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

Jean-Pierre Rosay University of Wisconsin, Madison, WI
Edgar Lee Stout University of Washington, Seattle, WA
Available Formats:
Electronic ISBN: 978-1-4704-0318-8
Product Code: MEMO/153/725.E
List Price: $54.00 MAA Member Price:$48.60
AMS Member Price: $32.40 Click above image for expanded view Strong Boundary Values, Analytic Functionals, and Nonlinear Paley-Wiener Theory Jean-Pierre Rosay University of Wisconsin, Madison, WI Edgar Lee Stout University of Washington, Seattle, WA Available Formats:  Electronic ISBN: 978-1-4704-0318-8 Product Code: MEMO/153/725.E  List Price:$54.00 MAA Member Price: $48.60 AMS Member Price:$32.40
• Book Details

Memoirs of the American Mathematical Society
Volume: 1532001; 94 pp
MSC: Primary 46; 32; Secondary 35; 42;

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.

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

• Chapters
• 1. Introduction
• 2. Preliminaries on analytic functionals and hyperfunctions
• 3. Analytic functionals as boundary values
• 4. Nonlinear Paley-Wiener theory
• 5. Strong boundary values
• 6. Strong boundary values for the solutions of certain partial differential equations
• 7. Comparison with other notions of boundary values
• 8. Boundary values via cousin decompositions
• 9. The Schwarz reflection principle
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Volume: 1532001; 94 pp
MSC: Primary 46; 32; Secondary 35; 42;

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.

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

• Chapters
• 1. Introduction
• 2. Preliminaries on analytic functionals and hyperfunctions
• 3. Analytic functionals as boundary values
• 4. Nonlinear Paley-Wiener theory
• 5. Strong boundary values
• 6. Strong boundary values for the solutions of certain partial differential equations
• 7. Comparison with other notions of boundary values
• 8. Boundary values via cousin decompositions
• 9. The Schwarz reflection principle
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