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Laplace ultradistributions on a half line and a strong quasi-analyticity principle

100%

Łysik G.

Annales Polonici Mathematici

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1996

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tom 63

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nr 1

13-33

EN

Several representations of the space of Laplace ultradistributions supported by a half line are given. A strong version of the quasi-analyticity principle of Phragmén-Lindelöf type is derived.

2

A Phragmén-Lindelöf type quasi-analyticity principle

100%

Łysik G.

Studia Mathematica

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1997

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tom 123

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nr 3

217-234

EN

Quasi-analyticity theorems of Phragmén-Lindelöf type for holomorphic functions of exponential type on a half plane are stated and proved. Spaces of Laplace distributions (ultradistributions) on ℝ are studied and their boundary value representation is given. A generalization of the Painlevé theorem is proved.

3

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Topological imbedding of Laplace distributions in Laplace hyperfunctions

89%

Szmydt Z., Ziemian B.

Instytut Matematyczny Polskiej Akademii Nauk

EN

CONTENTS Foreword..............................................................................................................................5 Introduction..........................................................................................................................6 1. Preliminaries....................................................................................................................7 1.1. Terminology and notation.............................................................................................7 1.2. Selected topics on complex topological vector spaces and their duals........................7 2. A review of basic facts in the theory of distributions.......................................................10 2.1. The spaces $D_K$ and $(D_K)'$..............................................................................10 2.2. The spaces D(Ω) and D'(Ω).......................................................................................11 2.3. The spaces D(A) and D'(A)........................................................................................12 2.4. The spaces $D^k(K)$ and $(D^k(K))'$.......................................................................14 3. Selected topics in the theory of holomorphic functions of one variable..........................15 3.1. Basic notions and theorems.......................................................................................15 3.2. The spaces A(K) and A'(K)........................................................................................16 3.3. Boundary values of holomorphic functions of one variable........................................19 4. Hyperfunctions in one variable.......................................................................................23 4.1. Definitions and basic properties of hyperfunctions....................................................23 4.2. Imbedding of analytic functions in hyperfunctions: A(Ω) ↪ B(Ω)................................26 4.3. Elementary operations on hyperfunctions..................................................................27 4.4. The Köthe theorem....................................................................................................28 4.5. Imbedding $D'_K ↪ B_K$, K compact in ℝ................................................................31 4.6. The distributional version of the Köthe theorem........................................................34 4.7. Imbedding D'(Ω)↪ B(Ω), Ω open in ℝ........................................................................35 4.8. Hyperfunctional boundary values of holomorphic functions.......................................39 4.9. Hyperfunctions supported by a single point...............................................................39 4.10. Substitution in a hyperfunction and in an analytic functional (distribution)...............40 5. Laplace hyperfunctions and Laplace analytic functionals in one variable......................42 6. Mellin hyperfunctions and Mellin distributions in one variable........................................50 6.1. Mellin hyperfunctions and Mellin analytic functionals.................................................50 6.2. Mellin distributions.....................................................................................................60 7. Laplace distributions L'(ω)(ℝ͞͞₊)......................................................................................64 7.1. Definitions and basic properties of Laplace distributions...........................................64 7.2. Imbedding of Laplace distributions in Laplace hyperfunctions...................................67 7.3. Imbedding of Mellin distributions in Mellin hyperfunctions..........................................79 References........................................................................................................................81

4

Harmonic functions on the real hyperbolic ball I: Boundary values and atomic decomposition of Hardy spaces

88%

Jaming P.

Colloquium Mathematicum

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1999

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tom 80

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nr 1

63-82

EN

We study harmonic functions for the Laplace-𝔹eltrami operator on the real hyperbolic space $𝔹_n$. We obtain necessary and sufficient conditions for these functions and their normal derivatives to have a boundary distribution. In doing so, we consider different behaviors of hyperbolic harmonic functions according to the parity of the dimension of the hyperbolic ball $𝔹_n$. We then study the Hardy spaces $H^p(𝔹_n)$, 0

Ograniczanie wyników

1 Annales Polonici Mathematici

1 Colloquium Mathematicum

1 Studia Mathematica

2 Łysik G.

1 Jaming P.

1 Szmydt Z.

1 Ziemian B.

1 1999

1 1998

1 1997

1 1996

Laplace ultradistributions on a half line and a strong quasi-analyticity principle

A Phragmén-Lindelöf type quasi-analyticity principle

Topological imbedding of Laplace distributions in Laplace hyperfunctions

Harmonic functions on the real hyperbolic ball I: Boundary values and atomic decomposition of Hardy spaces