Problems in Analysis

A SYMPOSIUM IN HONOR OF SALOMON BOCHNER PRINCETON MATHEMATICAL SERIES

Editors: PHILLIP A. GRIFFITHS, MARSTON MORSE, AND ELIAS M. STEIN

1. The Classical Groups, Their Invariants and Representation. By HERMANN WEYL. 2. Topological Groups. By L. PONTRJAGIN. Translated by EMMA LEHMER. 3. An Introduction to Differential Geometry with Use of the Tensor Calculus. By LUTHER PFAHLER EISENHART. 4. Dimension Theory. By WITOLD HUREWICZ and HENRY WALLMAN. 5. The Analytic Foundations of Celestial Mechanics. By AUREL WINTNER. 6. The Laplace Transform. By DAVID VERNON WIDDER. 7. Integration. By EDWARD JAMES MCSHANE. 8. Theory of Lie Groups: I. By CLAUDE CHEVALLEY. 9. Mathematical Methods of Statistics. By HARALD CRAMER. 10. Several Complex Variables. By SALOMON BOCHNER and WILLIAM TED MARTIN. 11. Introduction to Topology. By SOLOMON LEFSCHETZ. 12. Algebraic Geometry and Topology. Edited by R. Η. 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. · 18. Algebraic Geometry. By SOLOMON LEFSCHETZ. 19. Homological Algebra. By HENRI CARTAN and SAMUEL EILENBERG. 20. The Convolution Transform. By I. I. HIRSCHMAN and D. V. WIDDER. 21. Geometric Integration Theory. By HASSLER WHITNEY. 22. Qualitative Theory of Differential Equations. By V. V. NEMICKII and V. V. STEPANOV. Translated under the direction of SOLOMON LEFSCHETZ. 23. Topological Analysis, revised edition. By GORDON T. WHYBURN. 24. Analytic Functions. By NEVANLINNA, BEHNKE and GRAUERT, et al. 25. Continuous Geometry. By JOHN VON NEUMANN. Foreword by ISRAEL HALPERIN. 26. Riemann Surfaces. By L. AHLFORS and L. SARIO. 27. Differential and Combinatorial Topology. Edited by STEWART 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.

PROBLEMS IN ANALYSIS

A Symposium in Honor of Salomon Bochner

ROBERT C. GUNNING GENERAL EDITOR

PRINCETON, NEW JERSEY PRINCETON UNIVERSITY PRESS 1970 Copyright © 1970, by Princeton University Press All Rights Reserved L.C.CARD 76-106392 I.S.B.N. 0-691-08076-3 A.M.S. 1968: 0004

Printed in the United States of America by Princeton University Press, Princeton, New Jersey Foreword

A symposium on problems in analysis in honor of Salomon Bochner was held in Fine Hall, Princeton University, April 1-3, 1969, to celebrate his seventieth birthday, which took place on August 20, 1969. The symposium was sponsored by Princeton University and the United States Air Force Office of Scientific Research; the organizing com­ mittee consisted of W. Feller, R. C. Gunning, G. A. Hunt, D. Montgomery, R. G. Pohrer, and W. R. Trott. This volume contains some of the papers delivered by the invited speakers at the symposium, together with a number of papers contributed by former students of Professor Bochner and dedicated to him on this occasion. The papers were received by June 1, 1969.

Contents

PART I: LECTURES AT THE SYMPOSIUM

On the Group of Automorphisms of a Symplectic Manifold, by EUGENIO CALABI 1

On the Minimal Immersions of the Two-sphere in a Space of Constant Curvature, by SHIING-SHEN CHERN 27

Intersections of Cantor Sets and Transversality of Semigroups, by HARRY FURSTENBERG 41

Kahlersche Mannigfaltigkeiten mit hyper-^-konvexem Rand, by HANS GRAUERT and OSWALD RIEMENSCHNEIDER 61

Iteration of Analytic Functions of Several Variables, by SAMUEL KARLIN and JAMES MCGREGOR 81

A Class of Positive-Definite Functions, by J. F. C. KINGMAN 93

Local Noncommutative Analysis, by IRVING SEGAL 111

PART II: PAPERS ON PROBLEMS IN ANALYSIS

Linearization of the Product of Orthogonal Polynomials, by RICHARD ASKEY 131

Eisenstein Series on Tube Domains, by WALTER L. BAILY, Jr. 139

Laplace-Fourier Transformation, the Foundation for Quantum Information Theory and Linear Physics, by JOHN L. BARNES 157

An Integral Equation Related to the Schroedinger Equation with an Application to Integration in Function Space, by R. H. CAMERON and D. A. STORVICK 175

A Lower Bound for the Smallest Eigenvalue of the Laplacian, by JEFF CHEEGER 195 χ CONTENTS The Integral Equation Method in Scattering Theory, by C. L. DOLPH 201

Group Algebra Bundles, by BERNARD R. GELBAUM 229

Quadratic Periods of Hyperelliptic Abelian Integrals, by R. C. GUNNING 239

The Existence of Complementary Series, by A. W. KNAPP and E. M. STEIN 249

Some Recent Developments in the Theory of Singular Perturbations, by P. A. LAGERSTROM 261

Sequential Convergence in Lattice Groups, by SOLOMON LEADER 273

A Group-theoretic Lattice-point Problem, by BURTON RANDOL 291

The Riemann Surface of Klein with 168 Automorphisms, by HARRY E. RAUCH and J. LEWITTES 297

Envelopes of Holomorphy of Domains in Complex Lie Groups, by O. S. ROTHAUS 309

Automorphisms of Commutative Banach Algebras, by STEPHEN SCHEINBERG 319 Historical Notes on Analyticity as a Concept in Functional Analysis, by ANGUS E. TAYLOR 325

««/-Almost Automorphic Functions, by WILLIAM A. VEECH 345 Problems in Analysis

A SYMPOSIUM IN HONOR OF SALOMON BOCHNER

On the Group of Automorphisms of a Symplectic Manifold

EUGENIO CALABI1

1. Introduction

Let X be a connected, differential manifold of 2« dimensions. A symplectic structure on X is the geometrical structure induced by a differentiable exterior 2-form ω defined on X, satisfying the following conditions:

(i) The form ω is closed: dw = O; (ii) It is everywhere of maximal rank; this means that the 2«-form ω" (wth exterior power of ω) is everywhere different from zero, or equivalently, the skew-symmetric (In) χ (2w) matrix of coefficients of ω, in terms of a basis for the cotangent space, is everywhere nonsingular.

A classical theorem, ordinarily attributed to Darboux, states that a 2«-dimensional symplectic manifold (i.e., a manifold with a symplectic structure) can be covered by a local, differentiable coordinate system {U; (x)} where (x) = (x1,..., x2n): l/-> R2n, in terms of which the local representation of the structural form ω becomes

1 2 3 l 2 1 2n ω\υ = dx Λ dx + dx A dx +•••+ dx "' A dx (1.1) - 2l 1 21 = Jjdx ~ Λ dx ; J = I such a system of coordinates is called a canonical system. The purpose of this study is to describe the group G of automorphisms of a symplectic manifold, i.e., the group of all differentiable automorphisms of X which leave the structural 2-form ω invariant, and the invariant sub­ groups of G. The group G can also be characterized as mapping canonical coordinate systems into canonical systems.

1 The research reported here was supported in part by the National Science Foundation. 1 2 EUGENIO CALABI Two normal subgroups of G are distinguished immediately as follows: DEFINITION 1.1. Let (X, ω) be a 2«-dimensional symplectic manifold and let G be the group of all symplectic transformations of X. If X is not compact, we denote by G0 the subgroup of G consisting of all symplectic transformations of X that have compact support; that is to say, a sym­ plectic transformation g e G belongs to G0 if and only if g equals the identity outside a compact region of X. DEFINITION 1.2. Let (X, ω) be a 2n-dimensional symplectic manifold and let G be the group of all symplectic transformations of X. We denote by

G0io the subgroup of G called the minigroup generated by the so-called locally supported transformations, defined as follows: a transformation g e G is called locally supported if there exists a canonical coordinate system {U; (x)} defined in a contractible domain U with compact closure, such that the support of g lies in JJ.

The minigroup G0,0 and its corresponding Lie algebras are introduced here merely for expository convenience. In Section 3 it will be shown that the commutator subgroup of the arc-component of the identity in G0 coincides either with G0>0 or with a normal subgroup of codimension 1 in

G0>0 (see Theorem 3.7, Section 3). An elementary example of a locally supported transformation is the following: let {U; (x)} be a canonical coordinate system in X; let its range 2n V = Oc)(C/) c R2» contain the ball Ut) 1 2 J1 (t ) < A >j for some A > 0; i = l choose a real-valued differentiable function #(r) of a real variable r ^ 0 with support contained in a closed segment [0, A'] with A' < A. Then it is easily verifiable that the transformation f in R2n (with the 2-form ω = J dt*-1 A dt2S), (0-+(O = f(0with

t'*' = i2''-1 sin#(/) + /2> cos #(/·), ('-=201)2;! UjU^, is a symplectic transformation which equals the identity for r ^ A'. There­ fore its restriction to V defines via the coordinate map (x) a symplectic transformation in U that can be trivially extended by the identity map in X — U to a locally supported symplectic transformation in X. We shall state here the main results of this study in a preliminary form; more precise and stronger versions of these are repeated as theorems in the later sections.