the //^-spaces. In particular, we showed in [60]
that singular integral operators are bounded on for We
prove
d this by finding a good viewpoint, and we found our viewpoint by repeatedly
changin
g 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
arrive
d at a definition of Hp with the following excellent properties. First of all, it
was easy to prove that the new definition of was equivalent to the Stein-Weiss
definitio
n and its extensions below Secondly, the basic theorems of Fourier analysis, which seemed very hard to prove from the original definition of becam
e 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 on the disc belongs to if and only if the maximal function belongs to Lp (Unit Circle). Thus, Hp can be defined purely in terms of harmonic functions u, without recourse to

SELECTED THEOREMS BY ELI STEIN 13
the harmonic conjugate v. Stein and I showed in [60] that the same thing happens
i
n n dimensions. That is, a Cauchy-Riemann system on be-
longs to the Stein-Weiss Hp space if and only if the maximal function
belong
s to (Here, the nth function u„
plays a special role because is defined by Hence, Hp may be
viewed as a space of harmonic functions u(x, t) on The result extends below
if we pass to higher gradients of harmonic functions.
The next step is to view Hp as a space of distributions / on the boundary
An
y reasonable harmonic function u(x, t) arises as the Poisson integral of
a
distribution /, so that Poisson kernel. Thus, it is
natural to say that if the maximal function
(19
)
belongs to Stein and I found in [60] that this definition is independent of the
choice of the approximate identity and that the "grand maximal function"
(20
)
belongs to provided Here A is a neighborhood of the origin in a
suitable space of approximate identities. Thus, if andonly if
for some reasonable approximate identity. Equivalently, if and only
i
f the grand maximal function belongs to The proofs of these various
equivalencie
s are not hard at all.
We have arrived at the good definition of mentioned above.
T
o transplant basic Fourier analysis from to
oo)
, there is a simple algorithm. Take Calderon-Zygmund theory, and replace
ever
y application of the standard maximal theorem by an appeal to the grand
maxima
l function. Only small changes are needed, and we omit the details here.
Our paper [60] also contains the duality of and BMO. Before leaving [60], let
me mention an application of theory to Lp-estimates. If a denotes uniform
surfac
e measure on the unit sphere in then is bounded
on provided and . Clearly, this result gives
informatio
n on solutions to the wave equation.
The proof uses complex interpolation involving the analytic family of operators
(
a complex),
as is clear to anyone familiar with the Stein interpolation theorem. The trou-
ble here is that fails to be bounded on L1 when ais imaginary. This
make
s it impossible to prove the sharp result using

14 CHAPTER 1
Lp alone. To overcome the difficulty, we use Hx in place of L1 in the inter

polation argument. Imaginary powers of the Laplacian are singular integrals,
which we know to be bounded on H1. To
show that complex interpolation works
on Hx, we combined the duality of H1 and BMO with the auxiliary function
f*(x)
= supgM J^i JQ {y) — (meang
f)\dy. We refer the reader to [60] for
an explanation of how to use /#, 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, M) to more general semisimple Lie groups, with
profound consequences for representation theory. The results we discuss are con
tained 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 Un be the unitary principal series representations
of G, or one of its degenerate variants. The Un all act on
a common Hilbert space,
whose inner product we denote by (ξ, η). We
needn't write down Un 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 U7r and
Uwn are unitarily equivalent for w e W. Thus there is an intertwining operator
A(w, π) so that
(21) A(w, n)U™" = U*A(w,n) for g e G, w € W, and
for all π.
If Un is irreducible (which happens for most π), then A(w, π) is uniquely de

termined by (21) up to multiplication by an arbitrary scalar a(w, π). The crucial
idea is as follows. If the A(w, π) are correctly normalized (by the correct choice
of a(w, jr)), then A(w, π) continues analytically to complex parameter values π.
Moreover,
for certain complex (u>, π), the quadratic form
(22) ((ξ, n))w „ = ((TRIVIAL
FACTOR)A(u;, π) ξ, η)
is positive definite.
In
addition, the representation Un (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 S L (2, R).
Applications of this basic construction are as follows.
(1) Starting with the unitary principal series, one obtains understanding of the
previously discovered complementary series, and construction of new ones,

SELECTED THEOREMS BY ELl STEIN 15
e.g.,
on 5/j(4, C). Thus, Stein exposed a gap in a supposedly
complete list of
complementary series representations of SpiA, C) [GN]. See [33].
(2) Starting with a degenerate unitary principal series; Stein constructed new ir
reducible unitary representations of SL(n, C), in startling contradiction to
the standard, supposedly complete list [GN]
of irreducible unitary represen
tations 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) The analysis of intertwining operators required to carry out analytic continua
tion also determines which exceptional values of π lead to reducible principal
series representations. For example, such reducible principal series represen
tations exist already for SL(η, M), 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
intertwining 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
with K (y ) smooth away from the identity and homogeneous of the critical degree
with respect to "dilations"
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
cancellation
condition. Hence we assume fB^Bo K(y)dy — 0,
where the Bi are
dilates (Bi = δ,. (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.
(23) K(xy-l)f(y)dy,
(<5f)r>o : N N.
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 the
ory
provides a necessary and sufficient condition for L2-boundedness that no one

16 CHAPTER 1
know
s how to check. This fundamental analytic difficulty might have proved
fatal to the study of intertwining operators. Fortunately, Stein was working si-
multaneousl
y 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 the-
ore
m on commutator integrals [Ca], Stein proved a simple, powerful lemma in
functiona
l 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 Z.2-boundedness of singular in-
tegral
s 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 sim-
ple proof. Then we will return to its application to singular integrals on nilpotent
groups
.
The Cotlar-Stein lemma deals with a sum of operators on a Hilbert
space
. The idea is that if the are almost orthogonal, like projections onto the
variou
s coordinate axes, then the sum T will have norm no larger than ma
Th
e precise statement is as follows.
Cotlar-Stein Lemma. Suppose is a sum of operators on Hilbert
space.
Assume
Proof. so
(24
)
We can estimate the summand in two different ways.
Writin
g , we get
(25)
On the other hand, writing
we see that
(26)

SELECTED THEOREMS BY EL! STEIN 17
Taking the geometric mean of (25), (26) and putting the result into (24), we
conclud
e that
Thus. . Letting we obtain
the conclusion of the Cotlar-Stein Lemma.
To
apply the Cotlar-Stein lemma to singular integral operators, take a partition
o
f unity on iV, so that each is a dilate of a fixed function
that vanishes in a neighborhood of the origin. Then may be
decomposed into a sum , with 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.
Almos
t immediately after this work, the Cotlar-Stein lemma became the stan-
dar
d method to prove L2 boundedness of operators. Today one knows more, e.g.,
the T(l) theorem of David and Journe. Still it is fair to say that the Cotlar-Stein
lemm
a remains the most important tool for L2-boundedness.
Singular integrals on nilpotent groups were soon applied by Stein in a context
seemingl
y far from representation theory.
0-PROBLEM
S
We prepare to discuss Stein's work on the problems of several complex variables
an
d related questions. Let us begin with the state of the subject before Stein's
contributions
. Suppose we are given a domain with smooth boundary. If
we try to construct analytic functions on D with given singularities at the boundary,
the
n we are led naturally to the following problems.
I. Given a (0, 1) form on D, find a function u on D that solves
wher
e Naturally, this is possible only if a
satisfie
s the consistency condition ~ ~, i.e. Moreover,
u
is determined only modulo addition of an arbitrary analytic function on D.
T
o make u unique, we demand that u be orthogonal to analytic functions in
L2{D).

18 CHAPTER 1
II. There is a simple analogue of the operator for functions defined only on
the boundary In local coordinates, we can easily find > linearly
independent complex vector fields of type
for smooth, complex-valued ajk) whose real
and imaginary parts are all tangent to 3D. The restriction u of an analytic
functio
n to 3D clearly satisfies where in local coordinates
Th
e boundary analogue of the problem (I) is the
inhomogeneous equation Again, this is possible only if a
satisfie
s a consistency condition , and we impose the side condition
tha
t u be orthogonal to analytic functions in
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 , the
Neuman
n 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
Laplacia
n in the interior of D, but it is subject to non-elliptic boundary conditions.
On the other hand, is a non-elliptic system of partial differential operators on
3 D, with no boundary conditions (since 3 D has no boundary). Modulo lower-order
v terms (which, however, are important). is the scalar operator
wher
e and are the real and imaginary parts of the basic complex vector
fields
At a given point in 3 D, the Xk and Yk are linearly independent, but
the
y don't span the tangent space of , This poses the danger that C will behave
like a partial Laplacian such as acting on function
Th
e equation is very bad. For instance, we can take to
depend on z alone, so that with u arbitrarily rough. Fortunately, C is
more like the full Laplacian than like because the and together with
thei
r commutators span the tangent space of for suitable D. Thus,
is a well-behaved operator, thanks to the intervention of commutators of vector
fields.
It was Kohn in the 1960's who proved the basic regularity theorems for ,
an
d on strongly pseudoconvex domains (the simplest case). His proofs
wer
e based on subelliptic estimates such as , and
brough
t to light the importance of commutators. Hormander proved a celebrated
theorem on regularity of operators,
wher
e are smooth, real vector fields which, together with their
repeate
d commutators, span the tangent space at every point.

SELECTED THEOREMS BY ELI STEIN 19
If we allow Xo to be a complex vector field, then we get a very hard problem
tha
t is not adequately understood to this day, except in very special cases.
Stein made a fundamental change in the study of the the problems by bring-
ing 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
explicit solutions, it is then possible to prove sharp regularity theorems. Thus, the
equation
s on strongly pseudoconvex domains are understood completely. It is a
majo
r 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
pseudoconve
x case. We begin with the work of Folland and Stein [67]. The
simples
t 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
I
m Its boundary DSiegei has
an important symmetry group, including the following.
(a
) Translations for
(b) Dilations for
(c
) Rotations for unitary matrices U.
The multiplication law in (a) makes H into a nilpotent Lie group, the Heisenberg
group
. Translation-invariance of the Siegel domain allows us to pick the basic
complex vector-fields to be translation-invariant on H. After we
mak
e a suitable choice of metric, the operators C and become translation- and
rotation-invariant
, and homogeneous with respect to the dilations Therefore,
the solution1 of should have the form of a convolution
on the Heisenberg group. The convolution kernel K is homogeneous with respect
to the dilations 8, and invariant under rotations. Also, since K is a fundamental
solution
, it satisfies 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 equation for the Siegel domain.
To derive sharp regularity theorems for we combine the explicit fundamental
solutio
n with the Knapp-Stein theorem on singular integrals on the Heisenberg
group
. For instance, if , then and all
belong to To see this, we write
'Kohn's work showed that has a solution if we are in complex dimension > 2. In two
comple
x dimensions, has no solution for most We assume dimension > 2 here.

20 CHAPTER 1
an
d note that has the critical homogeneity and integral 0. Thus
is a singular integral kernel in the sense of Knapp and Stein, and it follows that
Fo
r the first time, nilpotent Lie groups have entered into the
stud
y of 3-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 . For simplicity we will take 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
Mor
e precisely, 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
. Therefore, if is our known solution of on the
Heisenber
g group, then it is natural to try
(27)
a
s an approximate solution of _ on M. (Since w and a are sections of
bundles, one has to explain carefully what (27) really means.) If we apply to
th
e w defined by (27), then we find that
(28)
where £ is a sort of Heisenberg version of x , .In particular, gains smooth-
ness
, so that can be constructed modulo infinitely smoothing operators
b
y means of a Neumann series. Therefore (27) and (28) show that the full solution
of is given (modulo infinitely smoothing errors) by
(29
)
fro
m which one can deduce sharp estimates to understand completely on M.
Th
e process is analogous to the standard method of "freezing coefficients" to
solv
e variable-coefficient elliptic differential equations. Let us see how the sharp
result
s are stated. As on the Heisenberg group, there are smooth, complex vector
fields
that span the tangent vectors of type (0,1) locally. Let Xj be the real
and imaginary parts of the In terms of the we define "non-Euclidean"
versions of standard geometric and analytic concepts. Thus, the non-Euclidean
bal
l may be defined as an ellipsoid with principal axes of length in the
codimension 1 hyperplane spanned by the , and length perpendicular to
tha
t hyperplane. In terms of , the non-Euclidean Lipschitz spaces

SELECTED THEOREMS BY ELI STEIN 21
are defined as the set of functions u for which for
(Here
, There is a natural extension to all The
non-Euclidean Sobolev spaces consist of all distributions u for which all
for
The
n the sharp results on are as follows. If and
the
n for If and
then I for (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
crucia
l results on L2-boundedness of singular integrals. Additional difficulties
aris
e 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
difficul
t the passage from Lp estimates to the Sobolev spaces
Before we continue with Stein's work on 3, let me explain the remarkable paper
o
f Rothschild-Stein [72]. It extends the Folland-Stein results and viewpoint to
genera
l Hormander operators Actually, [72] deals with
systems whose second-order part is , but for simplicity we restrict attention
her
e to £. In explaining the proofs, we simplify even further by supposing
The goal of the Rothschild-Stein paper is to use nilpotent groups to write down
an explicit parametrix for £ and prove sharp estimates for solutions of
Thi
s ambitious hope is seemingly dashed at once by elementary examples. For
instance, take with
(30
) on
Then X i and span the tangent space, yet £ clearly cannot be approximated
by translation-invariant operators on a nilpotent Lie group in the sense of Folland-
Stein
. The trouble is that £ changes character completely from one point to another.
Awa
y from the y-axis is elliptic, so the only natural nilpotent group
we can reasonably use is On the y-axis, £ degenerates, and evidently cannot
b
e approximated by a translation-invariant operator on The problem is so
obviousl
y fatal, and its solution by Rothschild and Stein so simple and natural,
tha
t [72] must be regarded as a gem. Here is the idea:
Suppos
e we add an extra variable t and "lift" X\ and X2 in (30) to vector fields
(31) on

22 CHAPTER 1
Then the Hormander operator C = X + X looks the same at every point of
M3, 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 £_1. On the other hand, C reduces to C when acting
on functions u(x, y, t) that do not depend on t. Hence, sharp results on Cu = /
imply sharp results on Cu = /.
Thus we have the Rothschild-Stein program: First, add new variables and lift
the given vector fields Xi · · • Xn to new vector fields X - -Xn whose underly
ing structure does not vary from point to point. Next, approximate C = Σ" X2j
by a translation-invariant operator C = Yj on a nilpotent Lie group Af.
Then
analyze the fundamental solution of C, and use it to write down an approx
imate fundamental solution for C. From the approximate solution, derive sharp
estimates for solutions of Cu — f. Finally, descend to the original equation
Cu = 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 X -Xn be smooth vector fields on a neighborhood of the ori
gin in R
n. Assume that the Xj and their commutators [ [ [Xji, Xj2], Xj3 ] ·• •, Xjs ]
of order up to r span the tangent space at the origin. Then we can find
smooth vec
tor fields Xi •• - Xn on a neighborhood U of the origin in R"+m with the following
properties.
(a) The Xj and their commutators up to order 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.
(b) The Xj and their commutators up to order r span the tangent space of U.
(c) Acting on
functions on R"+m that do not depend on the last m coordinates,
the Xj reduce to the given
Xj.
Next we need a nilpotent Lie group Af appropriate to the vector fields Xi - ·· XN.
The natural one is the free nilpotent
group AfNr of step r on N generators. Its Lie
algebra is generated by Fj · · · Y
n 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 AfNr. It is convenient
to pick a basis {Κα}αεΛ
for the Lie algebra of AfNr, consisting of Y^--Yn and some of their commutators.
OnAW we
form the Hormander operator C - Yj. Then C is translation-
invariant and homogeneous under the natural dilations on ΛfNr. Hence C 1 is
given by convolution on AfNr with a homogeneous kernel K(-) having a weak

SELECTED THEOREMS BY ELI STEIN 23
singularity at the origin. Hypoellipticity of shows that K is smooth away from
the origin. Thus we understand the equation very well.
We want to use to approximate at each point To do so, we
havet
o identify a neighborhood of y in with a neighborhood of the origin
in This has to be done just right, or else will fail to approximate
Th
e idea is to use exponential coordinates on both and Thus, if x
(identity) then we use as coordinates for x.
Similarly
, let be the commutators of analogous to the , and
le
t be given. Then given a nearby pointwe
us
e (ta)aeA as coordinates for x.
Now we can identify with a neighborhood of the identity in , simply
b
y identifying points with the same coordinates. Denote the identification by
, and note that identity.
In view of the identification the operators and live on the same space.
Th
e next step is to see that they are approximately equal. To formulate this, we
nee
d some bookkeeping on the nilpotent group Let be the natural
dilations on . If , then write for the function
When is fixed and t is large, then is supported in a tiny neighborhood of the
identity. Let V be a differential operator acting on functions on We say that
V
has "degree" at most k if for each we have
fo
r large, positive t. According to this definition, have degree 1 while
has degree 2, and the degree of a(x) depends on the behavior of
a(x)
near the identity. Now we can say in what sense and are approximately
equal. The crucial resul
t is as follows.
Theore
m B. Under the mapthe vector field pulls back to
where is a vector field on of "degree"
Using Theorem B and the map ©j,, we can produce a parametrix for £ and prove
that it works. In fact, we take
(32
)
where K is the fundamental solution of For fixed y, we want to know that
(33
)
where is the Dira
c delta-function and has only a weak singularity at
. T
o prove this, we use to pull back to Recall that
while . Hence by Theorem B, pulls back to an operator of the form
, with ~ having "degree" at most 1. Therefore (33) reduces to proving

24 CHAPTER 1
that
(34) (£ + V
y) K (χ) = <5ld.(.r) + £(x),
where £ has only a weak singularity at the identity. Since
CK(x) = <5ld,(x), (34)
means simply that VyK (x) has only a weak singularity
at the identity. However,
this is obvious from the smoothness and homogeneity of K (x), and from the fact
that Vy has
degree < 1. Thus, K (JC . y) is an approximate fundamental solution
for £.
From the explicit fundamental solution for the lifted operator £, one can
"descend" to deal with the original Hormander operator £ in two different ways.
a. Prove sharp estimates for the lifted problem, then specialize to the case of
functions that don't depend on the extra variables.
b. Integrate out the extra variables from the fundamental solution for £, to obtain
a fundamental solution for £.
Rothschild and Stein used the first approach. They succeeded in proving
the
estimate
(35) IlXquIlLPfi/) + \XjXku\Lnu) < C
p
Lf(V)
Cp\u\LP{V) for 1 < ρ <
oo and U CC V.
This is the most natural and the sharpest estimate for Hormander operators. It
was new even for ρ = 2. Rothschild and Stein also proved sharp estimates in
spaces
analogous to the Ta 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 Hormander 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 Hormander
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 Hormander operators cannot be modelled directly on
nilpotent Lie groups.
So it isn't even clear how to state a theorem on the fundamen
tal solution of a Hormander operator. Nagel, Stein and Wainger [119] realized that
a family of non-Euclidean "balls" Bc(x. P) associated
to the Hormander operator
£ plays the basic role. They defined the Bc(x. p) and
proved their essential proper
ties. 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 Lie group
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
YtXj+ X0 U

SELECTED THEOREMS BY ELI STEIN 25
balls. For general Hormander operators C the problem of defining and controlling
non-Euclidea
n balls is much more subtle. Closely related results appear also in
[FKP]
, [FS],
Let us look first at a nilpotent group such as with its family of dilations
The
n the correct family of non-Euclidean balls is essentially
dictated by translation and dilation-invariance, starting with a more or less arbitrary
harmless "unit ball" B.vNr (identity, 1). Recall that the fundamental solution for
on is given by a kernel K(x) homogeneous with respect to
the 8t. Estimates that capture the size and smoothness of may be phrased
entirel
y in terms of the non-Euclidean balls In fact, the basic estimate
i
s as follows.
(36)
for and
Next we associate non-Euclidean balls to a general Hormander operator. For
simplicity, take as in our discussion of Rothschild-Stein [72]. One
definition of the balls involves a moving particle that starts at x and
travel
s along the integral curve of for time t\. From its new position x' the
particl
e then travels along the integral curve of for time 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 The ball consists of
all y that can be reached in this way in time For instance, if is elliptic,
the
n is essentially the ordinary (Euclidean) ball about x of radius p. If
we take on , then the balls behave naturally under
translation
s and dilations; hence they are essentially the same as the
appearing in (36). Nagel-Wainger-Stein analyzed the relations between
an
d for an arbitrary Hormander operator (Here and are
as in our previous discussion of Rothschild-Stein.) This allowed them to integrate
ou
t the extra variables in the fundamental solution of to derive the following
shar
p estimates from (36).
Theorem. Suppose and their repeated commutators span the tangent
space.
Also, suppose we are in dimension greater than 2. Then the solution of
is given by dy with

26 CHAPTER 1
Here the Xji act either in the x- or the y-variable.
Let us return from Hormander
operators to the 3-problems on strongly pseu-
doconvex domains D C C". Greiner and Stein derived sharp estimates for the
Neumann Laplacian DID = a in their book [78]. This problem is hard, because
two different 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 3D,
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 boundary 3D lies in the span
of the complex vector fields L • - • Ln_u L1 • · • Ln-\. Here we have retained
the
notation of our earlier discussion of 3-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 3D. 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 Du; — a on a strictly pseudoconvex domain D C C". If
a belongs to the Sobolev space Lf1 then w belongs
to Lpk+X (1 < ρ < oo).
Moreover, if X and Y are allowable vector fields, then XYw belongs to L%. Also,
L w belongs to Z.£+, if L
is a smooth complex vector field of type (0, 1). Similarly,
if a belongs to the Lipschitz space Lip(/3) (0 < β < 1), then the gradient of w
belongs to Lip(/3) as well. Also the gradient of Lw belongs to Lip(/3) if L is a
smooth complex vector field of type (0, 1); and XYw belongs to Lip($) for X and
Y allowable vector fields.
These results for allowable vector fields were new even for L2. We
sketch the
proof.
Suppose Du; = a. Ignoring the boundary conditions for a moment, we have
Aw = a in D, so
(37) w = Ga
+ P.I.(u5)
where w is defined on
3D, and G, P.I. denote the standard Green's operator and
Poisson integral, respectively. The trouble with (37) is that we know nothing
about w so far. The next step is to bring in the boundary condition for Dui = a.
According to Calderon's work on general boundary-value problems, (37) satisfies
the
3-Neumann boundary conditions if and only if
(38) Aw =• {B(Got)}

SELECTED THEOREMS BY ELl STEIN 27
for
a certain differential operator BonD, and a certain pseudodifferential operator
A on 3D. Both A and B can be determined explicitly from routine computation.
Greiner and Stein [78] derive sharp regularity theorems for the pseudodifferen
tial equation Aw = g, and then apply those
results to (38) in order to understand
w in terms of a. Once they know sharp regularity theorems for w, formula (37)
gives the behavior of w.
Let us sketch how Greiner-Stein analyzed Aw — g. This is really a system
of η pseudodifferential equations for η unknown functions (η = dim C"). In a
suitable frame, one component of the system decouples from the rest of the prob
lem (modulo negligible errors) and leads to a trivial (elliptic) pseudodifferential
equation. 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
(39) • + w* = a#.
Here a* consists of the non-trivial components of {B(Ga)) |aD, w* is the unknown,
and •+ may be computed explicitly.
Greiner and Stein reduce (39) to the study of the Kohn-Laplacian In fact,
they produce a matrix EL of first-order pseudodifferential operators similar to •+,
and
then show that •_•+ = D6 modulo negligible errors.2 Applying •_ to (39)
yields
(40) DfcU)* = D_a# + negligible.
From Folland-Stein [67] one knows an explicit integral operator K that inverts
modulo negligible errors. Therefore,
(41) w* = £•_</ -(- negligible.
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 Gui = a, 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 this volume
for a discussion of singular Radon transforms; and to Nagel-Rosay-Stein-Wainger
2This procedure requires
significant changes in two complex variables, since then D6 isn 't invertible.

28 CHAPTER 1
[131], D.-C. Chang-Nagel-Stein [132], and [McN], [Chr], [FK] for
the solution of
the 3-problems on weakly pseudoconvex domains of finite type in C2.
Particularly in several complex variables are we 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 article 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 R"
Differentiability of functions on sets of positive measure
Fourier analysis on TRn when N —• oo
Function theory on tube domains
Analysis of diffusion semigroups
Pseudodifferential calculus for subelliptic problems.
The
list continues to grow.
Princeton University
BIBLIOGRAPHY OF Ε. M. STEIN
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 action 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," Annuli Scula 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 τι-dimensional Euclidean space." J. Math.
Me eh. 77(1958), 503-514.
9. (with G. Weiss) "Interpolation of operators with change of measures." Trans. Amer.
Math. Soe. 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.
Soe. 88 (1958), 430-466.

SELECTED THEOREMS BY ELl STEIN 29
12. (with G. Weiss) "An extension of a theorem of Marcinkiewicz and some of its
applications."/. 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
2x2 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; Π, 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.
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. Ti
(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 Π. Analytic continuation of
the principal series of representations of the η χ η complex unimodular groups." Amer.
J. Math. 83 (1961), 723-786.
23. (with A. Zygmund) "Smoothness and differentiability of functions." Ann. Univ. Sci.
Budapest,
Sectio Math., JlI-IV (1960-61), 295-307.
24. "Conjugate harmonic functions in several variables." Proceedings of the International
Congress of Mathematicians, Djursholm-Linden, Instut 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) H''-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 of holomorphic functions." Awn. 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.
34. "Analysis in matrix spaces and some new representations of SL(N, C)." Ann. of Math.
86(1967),461^190.
35. (with
A. Zygmund) "Boundedness of translation invariant
operators in Holder spaces
and Lp spaces "Ann. of
Math. 85 (1967), 337-349.

30 CHAPTER I
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 representa
tions 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 L log L." 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 offunctions. Princeton Mathemati
cal 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."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 Π." 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.
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. (withC. 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 Har
monic 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.

SELECTED THEOREMS BY ELl STEIN 31
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 3-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 db 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 db complex
and analysis on the Heisenberg
group." Comm. Pure and Appl. Math. 27 (1974), 429-522.
67. "Singular integrals, old and new." In Colloquium Lectures of the 79th SummerMeeting
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), 4295^296.
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 3-Neumann problem." Mathematical Notes 19.
Princeton University Press, 1977.
78. (with D. H. Phong) "Estimates for the Bergman and Szego projections." Duke Math.
J.
44(1977), 695-704.
79. (with N. Kerzman) "The Szego kernels in terms of Cauchy-Fontappie kernels." Duke
Math. J. 45 (1978), 197-224.
80. (with N. Kerzman) "The Cauchy kernels, the Szego kernel and the Riemann mapping
function." Math. Ann. 236 (1978), 85-93.
81. (with A. Nagel and S. Wainger) "Differentiation in lacunary direction." 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.

32 CHAPTER 1
83. (with
Ν. Kerzman) "The Szego kernel in terms of the Cauchy-Fontappie kernels." In
Proceedings
of the Conference on Several Complex Variables,
Cortona, 1977. 1978.
84. (with P. Greiner) "On the solvability of some differential operators of the type In
Proceedings
of the Conference on Several Complex Variables,
Cortona, 1977. 1978.
85. (with S. Wainger) "Problems in harmonic analysis related to curvature." Bull. Amer.
Math. Soc.
84 (1978), 1239-1295.
86. (with R. Grundy)
"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 andAppl. 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
Π, 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 Π." Invent. Math. 60
(1980),9-84.
93. "The differentiability of functions
in R"." Ann. of Math. 113 (1981), 383-385.
94. "Compositions of pseudo-differential operators."
In Proceedings of Journees Equa
tions aux derivees partielles, Saint-Jean de Monts, Juin 1981, Societe Math, de France,
Conference #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.
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.

SELECTED THEOREMS BY ELl STE/N 33
105. (with R. Fefferman, R. Gundy, and M. Silverstein) "Inequalities for ratios of
functional of harmonic functions." Proc. Nat. Acad. Sci. 79 (1982), 7958-7960.
106. (with D. H. Phong) "Singular integrals with
kernels of mixed homogeneites." (Confer
ence in Harmonic Analysis in honor of Antoni Zygmund, Chicago, 1981), W. Beckner,
A. Calderon, R. Fefferman, R Jones, eds. Wadsworth, 1983.
107. "Some results in harmonic analysis in Rn, for η —
> oo." Bull. Amer. Math. Soc. 9
(1983),71-73.
108. "An H1 function with non-summable Fourier expansion." In Proceedings of the Con
ference
in Harmonic Analysis, Cortona, Italy, 1982. Lecture Notes in Mathematics,
no. 992. Springer
Verlag, 1983.
109. (with R. R. Coifrnan and Y. Meyer)"Un
nouvel espace fonctionel adapte a 1'etude des
operateurs definis pour des integrates singulieres." 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. 0. Stromberg) "Behavior of maximal functions in K" 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-liZiWc. Nat. Acad. Sci. 80 (1983), 7697-7701.
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."/. 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. Rubiode 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 1."
Acta Math. 157 (1986), 99-157.
125. (with
C. D. Sogge) "Averages over hypersurfaces: II." Invent. Math. 86 (1986), 233-
242.

34 CHAPTER 1
126. (with Μ. Christ) "A remark on singular Calderon-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-
Fund. Anal. 73 (1987), 179-194.
129. (with F. Ricci) "Harmonic analysis on nilpotent groups
and singular integrals II." J.
Fund. Anal. 78 (1988), 56-84.
130. (with A. Nagel, J. R Rosay and S. Wainger) "Estimates for the Bergman and Szego
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 3-Neumann problem for pseudo-
convex 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 Szego
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 nilpo
tent Lie groups in Analysis and P.D.E." A collection of papers dedicated to Mischa
Codar. 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 prob
lems." In the Madison Symposium on Complex Analysis, Contemp. Math., 137 (1992),
119-131.
OTHER REFERENCES
[BGS] D. Burkholder, R. Gundy, and M. Silverstein. "A maximal function characterization
of the class HT Trans. Amer. Math. Soc. 157 (1971): 137-153.
[Ca] A. P. Calderon. "Commutators of singular integral operators." Proc. Nat. Acad. Sci.
53(1965): 1092-1099.

SELECTED THEOREMS BY ELl STEIN 35
[Chr] M.
Christ. "On the 3fc-equation and Szego projection on a CR manifold." In Pro
ceedings, 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
/(1955): 105-167.
[CZ] A. P. Calderon 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. Unitare Darstellungen der Klassischen Gruppen.
Akademie Verlag, 1957.
[H] 1.1. Hirschman, Jr."Multiplier transformations I."
Duke Math. J. 26 (1956): 222-
242;"Multiplier transformations Π." Duke Math. J. 28 (1961): 45-56.
[McN] J. McNeal, "Boundary behavior of the Bergman kernel function in C2." Duke
Math.
J. 58 (1989): 499-512.
[Z] A. Zygmund, Trigonometric Series. Cambridge University Press, 1959.

2
Geometric Inequalities
in Fourier Analysis
William Beckneff
1 INTRODUCTION
Geometric ideas occur in almost every aspect of Fourier analysis. Beginning with
the symmetry structure of the domain and the product structure of the operator,
geometric concepts control deep facts about the Fourier transform.
The symmetry
structure of a Riemannian manifold defines not only the natural objects of analysis
for the domain such as the Laplace-Beltrami operator, Green's function, global
transforms, and boundary operators, but also determines intrinsic ways to compare
the "size" of such objects as measured by the classical function spaces. The
geometric structure of a manifold is manifest in the character of analytic operator
and variational inequalities. Such estimates are the building-blocks of "everyday
analysis."
Convolution is a natural object viewed
both as an averaging process for a
translation-invariant measure and as the dual operator to the Fourier transform.
Fractional integration is even more natural arising in the context of Green's
functions and potential theory, restriction phenomena for the Fourier transform,
intertwining operators for representations of the Lorentz groups and correlation
functions in conformal field theory and statistical mechanics. A framework for the
analysis of convolution inequalities
on a manifold is developed here, especially in
terms of (1) conformal invariance; (2) geometric symmetrization and equimeasur-
able rearrangement of functions; (3) complex symmetry structure on Lie groups;
and (4) embedding of geometric and probabilistic information in exact constants
"This
research was supported in part by the National Science Foundation.

GEOMETRIC INEQUALITIES IN FOURIER ANALYSIS 3 7
fo
r variational problems. Several themes from E. M. Stein's work have influenced
thi
s program, including his treatment of Riesz transforms, spherical harmonics, and
the Hardy-Littlewood-Sobolev theorem on fractional integration; his viewpoint on
Li
e groups and boundary manifolds including SL (2, R) and the Heisenberg group;
and his emphasis on integral transforms and boundary behavior in several comple x
variables
. Philosophically the roots for the overall approach go back to Hardy and
Littlewood
. In addition, a strong connection can be drawn to problems in physics,
including the Bargmann-Fock analysis of the hydrogen atom, ideas of Bargmann,
Fock
, and Segal on quantum field theory, and Polyakov's quantum string theory.
Thre
e important calculations for sharp inequalities due to J. Moser, E. Onofri, and
E. H. Lieb directly motivate much of this program. The interaction of ideas and
method
s from differential geometry, Fourier analysis, and quantum string theory
has led to a richer understanding of the role of algebraic invariance and geometric
structur
e in analysis on manifolds.
2 CLASSICAL INEQUALITIES
The basic convolution inequality on a unimodular Lie group is Young's inequality
(1)
wit
h Using product structure and
radia
l symmetry, one can show that on the Euclidean space this inequality can
b
e improved ([8],[16]) with the extremal result holding only for gaussian functions
(2)
wit
h
and primes always denoting dual exponents, It is relatively
easy to see that this inequality extends to include weak Lorentz classes and in fact
corresponds to the Hardy-Littlewood-Sobolev theorem for fractional integration
([68]
)
(3)
with for related as above). The function is
characteristi
c for the Lorentz class so this inequality has the equivalent

38 CHAPTER 2
form
(4)
Th
e linking step between these inequalities involves a symmetrization argument
o
f Riesz-Sobolev type.
Riesz-Sobolev Lemma.
(5)
Here
denotes the equimeasurable radial decreasing rearrangement applied to
the modulus of a function.
This technical result is central to the analysis of positive convolution operators
an
d Sobolev inequalities ([70]), and is geometric in nature being equivalent to the
Brunn-Minkowski inequality. The symmetry structure intrinsic to these inequali-
ties includes: translation invariance, rotational symmetry, dilation invariance, and
Euclidea
n product structure.
It is useful to take into account this product structure even though it does not
provid
e good control of constants. Using the relation between arithmetic and geo-
metric means, the one-dimensional Hardy-Littlewood-Sobolev inequality controls
no
t only fractional integration on classical n-dimensional Riemannian manifolds
bu
t also on nilpotent Lie groups. Moreover, the Riesz-Sobolev symmetrization
lemm
a can be applied on a nilpotent Lie group to remove the non-abelian structure
an
d give the following extension for the inequalities that appear above.
Theorem 1. Let G be a nilpotent Lie group of dimension n, homogeneous dimen-
sion m with |x| denoting the canonical distance on G. Then for
(6)
and for
(7)
(8
)
No extremal functions exist for inequality (6) when the constant is less than one.
Interpolation arguments are the standard method used to prove weak Lorentz
class inequalities, but dilation invariance and the Riesz-Sobolev lemma provide a

GEOMETRIC INEQUALITIES IN FOURIER ANALYSIS 39
more elementary reduction to the one-dimensional form of Young's inequality on
the real line. To illustrate this point consider the inequality
(9
)
for and Using symmetrization the
proble
m of verifying this inequality is reduced to non-negative radial decreasing
function
s which satisfy the estimates
For the most part, the symbol C will denote a generic constant. Since the problem is
one-variable
, dilation invariance provides a formulation as a convolution inequality
on the line. Set exp t and exp s
the
n the inequality above becomes
but under the conditions
and The estimate now follows immediately
from Young's inequality (1).
E. H. Lieb recognized ([48]) that when the map
is a map from a space to its dual, then two additional types of symmetry are evident
whic
h can be used to calculate the best constant for the inequality
(10
)
with and . There is a quadratic functional symmetry which
make
s the operator positive-definite self-adjoint so that one may take More
importantly, the inequality is conformally invariant. Suppose r is a conformal
transformatio
n and let J denote the modulus of the Jacobian determinant for this
change of variables. Then under the transformation
the functional inequality (10) is invariant. This application of conformal invariance
t
o convolution problems has a rich history in mathematical physics. It was used
by Bargmann and Fock to give a group-theoretical analysis of the spectrum of the
hydrogen atom and more recently by Onofri ([57]) to study the variation of the
zeta-functio
n determinant of the Laplacian in two dimensions under conformal
deformation. The critical issue here is the algebraic invariance of the metric. In

40 CHAPTER 2
addition
, conformal invariance implies that equivalent forms of inequality (10)
exist on any domain conformally equivalent to the plane including the sphere
an
d the two-sheeted hyperboloid
Using the Riesz-Sobolev lemma twice, one can easily observe that extremal
function
s exist for inequality (10). Symmetrization on reduces the problem to
on
e variable. Then the dilation structure provides a one-dimensional inequality
whic
h can be put on the multiplicative group R+. That is, let
no
w inequality (10) becomes
(11
)
But the kernel is symmetric decreasing away from the origin t = 1 so one can
apply the Riesz-Sobolev lemma a second time to insure that u and v are radial
decreasin
g functions on and as observed above they must also be uniformly
bounded. Now choose a sequence with such that
wher
e is the best constant in (11). Since the un's are decreasing functions,
on
e can use the Helly selection principle to choose a subsequence that converges
almost everywhere to a function By Fatou's lemma But
and so applying Lebesgue's dominated
convergenc
e theorem gives
Since Cp is the best constant, and u must be an extremal function for
inequalit
y (11) which then provides an extremal function for the fractional integral
inequalit
y (10).
The different symmetries of the conformal structure determine the form of the
extremal functions. This is clearly realized in terms of the conformal transfor-
matio
n (stereographic projection) from the plane to the sphere Sn where the
Jacobian determinant is proportional to Since the sphere is a compact
manifold
, one expects constants to be extremal on that domain. This would then
impl
y that on the extremal functions are given by the function
up to conformal automorphism.

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Er stand plötzlich auf, warf dabei seinen Stuhl hin und umarmte
mich noch einmal:
„Wie schön, daß du hier bist!“
Natürlich errötete er, sprang an die Tür und schrie, der Tisch sei
schlecht gedeckt. Der Diener kam und Wolfgang schlug sich an den
Kopf.
„Ich Esel! Willst du ein Beefsteak?“
„Ein Beefsteak?“
„Es dauert gar nicht lange. Fritz, wie lange dauert ein Beefsteak?“
„Eine Viertelstunde“, war die Antwort.
„Ach, Unsinn“, protestierte ich. „Was soll ich denn jetzt um halb
sechs mit einem Beefsteak?“
Wolfgang lachte und goß sich ein Glas Fachinger ein.
„Prost, Walter! Du kennst unsern Stil noch nicht. Wir leben
nämlich hier den Stil englischer Peers. Morgens you take your steak,“
– er bediente sich hierbei einer manirierten Aussprache, – „mittags
hungert man, das nennt man luncheon und abends ißt man im
dinnerjackett alles das, was man am Mittag versäumt hat. Das hat
Nina hier so eingeführt.“
Nina, immer Nina!
Ich fragte unvermittelt:
„Aus welcher Familie stammt sie eigentlich? Hat sie noch Eltern?“
Wolfgang warf nachdenklich zwei Stück Zucker in seine Teetasse.
„Weißt du, bei Nina muß man nicht fragen, woher sie kommt und
wohin sie geht. Nina ist einfach da, – verstehst du? – einfach da.“
Ich sah Wolfgang aufmerksam an. Schau an, dachte ich, wie klug
er ist! Was er da eben gesagt hatte, war mir nicht fremd. Nina war
einfach da, ... sie war eigentlich ... seelenlos.
„Sie ist eigentlich seelenlos,“ sagte ich.
Wolfgang trank seinen Tee. Er stöhnte einige Male wie ein Kind in
die Tasse hinein, setzte sie dann ab, sprang vom Tische auf und
sagte:
„Jawohl, seelenlos, aber herrlich! – Bist du fertig?“
„Ja.“
„Gut. Wie wäre es, wenn wir jetzt aufs Feld gingen und
arbeiteten? Ich lasse mir nämlich jeden Abend von unserm Inspektor

ein Feld anweisen.“
Ich willigte in diesen Vorschlag ein. Wir zündeten uns jeder eine
Zigarette an und gingen in den Hof. Dort holten wir uns aus einem
Schuppen lange Forken und zogen darauf munter durch den Park.
Einmal wandte ich mich um und blickte zu Ninas Fenstern hinauf.
Sie waren fest verschlossen und die Vorhänge heruntergelassen.
„Das gnädige Fräulein pflegt bis neun Uhr zu schlafen,“ sagte
Wolfgang, der meinen Blick bemerkt hatte.
Ich errötete und schwieg.
*
Wir sind auf dem Feld angelangt und ziehen unsere Jacken aus. Die
Kornfelder stehen in der jungen gelbstrahlenden Sonne. Auf den
heiteren grünen Wiesen und Weidegründen grasen die roten und
braunen Kühe des Gutes und senden den Ton von tiefen Glocken
durch das flüssige Licht. Am Horizont suchen auf noch beschattetem
Hügel Schafe ihr Futter. Ein Schäfer mit einem großen Hut steht
neben ihnen. Er hält den Hirtenstab in der ausgestreckten Hand auf
die Erde gestützt, als sei er der Wächter dieses Tales und behüte
seine Unschuld. Eine Wolke zieht langsam über den bleichen
westlichen Himmel.
„So, nun stellen wir hier die Garbenbündel auf,“ sagt Wolfgang.
„Du bist ja früher auf dem Land gewesen und weißt, wie man das
macht. Immer zu sechs auf einen Haufen.“
„Bei uns nahm man acht.“
„So ... na ja, wir nehmen immer sechs. Weiß der Teufel, warum.
Bald kommen die ersten Leiterwagen vom Gut. Dann gehen wir dort
auf das Feld, – siehst du es? – und packen das Korn auf. Das macht
immer sehr viel Spaß.“
Wir arbeiten schweigend und mit gesammeltem Eifer. Die Ähren
stechen unsere Hände wund und ihre Körner rieseln uns in Hemd
und Hose. Wolfgang macht manchmal eine Bewegung, als habe ihm
jemand kaltes Wasser in den Nacken gegossen.
Später singt er mit klarer Stimme und deutlicher Aussprache einen
altfranzösischen Chanson. Da ist von einem Grafen die Rede, dem es

nicht wohl erging, weil seine Gemahlin der Majestät von Frankreich
allzusehr gefiel.
*
Bald vernehmen wir das Rollen und Klappern von Wagen, die über
die Landstraße zu uns herauffahren. Wir haben unsere Arbeit gerade
beendet, als wir die Rufe der Bauern hören, die mit ermunterndem
Einsprechen ihre Pferde einige schwere Hügel erklimmen lassen.
Dann ertönt das Dröhnen von Wagen, die über eine hölzerne Brücke
fahren, und gleich darauf ziehen sie alle an uns vorbei. In einem der
Wagen sind nur Frauen. Sie haben alle rote Tücher um die Köpfe
geschlungen. Jedermann wünscht uns: „Guten Morgen!“ worauf wir
beinahe feierlich unsere Mützen lüften und den Gruß erwidern. In
einem Gefährt sitzt ein hübsches junges Mädchen. Ich nicke ihr zu,
worauf sie verlegen zu Boden sieht. Ich bin sehr stolz, das erreicht
zu haben.
Der letzte Leiterwagen wird von einem Bauernjungen gelenkt, der
auf dem linken Pferde sitzt. Er grüßt uns, wie ein Souverain zu
grüßen pflegt.
„He Hans!“ ruft Wolfgang. „Bleib du bei uns!“
Hans steigt vom Pferd. Wolfgang legt seinen Arm auf die Schultern
des Jungen und führt ihn zu mir heran. Die beiden stehen der Sonne
entgegen, blinzeln, sind wohlgestaltet, blond, und – seltsam – sie
sehen einander ähnlich.
„Ich stelle dir hier meinen Freund Hänschen Kietschmann vor.“
Der Junge macht eine Verbeugung, eine leichte, weltmännische,
garnicht zu tiefe Verbeugung, und bietet mir die Hand, die ich
schüttle.
Er geht fort, um noch einige Bauern zu holen. Ich sehe ihm nach.
Er ist schlank und groß gewachsen.
Wolfgang macht ein sonderbares Gesicht und lächelt.
„Nun?“
„Wie?“
„Ist dir etwas ... wie soll ich sagen ... aufgefallen?“
„Aufgefallen? ... Nein, ... das heißt ...“

Ich bin mit einem Male verwirrt.
„Er sieht dir ähnlich.“
Wolfgang nickt, sieht zum Himmel, zieht die Nase kraus, blinzelt,
schluckt herunter und sagt:
„Er ist mein Halbbruder.“
„Wie –?“
Wolfgang bewegt seine Hand in einer sehr sprechenden, etwas
frivolen Art.
„Mein Gott, ... wir vergessen, daß unsere Väter auch jung waren
... Mein Vater lebte hier allein ... na und ... wie das so kommt.“
Er geht mit graziösem Schritt fort, um die Gabeln vom Graben zu
holen.
Ich schüttle den Kopf, wundere mich und vergesse im nächsten
Augenblick alles.
Wir arbeiten schweigsam fort.
Hans Kietschmann steht zusammen mit einem Bauern oben auf
dem Wagen und packt das Korn auf. Neben uns sind Weiber, die von
Zeit zu Zeit miteinander sprechen. Ein leichter, von der
aufsteigenden Sonne gewärmter Wind trägt aus der Richtung der
anderen Wagen den Schall von Reden und Gelächter zu uns herüber.
Es beginnt allmählich heiß zu werden. Die Augen schmerzen ein
wenig; ich sehe nichts als flimmerndes Gelb. Die Weiber riechen
nach Schweiß. Die Ochsen sind von Fliegen geplagt und schlagen
mit den Schwänzen kräftig umher. Ich fühle mich sehr wohl. Nina ist
vergessen, vollkommen vergessen. Wie süß es ist, daran zu denken,
daß ich Nina so völlig vergessen habe.
Es schlägt zwölf Uhr, wir hören mit der Feldarbeit auf, trinken
Wasser und ziehen die Jacken an.
Ich gebe Wolfgang die Hand.
„Danke für den Vormittag, Wolfgang.“
Wolfgang lächelt und nimmt meinen Arm. Wir gehen als Freunde
zum Schloß. Wolfgang ist zärtlich und spricht sehr viel.

N
10
achdem wir in unsern Zimmern Gesicht und Hände erfrischt
hatten, betraten wir die Veranda, um dort zu lunchen.
Nina saß am Tisch. Sie schien sich zu langweilen und benahm
sich wie ein kleines Mädchen, das auf seine Mahlzeit wartet.
Ich betrachtete Nina von der Seite. Sie hatte ein steifes weißes
Kattunkleid an. Ihr Hals und ihre Arme waren nackt. Auf ihrer Brust
trug sie eine Brillantenbrosche, an der linken Hand, der
elfenbeinernen mit den langen schmalen Fingern, leuchteten vier
herrliche Saphire von mildem Blau. Das kastanienbraune Haar war
eine Pracht, eine Krone, ein Akkord von rauschenden, dunklen
Tönen.
‚Mein Gott und dennoch, was ist denn Nina? Ein kleines Mädchen,
das sich langweilt! Aber ein Mädchen, das ich liebe? Nun ja, was ist
schon dabei? Viele Jungens lieben viele Mädchen. Da ist gar nichts
dabei.‘
Ich fühlte mich Nina überlegen.
Ich setzte mich an den Frühstückstisch. Obwohl es sehr heiß war,
hatte Nina einen Schnupfen, was mir ganz sonderbar vorkam.
Sie führte ihr Tuch an den Mund und fragte mit einer Stimme, die
heute noch näselnder klang als sonst:
„Wo habt ihr denn eigentlich so lange gesteckt?“
In diesem Augenblicke wurde es mir recht deutlich, daß Nina gar
nichts anderes war als eine große faule schöne Katze. Ich beugte
mich spöttisch vor bis auf die Tischplatte und sagte von unten zu ihr
aufblickend:
„Wir haben gearbeitet, – und Sie, was haben Sie getan?“
„Ich habe geschlafen.“
„Ah, Sie haben geschlafen ...“

„Jawohl; ich bin nämlich kein Troubadour, der wie ein Hase mit
offenen Augen nachts im Felde schläft.“
Hier betrat Frau Seyderhelm die Veranda. Sie begrüßte mich sehr
herzlich, schalt auf das freundlichste, daß ich die Nacht draußen
zugebracht hatte, und sprach die Erwartung aus, daß ich nun doch
die Ferien auf Wiesenau verleben würde.
Man frühstückte.
Es stellte sich im Lauf des Gesprächs heraus, daß Frau Seyderhelm
mir am Tag nach der Gesellschaft einen Brief mit der Einladung nach
Wiesenau in die Wohnung geschickt hatte, der nicht mehr in meine
Hände gekommen war.
Nina begann mit einer Geschichte, die so komisch war, daß wir alle
fürchterlich lachen mußten. Sie sprach lebhaft, mit vielen Gesten,
erzählte vorzüglich und ward durch ihren Erfolg so angeregt, daß
sich der Schnupfen zu verlieren schien.
Wolfgang machte seiner Mutter kopfschüttelnd Vorwürfe, daß die
Gänseleberpastete schon seit einigen Tagen nicht mehr genügend
auf Eis liege. Dann wandte er sich zu mir und fragte mit einer
kindlich hohen, liebenswürdigen Stimme:
„Ißt du Radieschen gern?“
Man hörte von Frau Seyderhelm, daß die Gräfin Königsmarck
heute morgen dagewesen sei; man sprach dann sehr lange über die
Gräfin Königsmarck. Nina schien sie nicht zu lieben. Wolfgang
behauptete, diese Dame röche nach wilden Tieren.
„Wolfgang, so spricht man nicht von einer Dame!“ sagte Frau
Seyderhelm.
Nina jubelte und begann ohne den mindesten Zusammenhang
eine Schilderung zu entwerfen, wie sie auf der Treppe meinen
Ranzen gefunden und aufgemacht habe.
„Stellen Sie sich vor, Frau Seyderhelm: er reist mit einem
zerrissenen Hemde, einer Zahnbürste, zwei alten Brötchen und dem
Werther; den Werther hat er in seine Socken gepackt!“
Man lachte sehr. Mich erfaßte mit einem Mal der unbezähmbare
Drang, Ninas Hand, die elfenbeinerne mit den spitzen Nägeln und
der kühlen Haut, zu küssen. Ich bückte mich nach einer Serviette

und berührte wie zufällig Ninas Finger mit meinen Lippen. Nina ließ
es ruhig geschehen; sie tat, als habe sie nichts gespürt.
„Es war übrigens gar nicht der Werther,“ sagte ich, als ich wieder
aufrecht saß. „Es war die Versuchung des Pescara.“
Ich bediente mich mit einer kalten Reisspeise und war von
meinem Abenteuer so aufgeregt, daß ich kaum schlucken konnte.
„Oh, die Versuchung des Pescara,“ sagte Frau Seyderhelm. Und
sie fing an, sich des längeren über „Huttens letzte Tage“
auszulassen.
Wolfgang zog ein gelangweiltes Gesicht und schlug Nina für den
Nachmittag eine Tennispartie vor. Sobald er mit Nina sprach, war
seine Stimme zart und fast unterwürfig.
Frau Seyderhelm hob die Tafel auf.
„Schreiben Sie mir später den Namen Ihrer Wirtin auf, lieber
Walter,“ sagte sie. „Man soll uns Ihre Sachen nachschicken.“
Ich küßte Frau Seyderhelm die Hand und verbeugte mich vor
Nina.
„Spielen Sie Tennis?“ fragte Nina.
„Ja, ein wenig.“
Sie fuhr mit ihrer Zunge zwischen den Lippen einher.
„Du reitest heute nicht mehr, Wolfgang?“
„Nein; es ist zu heiß.“
Ich spürte plötzlich den Duft von Ninas Körper. Ich sah ihren
weißen Hals und erbebte.
Nina lächelte.
„Addio, meine Herren. Ich gehe in den Wald.“
„Addio.“
Wolfgang zog sich in die kühlen Räume zurück.
Ich blieb auf der Veranda und sah in den Park. Nina ging langsam
die kiesbedeckte Allee entlang, blieb zuweilen stehen, betrachtete
mütterlich ein Blättchen, das sie mit der kühlen Hand liebkoste,
pflückte eine Rose vom Blumenbeet und befestigte sie an ihrer
jugendlichen Brust. Darauf verlor sie sich – unvergleichlich
ebenmäßig ausschreitend – im mittäglichen Gehölz.
Die Gutsglocke schlug ein Uhr. Malatesta, der Hofhund, dehnte
sich schläfrig, beroch mißtrauisch seine Pfote und legte sich auf den

Rasen. Der Diener räumte den Frühstückstisch ab.
*
Am Nachmittag lag ich irgendwo im Wald auf dem Rücken und
träumte in den blauen Himmel hinein. Manchmal streichelte ich den
schönen Malatesta, der mich begleitet hatte. Es war sehr heiß. Der
Hund hob zeitweise den Kopf, stieß, von Wärme bedrückt, den Atem
aus der Kehle, ließ die Zunge hängen und hatte feurige Augen. Mich
plagten die summenden und stechenden Mücken. Ich begann
unruhig und gestört zu schlafen. Böse Träume von großer
Leidenschaft und überquellender Sehnsucht verfolgten mich. Ich
sah, wie Nina zu mir, dem Schlafenden, trat, ihr mokantes Lächeln
lächelte und mit einem Male mütterlich, mit drängenden Händen und
junger weißer Brust sich neigte.
Der nahe Gong, der zum Tee rief, weckte mich auf. Die Sonne war
tiefer herabgesunken; unter ihren schrägen Strahlen beruhigte sich
die Welt und wurde kühl. Ein Wind ging durch die Bäume, der in den
Blättern flüsterte und schluchzte. Der Hund war fortgelaufen. Ich
fühlte, daß alles nutzlos sei und ich ewig einsam bleiben müsse.
*
Gegen Abend spielten wir Tennis.
Nina war biegsam, schmal in den Fesseln und schnellfüßig. Ihre
Hand war sicher, der Schlag ihres Rackets ruhig.
Wolfgang, ihr Partner, war weißgekleidet, hatte den rechten Ärmel
seines Hemdes aufgeschlagen und zeigte einen braungebrannten,
schmalen und kräftigen Arm.
Ich gab streng auf das Spiel acht und hatte den brennenden
Ehrgeiz, mich gut zu halten. Ich verlor das erste Match, trat beim
Wechseln an das Netz, beglückwünschte Nina und küßte ihre Hand.
Wolfgang sah mich ein wenig befremdet an. Nina lächelte, war
unendlich liebenswürdig, legte einmal beim Gespräch ihre Hand auf
meinen Arm und nannte mich Walter. Ich war rasend vor Glück,

machte ein hochmütiges Gesicht und verdoppelte meine
Anstrengungen.
Mir war, als ständen Nina und Wolfgang in abendrotem Dunst und
rosafarbenem Nebel. Jedermann von uns spielte mit streng
geschlossenen Lippen. Nichts unterbrach das Schweigen als nur das
Aufschlagen des Balles, das Summen des festgespannten Rackets
und zeitweis ein kleiner Ausruf der Überraschung oder des Ärgers.
Niemand zählte laut, denn jeder von uns wußte, wie wir standen.
Frau Seyderhelm trat ans Gitter; wir grüßten flüchtig und spielten
weiter. Frau Seyderhelm sprach mit einem Gärtner, deutete einmal
mit der Hand auf ein Blumenbeet und wandte sich über unsern Eifer
lächelnd zum Gehen. Ich wurde gewahr, daß sich mein Spiel von
Minute zu Minute verbesserte. Im letzten entscheidenden Set
gewann ich alle sechs Spiele und war somit Sieger im Match. Nina
sagte uff und fächelte sich mit ihrem Tuch kühle Luft ins Antlitz. Als
wir uns die Hände schüttelten, sah sie mich wie zum erstenmal an.
In ihren Augen leuchtete mir etwas Verlockendes und Gefährliches
entgegen.
„Sie spielen gut,“ sagte Nina. „Reiten Sie?“
„Gewiß.“
„Wolfgang, wir werden morgen früh reiten.“
„O Nina, rede keinen Unsinn, das hast du schon zehnmal gesagt.
Du stehst ja doch nicht um sieben Uhr auf.“
„Doch, ich werde ganz bestimmt um sieben Uhr aufstehen.“
Sie sah mich wieder mit ihren lockenden Augen an, wobei sie die
Lider ein wenig zusammenzog. Mir war, als liebkosten mich die
goldfarbenen seidenen Wimpern.
„Was wird Herr Regnitz für ein Pferd reiten?“
O weh, sie sagte wieder Herr Regnitz!
„Willst du einen ruhigen Gaul, Walter?“
„Nein, im Gegenteil.“
„Gut, du sollst die Moissi haben. Eine Rappstute, weißt du. Du
bekommst den neuen Sattel, den mir Mama geschenkt hat.“
„Hören Sie zu, Walter, das ist eine unerhörte Gnade.“
O – sie sagte wieder Walter!

Ich spürte in diesem Augenblick den einzigartigen Duft von Ninas
mädchenhaftem Körper. Ich sog ihn wissend und gekräftigt ein.
Der Teufel wird mir an diesem Abend wenig anhaben können. Ich
habe mein Match gewonnen und morgen reite ich Moissi.
*
Die Damen zogen sich bald nach dem Abendessen zurück.
Wolfgang und ich, wir saßen noch eine Weile auf der Terrasse,
fühlten eine angenehme Ermüdung in unsern Gliedern und tranken
ein wenig Black and White mit sehr viel Sodawasser gemischt.
Wir sprachen nicht viel, sondern sahen zum reichbesternten
Himmel empor und beobachteten die Sternschnuppen. Der Diener
setzte einen Eiskühler neben den Tisch und verschwand.
„Nina reitet gut,“ sagte Wolfgang. „Ich werde ihr mal morgen den
‚Sekt‘ geben. Da kann sie was erleben.“
Und dann, nach einer Weile:
„Mama hat im vergangenen Jahr viel Sorge mit dem Stall gehabt.
Weißt du, der Rotz ... Na, jetzt ist es vorbei ...“
„So?“
„Ja, jetzt sind sie wieder alle gesund. Einer ging ein. Na,
meinetwegen, mir lag nichts an ihm. Ein Wallach.“
Ein Knecht schritt mit einer Laterne durch den Garten. Wir sahen
dem unruhigen Licht nach.
„Komisch,“ sagte Wolfgang plötzlich, „wir kennen uns erst seit
sechs Tagen.“
„Ja.“
Eine Stille.
„Du bist immer so hochmütig. Hast du was?“
„Nein. Garnichts.“
Eine Stille.
„Du mußt in den Herbstferien herkommen und hier mit uns
jagen.“
„Danke. Ja.“
Mir stieg ein Gedanke auf.
„Jagt Nina auch?“

„Ja, sie schießt sehr gut. Sie hat gar keine Angst.“
„Wie schön.“
Ich sah ein Bild vor mir: Nina mit dem unvergleichlichen Gang der
Kosakenmädchen durch den Wald schreitend, die Büchse in der
Hand, mit spähenden Augen und grausamen Lippen.
„Wie schön,“ wiederholte ich.
Ein Stern glitt in mächtiger und graziöser Bewegung durch den
erleuchteten Raum.
„Hast du dir etwas gewünscht?“ fragte Wolfgang.
„Ja.“
„Was denn?“
„Mehr Whisky.“
Wolfgang lachte und schenkte ein.
„Na, Mama wird morgen Augen machen über unsere Sauferei.
Prost!“
„Prost!“
Wir schwiegen lange.
„Man muß das Leben mit gesunden Händen anfassen.“
Wolfgang sah mich unsicher an. Dann sagte er verlegen:
„Ja.“
Wir beobachteten zwei Fledermäuse.
„Was denkst du über die Frauen?“ fragte ich.
„Über welche Frauen?“
„Ich meine ... fändest du etwas dabei, wenn Jungens wie wir ...
ein Verhältnis haben?“
„Nein ... ja, das heißt ... es kommt darauf an!“
Wolfgang lachte ein wenig hilflos.
Ich stand auf und bot ihm die Hand.
„Wir sollten recht lange Zeit Freunde bleiben,“ sagte ich sehr
herzlich.
Auch Wolfgang erhob sich. Er schüttelte meine Hand kräftig, und
es lag in dieser Bewegung etwas eigentümlich Ritterliches.
„Ja, das sollten wir wirklich,“ erwiderte er in demselben Ton.
„Gute Nacht, Wolfgang.“
„Gute Nacht, Walter, – und danke für alles.“
Ich ging in mein Zimmer.

W
11
ir reiten zu dritt im abgekürzten Galopp – von Hans
Kietschmann gefolgt – über eine jüngst gemähte Wiese, deren
Heu naß und ohne Duft ist. Wir reiten Schulter an Schulter und
achten streng darauf, daß die Linie eingehalten wird. Jeder von uns
beschäftigt sich schweigend mit seinem Pferde, beobachtet den
gebogenen Tierhals und übt auf jeden Druck den Gegendruck der
Schenkel aus.
Manchmal sehe ich zu Nina hin. Das feurige Haar lodert wie eine
Flamme, wie ein Triumph unter dem schwarzen Hut hervor; die
weißen Kinderzähne beißen auf die feuchte Unterlippe, die
unbedeckten Hände erfassen die Zügel des unruhigen Pferdes mit
freudiger Kraft. Unausgesetzt richtet Nina die verliebten Blicke auf
den Kopf des Pferdes, das in großzügiger Bewegung galoppiert. Ich
sehe mit Vergnügen, daß der schlanke Körper mit den säulenstarken
hohen Beinen und der jugendlichen weichen Brust sich entzückt der
Bewegung des schnaubenden und wiehernden Tieres hingibt und
niemals die Verbindung mit ihm verliert.
Es geschieht einige Male, daß Sekt sich nahe an meine Stute
drängt und Ninas Fuß den meinen berührt.
Hatte ich nicht die ganze Nacht von der einen Minute geträumt, in
der Nina ihren Fuß auf meine Hand setzen würde, um das Pferd zu
besteigen? Und war ich nicht, als sie es wirklich getan, verwirrt und
mit pochendem Herzen davongestürzt?
Sekts Gangart wird von Augenblick zu Augenblicke länger. Der
Schimmel und seine Herrin freuen sich des wie unbegrenzten
Raumes, der morgendlichen Luft und der würzigen Gerüche des
Feldes.

Ich sehe unsicher zu Wolfgang hin, der immerfort mit tiefer
Stimme auf den Schimmel einspricht:
„Ruhe! – Sekt! – Ruhe! – Ohlala – Ohlala!“
Meine Moissi geht leichtfüßig mit. Wolfgangs nicht so belebtem
Fuchs wird es schwer, die Linie einzuhalten.
„Ruhe, Fräulein Nina!“ sage auch ich jetzt. „Bitte abgekürzter
Galopp!“
Aber Nina hört nichts. Sie sieht verzückt, mit nassem, erregtem
Munde und blinkenden Augen auf den Schimmel und beißt mit den
weißen Zähnen auf die Lippe.
„Gib auf die Sporen acht!“
In diesem Augenblick tut Sekt, den irgend etwas erschreckt hat,
einen kleinen Sprung, Nina kommt mit den Sporen an die Weichen,
der Schimmel wirft den Kopf mit einer schmerzlichen Gebärde in die
Höhe und geht durch.
Moissi folgt sofort. Wolfgang und Hans Kietschmann bleiben
zurück.
*
„So, Fräulein Nina ... jetzt Ruhe, nur Ruhe!“
Die Pferde rasen über das Feld. Die Morgensonne erhebt sich
gelbstrahlend über einem Hügel und blendet uns.
„Rechte Kandare ziehen! ... Sekt, Ruhe!“
Nina richtet das Tier mit allen Kräften nach rechts.
Wenn ihr nur nichts geschieht! ... Nein, sie ist ruhig. Es geschieht
ihr nichts.
„Mehr rechts, immer mehr rechts! ... Fort vom Stall! ...“
Sieh da, sie ist zufrieden, sie ist hingegeben dieser einzigartigen
Geschwindigkeit, dieser goldenen Flucht durch den Morgendunst.
„Noch mehr rechts! ... Bravo, Fräulein Nina! Noch mehr!“
Wir beschreiben mit unserem Ritt eine Kurve.
„Reitpeitsche fortwerfen!“
Nina läßt die Peitsche fallen.
Ich bekomme über meine Stute Gewalt, meine Knie und Schenkel
sind unausgesetzt an den Sattel gepreßt. Ich drücke den Rappen an

Nina heran.
„Noch einmal nach rechts ... sehr gut! ... Noch einmal! ... Ah, er
läßt nach ...“
Ich beuge mich vor und greife in Ninas Zügel. Der Schimmel
erschrickt, bäumt sich, – ich packe den Halfter und der Schimmel
steht.
Nina lacht, ein nervöses, schreiendes, jubelndes Lachen.
Ich steige von meinem Pferd, um Sekt liebkosend zu beruhigen.
Ein unerklärlicher Gram erfaßt mich, ich spreche kein Wort, sehe
Nina nicht an und bebe vor Schmerz und Zorn ...
Wolfgang erreichte uns endlich. Er lacht.
„Bravo Nina! – Nichts geschehen?“
Nina schüttelt den Kopf.
„Ein schöner Unsinn, dieses Biest da mit Sporen reiten zu lassen!“
sage ich scharf und böse.
Wolfgang zieht ein beleidigtes Gesicht.
„Nehmen Sie die Sporen ab!“ herrsche ich Nina an, ohne
hinaufzusehen.
Wolfgang und Hans steigen von den Pferden.
„O – Sie sind zornig, Walter!“ ruft Nina.
Ich blicke auf. Ninas Augen lachen, aber sie ist blaß, sehr blaß,
und ihre Lippen zittern nervös.
„Nehmen Sie jetzt bitte die Sporen ab.“
Hans befreit Nina von den Sporen und reitet zurück, um auf der
Wiese die Reitpeitsche zu suchen. Ich stecke die Sporen in meine
Tasche.
Wir reiten im Schritt weiter und erreichen ein belichtetes Gehölz.
Unsere Tiere sind ermüdet und zufrieden. Sie gehen in großen
Schritten durch den Wald und spähen an den stolzen
Fichtenstämmen stolz vorbei. Wir sind schweigsam und schlecht
gelaunt.
Mit einem Male streckt Nina die Hand nach mir hin. Da ich nicht in
ihrer Nähe bin, fingert sie ungeduldig in der Luft herum. Ich nehme
ihre Hand, beuge mich tief nach unten und küsse sie lange.
Wie ich mich emporrichte, sehe ich, daß Nina mit lächelndem
Antlitz und feuchten goldenen Wimpern nach der andern Seite blickt.

Wolfgang ist blaß geworden und hält die Augen gesenkt. Hans reitet
irgendwo hinterher.
Wir erreichen, ohne ein Wort zu sprechen, nach einer Stunde den
Gutshof. Die Pferde sind naß und wollen ihr Futter. Ich grüße Nina
mit dem Hut und gehe ins Haus.

W
12
ir fuhren am Abend mit einem leichten Jagdwagen ins Gebirge.
Frau Seyderhelm war im Schloß geblieben, da sie Besuch
erwartete.
Wir saßen auf der Terrasse eines vornehmen und einsam am Fluß
gelegenen Hotels. Vor unseren Blicken zerflossen die kupferbraunen
Abhänge und goldenen Bergeshäupter, die ein unaufhörlich
gleitendes Licht belebte.
Ich stand, noch ehe die Mahlzeit bereitet war, im Stalle bei den
Pferden und sorgte dafür, daß sie ihr Futter bekamen. Mein Kopf war
benommen, und meine Augen brannten. Den ganzen Tag in Ninas
Kreise zu leben, den Hauch ihrer Lippen zu spüren, im Wagen ihren
Knieen nahe zu sein und ihrem duftenden Haar, zu sehen, wie der
Wind das helle, sich innig an den Körper schmiegende Sommerkleid
berührte, und mit verwirrten Sinnen zu ahnen, vieles zu ahnen, – ah,
das alles war nicht ganz leicht zu ertragen.
Ein Kellner meldete, das Essen sei angerichtet. Ich stieg die
steinerne Treppe der Terrasse langsam hinauf. Die unaufhörlich
wechselnden Farben des Abends quälten mich; ein drohendes
Verhängnis war in dieser Bewegung, eine Unruhe ohnegleichen, eine
süße und unsäglich schmerzliche Hast, eine Flucht und ein Jammer
ohne Trost ...
Als ich oben angelangt war, sah ich, wie Nina ihre Hand auf
Wolfgangs Arm gelegt hatte. Sie schien ihn etwas zu fragen. Er
beantwortete Ninas Frage, und sein Gesicht bekam den überaus
liebenswürdigen und ritterlichen Zug, den ich an ihm liebte. Ein
kindliches, verhaltenes Schluchzen stieg in mir empor.
Ich setzte mich an den Tisch, Nina und Wolfgang sahen mich an.
„Na Lieber? Wie gehts?“ fragte Wolfgang.

„Danke, die Pferde fressen.“
Nina lachte und blickte fort.
Ich wurde rot.
Nina sprach in näselndem Ton von Trüffeln.
„Sieh mal, Wolfgang, wie witzig, hier gibt es gefüllte Trüffel.
Raffiniert – nicht?“
„Nina, du redest wie ein Kavallerieoffizier,“ sagte Wolfgang,
wandte mir sein Gesicht schräg zu und fragte in seinem kindlichen
Ton:
„Spricht sie nicht wie ein Gardekürassier?“
Wir aßen danach Forellen. Nina verstand es gut, das zarte rosige
Fleisch der Fische von den Gräten loszulösen. Die weißen, nun der
Seele beraubten Tieraugen starrten ausdruckslos zu uns herauf. Nur
um die Mäuler lag ein böser Zug, der von Todespein und letztem
Kampf erzählte.
Um die Zeit der späten Dämmerung trat ein Hirsch aus dem Wald
des gegenüberliegenden Berges hervor, äugte mit einer kühnen
Gebärde des Kopfes nach dem Hotel hin und trank aus dem Fluß.
Der Geruch von Bergwasser und nassem Sand stieg zu uns empor.
Allmählich entfaltete der dunkelnde Himmel die Schönheit der
beginnenden Nacht vor unsern Augen. Die stolzen Gestirne wurden
sichtbar; vor ihrer urweltlichen Starrheit wichen die wechselnden
Farben des Abends besiegt zurück. Das Gebirge ward im funkelnden
Schein groß und ehern.
Wir standen nach beendetem Mahle auf und gingen über die
hölzerne Brücke des Flusses dem andern Ufer zu. Die Nacht gab mir
mitleidsvoll von ihrer Kühle und besänftigte mich wunderbar. Nina
schien mir schöner denn je, aber ihre Schönheit war meinen Sinnen
und meinem undeutlichen Verlangen entfernt. Sie ging mit ihrem
weißen Sommerkleid wie durchsichtig durch die Nacht dahin. Auf
ihren Schultern lag ein bläuliches Orenburger Tuch. Ihr Haar war
unbedeckt und bewegte sich ein wenig im Nachtwind.
Ein leises, sehnsüchtiges Tönen rief uns in den Wald. War es eine
Flöte oder eines Mundes Klage? Wir folgten neugierig der oft
entschwindenden und dann wieder genäherten Musik.

Vor einem Bretterverschlag, dem Sammelplatz der Tiere, machten
wir Halt. Wir sahen die Gestalt eines Mannes zwischen sternhellen
Bäumen einhergehen, wir sahen ihn in seine Schürze greifen und –
einem Sämann gleich – Eicheln und Kastanien mit einer weiten
Bewegung seines Armes über den Waldboden streuen. Dazu pfiff er
eine Melodie, eine kleine, sentimentale, unbeholfene und doch
unendlich rührende, süße, zärtlich lockende Melodie. Nach einer
Weile schien es, als bewege sich der Wald. Unhörbar, aber mit
großzügigen Bewegungen und bei jedem Schritt ein wenig mit den
Häuptern nickend, kamen wie aus einem dunkel gewebten Teppich
Hirsche und Rehe aus der Nacht hervor, beugten sich zu Boden und
näherten sich langsam dem lockenden Freund der Tiere. Allmählich
entfernte sich der Mann, umdrängt von seinen zärtlichen
Geschöpfen, ferner und ferner klang die Musik seines Mundes und
löste sich endlich auf im Rauschen des Waldes.
*
Wolfgang eilte voraus, um mit Hans die Pferde anzuschirren. Es
zeigten sich Wolken am Himmel.
Ich ging mit Nina langsam den jäh erleuchteten Waldweg entlang.
Nina hatte wieder ihren Schnupfen und führte das kleine Tuch
oftmals an den Mund.
„Walter.“
„Ja.“
„Wie alt sind Sie?“
„Siebenzehn Jahre.“
„Siebenzehn Jahre,“ wiederholte Nina.
Eine Stille.
„Walter.“
„Nina?“
„Sie werden morgen fortreisen, – nicht wahr?“
Und da sie mein Gesicht sah, hob sie beschwörend die bittenden
Hände empor und sagte in unvergleichlich rührendem Ton:
„Walter, – Sie sind siebenzehn Jahre!“
Ich hatte wieder solche Angst.

Ich werde mich töten, dachte ich.
Eine lange Stille.
„Sie werden reisen, Walter?“
„Ja.“
„Danke.“
Ich werde mich töten. Es wird noch diese Nacht geschehen.
*
Wir fuhren über Felder. Wolfgang kutschierte, wobei er manchmal
einige Worte mit Hans wechselte. Ich saß mit Nina in der Break.
Nina sprach viel und war nervös.
Es erhob sich ein Wind und trieb große, von den Sternen erhellte
Wolken über den Himmel. In der Ferne leuchteten Blitze.
Nina klagte über den Sturm, der ihr Kopfschmerzen verursachte,
und bat, man solle die Verschläge herunterlassen. Der Wagen hielt,
die Pferde stampften ängstlich auf dem undeutlichen Feldwege, und
Hans spannte die leinenen Gardinen auf.
Wir waren nun von den andern durch eine Wand getrennt und
sahen die Welt einzig durch die Öffnung über der Türe. Wir hörten
von irgendwoher kleine Bäche rauschen, den Wind im Korn und in
entfernten Wäldern blasen, und aufgescheuchte Enten, die schreiend
nach irgend einem wohlgeborgenen Teiche zogen.
„Sie frieren, Walter?“
„Nein. Danke.“
Nina hüllte sich fester in das weiche blaue Gewebe ihres Tuches.
Ein Blitz zuckte.
„Haben Sie den Hasen gesehen, Walter?“
„Ja.“
Wir fuhren über eine Brücke. Das Holz dröhnte.
„Sie haben noch einen Vater, Walter?“
„Ja.“
„Wo ist er?“
„In Skandinavien.“
„Allein?“
„Anny Döring ist bei ihm.“

„Wie? – Die Soubrette?“
„Ja.“
„Ach –!“
Nina blickte mich verwundert und ängstlich an.
Wie liebte sie in diesem Augenblick meinen Vater. O Nina, Nina!
Ich sah lange Zeit hinaus und träumte. Ich fühlte, daß mich Nina
unausgesetzt betrachtete. Später vergaß ich es.
Eine Hand lag auf der Decke. Es war Ninas Hand.
„Darf ich sie küssen?“ fragte ich.
Nina lachte mit einem hellen Ton. Es klang, als fiele ein kleiner
silberner Hammer schnell auf Metall.
Ich küßte die Hand und dachte dabei an den Förster, der durch
den Wald ging und Eicheln über die Erde streute. Ich küßte keine
lebendige Haut, sondern Wildleder, dänisches Wildleder. Ich küßte
dieses Leder noch einige Male und ließ die Hand dann fahren. Ich
empfand kein besonderes Vergnügen dabei und wunderte mich.
Wahrscheinlich träumte ich dies alles nur, sonst wäre ich doch wohl
anders gewesen. Ich hätte vielleicht geschrieen ...?
Es begann langsam zu regnen. Ich streckte die Hand hinaus.
Große warme Tropfen fielen hernieder.
„Wir werden morgen nicht Tennis spielen können,“ sagte ich
schläfrig.
„Ja,“ erwiderte Nina verwundert.
Ach so, ich reise ja morgen fort, dachte ich. Wie ungeschickt!
Ich träumte fort, sah Steine, Wolken und Bäume vorbeieilen; oben
sprach Wolfgang irgend etwas, was ich nicht verstand, und der
Donner wurde stärker, immer stärker.
Nein, ich werde morgen nicht fortreisen. Ich werde mich heute
Abend töten.
Schafe standen zusammengedrängt und fürchteten sich ... Sieh
da, Schafe ... „Und es waren Hirten in derselbigen Gegend auf dem
Felde bei den Hürden, die hüteten des Nachts ihrer Herde. Und
siehe, des Herrn Engel trat zu ihnen, und die Klarheit des Herren
leuchtete um sie; und sie fürchteten sich sehr. Und der Engel sprach
zu ihnen: Fürchtet euch nicht, siehe, ich verkündige euch große
Freude ...“ wie schön, – siehe, ich verkünde euch große Freude! Mir

war mit einem Male, als sei mein Körper durchströmt von gutem
warmem Blut. Es war ja alles gar nicht so schlimm! Denn ich
verkünde euch große Freude ...
Da – was war das? Eine bebende Hand griff nach meiner. Mein
Traum zerriß – –
„Nina!“
Ich schrie.
„Sei still, um Gottes willen ...“
„Hallo, was gibt’s?“ fragte Wolfgang.
„Nichts. Ninas Haar im Wind ...“
Ich riß Nina an mich, überflutete ihr Antlitz mit Küssen, umarmte
ihre Kniee und biß in ihre Lippen und Hände ...
„Laß ... Laß ... Du bist verrückt.“
Sie stöhnte.
Ich flehte unverhüllt mit meinen fiebernden Lippen auf ihren
Lippen, auf ihren Händen, ihrem Haar, ihren Augen und ihrer jungen,
jungen Brust ...
O unerhörtes Glück des Aneinanderschmiegens, der
verschlungenen Finger, der wirren, in die dunkle Luft
hineingesprochenen Reden!
Und dann dieses wunderbare, einzigartige Ermatten, diese
tränenreiche, gütige Müdigkeit, ... dieses bekümmerte Suchen der
Hände, ... und endlich diese Ruhe, diese tiefe, tiefe Ruhe! ...
Wie wir einst so glücklich waren!
*
Um Mitternacht stürmten die gepeitschten nassen Pferde mit
rasselndem Wagen in den Schloßhof. Frau Seyderhelm empfing uns
in der Türe. Sie war ein wenig müde, aber freundlich und besorgt.

I
13
ch stellte mich an das Fenster meines Zimmers und sah hinaus.
Blitze spalteten Eichen und Kiefern, und über Wälder und weite
Ebenen rollten ihre Donner. Aus den Ställen brüllten und
wieherten geängstigte Tiere, und Malatesta saß mit glühenden
Augen in seiner Hütte vor meinem Fenster und heulte.
Auch dies ging vorbei. Ein stetig und kühl strömender Regen
spendete uns, den Fiebernden, Genesung. Gerüche von niegeahnter
Kraft erfüllten die Luft, und die Tiere in den Ställen begannen ihren
Schlaf. Zwei Uhr schlug die Glocke, aber der trübe Morgen war noch
fern.
Ich setzte mich an den Tisch. Ich wollte etwas Unerhörtes
schreiben, aber ach, – es wurden nur diese einfachen Zeilen:
Ist es denn möglich, daß wir diese Nacht
In einem Wagen über Felder fuhren?
Hab’ ich geträumt? Ich sah doch einen Wald!
Eilten nicht Steine, Wolken, Bäume, Sterne
An uns vorbei, und hast du später nicht
– So hab’ ich doch geträumt, – und hast du nicht
Mir abgewandten Blicks die Hand gereicht?
... Und küßte ich sie nicht?
Ich habe nicht geträumt. Wir fuhren nachts
In einem Wagen über weite Felder,
Es eilten stille Wolken, Bäume, Sterne
An uns vorbei ... Du gabst mir deine Hand ...
... Ich küßte sie ... So hab’ ich doch geträumt?
Ich packte meinen Ranzen, nahm das Blatt, stieg zu Ninas Zimmer
hinauf, öffnete die erste ihrer beiden Türen und legte mein Gedicht

auf ihre Diele. Dann schlich ich mich hinunter.
Ich trat auf den Hof, streichelte Malatesta und dachte: Frau
Seyderhelm und Wolfgang ... ach, Frau Seyderhelm und Wolfgang!
Ich wanderte die Straße hinab, bis sich im Osten der bewölkte Tag
ankündete. Auf einem Hügel blieb ich stehen und sah die verlassene
bleiche Landschaft unter mir. Eine Starenkette flog durch die
gereinigte Luft des Morgenrots.
Da schlug ich mit der Stirn auf einen Baum und stürzte nieder.

Albert Langen, Verlag für Litteratur und Kunst, München
Karl Borromäus Heinrich
Karl Asenkofer
Geschichte einer Jugend
Zweites Tausend
Geheftet 3 Mark 50 Pf., geb. 5 Mark
Süddeutsche Monatshefte, München: Wenn ich aber sagen sollte,
welches erzählende Buch des letzten Jahres den stärksten und
nachhaltigsten Eindruck auf mich gemacht hat, so müßte ich Karl
Asenkofer von Karl Borromäus Heinrich nennen. Das ist mehr als
Litteratur: jede Zeile ist erlebt, und was noch wichtiger, jedes
Erlebnis ist behutsam aufbewahrt! noch hängt der ganze
Flügelstaub an den leichten Schwingen. Ein Buch von packender
Ehrlichkeit, die nichts hinzu tut, und so niemals den Eindruck des
Beabsichtigten, Arrangierten aufkommen läßt. Die letzten
Gymnasial-, die ersten Universitätsjahre sind kaum je so
unmittelbar und überzeugend wahrhaftig dargestellt worden. Als
Heldin steht von der ersten bis zur letzten Seite eine der
ergreifendsten Muttergestalten da. Dies Buch ist so ausgezeichnet,
daß man vor der Fortsetzung ganz Angst hat. Man möchte den
Verfasser inständig bitten, mit dem zweiten Teile zu warten, bis er
sich dem ersten an die Seite stellen kann: ja nicht zu früh, ja nicht
zu viel über seine augenblicklichen Erlebnisse zu berichten, sondern
in Gelassenheit und Demut geduldig zu warten, bis zum ersten
meisterlichen Bande ein zweiter von selber in Stille und Sturm reif
geworden ist. An dem Tag aber wollen wir uns mit ihm freuen,
denn an dem Tag ist unsere Litteratur um ein bleibendes Werk
reicher: um ein solches, das eine Generation weiter gibt an die
andere.

Albert Langen, Verlag für Litteratur und Kunst, München
Korfiz Holm
Thomas Kerkhoven
Roman
Vierte Auflage
Flexibel geb. 5 Mark, steif geb. 6 Mark
„The Times“, London: „Thomas Kerkhoven“ belongs almost to the
rank of classics like „Tom Jones“ or „David Copperfield“ or
„Pendennis“.
Rudolf Herzog in den „Neuesten Nachrichten“, Berlin: Sicher ist,
daß dieses Werk den besten Büchern beizuzählen ist, die in den
letzten Jahren erschienen sind.
Wilhelm Hegeler im „Litterarischen Echo“, Berlin: Auf jeder Seite
ist das Buch voll sprühender Lebendigkeit, von müheloser
Anschaulichkeit, amüsant und glänzend von Anfang bis zu Ende.
„Münchener Neueste Nachrichten“: Es wird seinen Weg machen;
denn es ist wert, den besten Dichtungen unserer Zeit an die Seite
gestellt zu werden.
„Berner Bund“: Ganz „verflixt gut geschrieben“ ist es, mit einer
geradezu bewunderungswürdigen Sicherheit in der Technik.
Druck von Hesse & Becker in Leipzig

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Essays On Fourier Analysis In Honor Of Elias M Stein Pms42 Course Book Charles Fefferman Editor Robert Fefferman Editor Stephen Wainger Editor

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Essays On Fourier Analysis In Honor Of Elias M Stein Pms42 Course Book Charles Fefferman Editor Robert Fefferman Editor Stephen Wainger Editor

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