This book is dedicated to two problems. The first concerns the description of maximal exponential growth of functions or distributions for which the Cauchy problem is well posed. The description is presented in the language of the behavior of the symbol in a complex domain. The second problem concerns the structure of and explicit formulas for differential operators with large automorphism groups. It is suitable as an advanced graduate text in courses in partial differential equations and the theory of distributions.
Readership: Mathematicians and graduate students interested in partial differential equations and several complex variables.
Author(s): S G Gindikin
Series: Translations of Mathematical Monographs, Vol. 111
Publisher: American Mathematical Society
Year: 1992
Language: English
Pages: C+VI+132+B
Cover
S Title
Titles in This Series
Tube Domains and the Cauchy Problem
Copyright
©1992 by the American Mathematical Society.
ISBN 0-8218-4566-7
ISBN-13: 978-0-8218-4566-0
QA377.G5313 1992 515'.35-dc20
LCCN 92-19406
Contents
Introduction
CHAPTER 1 The Cauchy Problem in Spaces of Distributions with Exponential Estimates
§1. Spaces of functions (distributions) of exponential decrease (growth)
1.1. Convex functions
1.2. Holder scales corresponding to exponentially growing weight.
1.3. Hilbert norms in Sµ and imbedding theorems
1.4. Limit spaces for scales and the Fourier transform in them
1.5. Spaces of exponentially growing functions and distributions.
§2. Convolution operators and convolution equations
2.1. Preliminary facts about multipliers
2.2. Convolution operators and convolutors.
2.3. The description of convolutors on Sµ
2.4. Description of convolutors on Oµ
2.5. Convolutors in spaces of functions and distributions of exponential growth
2.6. Equations in convolutions in the spaces (\Phi_µ , '\Psi_µ
2.7. Relation to the homogeneous Cauchy problem
2.8. Exponentially correct differential operators
§3. Convolution equations in a strip and the nonhomogeneous Cauchy problem for convolution equations
3.1. Convolutors in a strip
3.2. Convolution and differential equations in a strip
3.3. Isotropic spaces
3.4. Certain auxiliary results
3.5. Function spaces associated to the nonhomogeneous Cauchy problem
3.6. Exponential growth distributions with smoothness conditions
3.7. Convolution and differential equations in (xD, l-'}' . The nonhomogeneous Cauchy problem
§4. The Cauchy problem for exponentially correct differential operators with variable coefficients
4.1. Exponentially correct differential operators of constant strength.
4.2. Energy estimates for exponentially correct differential operators with variable coefficients.
§5. Special classes of exponentially correct differential operators
5.1. Hyperbolic operators
5.2. 2b-parabolic operators
5.3. N-parabolic operators
5.4. Dominantly correct operators
5.5. Pluriparabolic operators
5.6. (2b + 1)-hyperbolic operators
CHAPTER 2 Strongly Homogeneous Differential Operators
§1. The structure of affine-homogeneous domains
§2. Compound power functions and Siegel integrals
§3. Riemann-Liouville operators and differential operators associated to homogeneous cones
§4. Analysis of fundamental solutions of differential operators related to linear-homogeneous cones
§5. Pluriparabolic strongly homogeneous differential operators
References
Subject Index
Notation Index
Titles in This Series
Back Cover
