**Memoirs of the American Mathematical Society**

2001;
94 pp;
Softcover

MSC: Primary 46; 32;
Secondary 35; 42; 58

Print ISBN: 978-0-8218-2712-3

Product Code: MEMO/153/725

List Price: $54.00

Individual Member Price: $32.40

**Electronic ISBN: 978-1-4704-0318-8
Product Code: MEMO/153/725.E**

List Price: $54.00

Individual Member Price: $32.40

# Strong Boundary Values, Analytic Functionals, and Nonlinear Paley-Wiener Theory

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*Jean-Pierre Rosay; Edgar Lee Stout*

We introduce a notion of boundary values for functions along real analytic boundaries, without any restriction on the growth of the functions. Our definition does not depend on having the functions satisfy a differential equation, but it covers the classical case of non-characteristic boundaries. These boundary values are analytic functionals or, in the local setting, hyperfunctions. We give a characterization of nonconvex carriers of analytic functionals, in the spirit of the Paley-Wiener-Martineau theory for convex carriers. Our treatment gives a new approach even to the classical Paley-Wiener theorem. The result applies to the study of analytic families of analytic functionals. The paper is mostly self contained. It starts with an exposition of the basic theory of analytic functionals and hyperfunctions, always using the most direct arguments that we have found. Detailed examples are discussed.

#### Readership

Graduate students and research mathematicians interested in functional analysis, several complex variables, analytic spaces, and differential equations.

#### Table of Contents

# Table of Contents

## Strong Boundary Values, Analytic Functionals, and Nonlinear Paley-Wiener Theory

- Contents vii8 free
- 1. Introduction 110 free
- 2. Preliminaries on Analytic Functionals and Hyperfunctions 312 free
- 3. Analytic Functionals as Boundary Values 3140
- 4. Nonlinear Paley-Wiener Theory 4756
- 5. Strong Boundary Values 5564
- 6. Strong Boundary Values for the Solutions of Certain Partial Differential Equations 7079
- 7. Comparison With Other Notions of Boundary Values 7382
- 8. Boundary Values Via Cousin Decompositions 8291
- 9. The Schwarz Reflection Principle 8594
- References 91100
- Index of Notation 94103 free