LAPLACE TRANSFORMS FOR ANALYTIC FUNCTIONS IN TUBULAR DOMAINS

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LAPLACE TRANSFORMS FOR ANALYTIC FUNCTIONS IN TUBULAR DOMAINS

LAPLACE TRANSFORMS FOR ANALYTIC FUNCTIONS IN TUBULAR DOMAINS

LAPLACE TRANSFORMS FOR ANALYTIC FUNCTIONS IN TUBULAR DOMAINS

LAPLACE TRANSFORMS FOR ANALYTIC FUNCTIONS IN TUBULAR DOMAINS

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doi:10.1007/s10473-021-0610-6

数学物理学报(英文版) ›› 2021, Vol. 41 ›› Issue (6): 1938-1948.doi: 10.1007/s10473-021-0610-6

邓冠铁, 付倩, 曹辉

Guantie DENG, Qian FU, Hui CAO

收稿日期:2021-03-22 修回日期:2021-09-16 出版日期:2021-12-25 发布日期:2021-12-27
通讯作者: Qian FU,E-mail:fqian19@126.com E-mail:fqian19@126.com
作者简介:Guantie DENG,E-mail:denggt@bnu.edu.cn;Hui CAO,E-mail:18166038936@163.com
基金资助:
This work was partially supported by NSFC (11971045, 12071035 and 11971063).

Guantie DENG, Qian FU, Hui CAO

Guantie DENG, Qian FU, Hui CAO

Received:2021-03-22 Revised:2021-09-16 Online:2021-12-25 Published:2021-12-27
Supported by:
This work was partially supported by NSFC (11971045, 12071035 and 11971063).

摘要： Assume that $0< p<\infty$ and that $B$ is a connected nonempty open set in $\mathbb{R}^n$ , and that $A^{p}(B)$ is the vector space of all holomorphic functions $F$ in the tubular domains $\mathbb{R}^n+{\rm i}B$ such that for any compact set $K \subset B,$

‖ y \mapsto ‖ x \mapsto F (x + i y) ‖_{L^{p} (R^{n})} ‖_{L (K)} < \infty,

$\|y\mapsto \|x\mapsto F(x+{\rm i}y)\|_{L^p(\mathbb{R}^n)}\|_{L(K)}<\infty,$ so

A^{p} (B)

$A^p(B)$ is a Fréchet space with the Heine-Borel property, its topology is induced by a complete invariant metric, is not locally bounded, and hence is not normal. Furthermore, if

1 \leq p \leq 2

$1\leq p\leq 2$ , then the element

F

$F$ of

A^{p} (B)

$A^{p}(B)$ can be written as a Laplace transform of some function

f \in L (R^{n})

$f\in L(\mathbb{R}^n)$ .

关键词: Laplace transforms, Fourier transform, tubular domain, regular cone

Abstract: Assume that $0< p<\infty$ and that $B$ is a connected nonempty open set in $\mathbb{R}^n$ , and that $A^{p}(B)$ is the vector space of all holomorphic functions $F$ in the tubular domains $\mathbb{R}^n+{\rm i}B$ such that for any compact set $K \subset B,$

‖ y \mapsto ‖ x \mapsto F (x + i y) ‖_{L^{p} (R^{n})} ‖_{L (K)} < \infty,

$\|y\mapsto \|x\mapsto F(x+{\rm i}y)\|_{L^p(\mathbb{R}^n)}\|_{L(K)}<\infty,$ so

A^{p} (B)

$A^p(B)$ is a Fréchet space with the Heine-Borel property, its topology is induced by a complete invariant metric, is not locally bounded, and hence is not normal. Furthermore, if

1 \leq p \leq 2

$1\leq p\leq 2$ , then the element

F

$F$ of

A^{p} (B)

$A^{p}(B)$ can be written as a Laplace transform of some function

f \in L (R^{n})

$f\in L(\mathbb{R}^n)$ .

Key words: Laplace transforms, Fourier transform, tubular domain, regular cone

中图分类号:

42B30

邓冠铁, 付倩, 曹辉. LAPLACE TRANSFORMS FOR ANALYTIC FUNCTIONS IN TUBULAR DOMAINS[J]. 数学物理学报(英文版), 2021, 41(6): 1938-1948.

Guantie DENG, Qian FU, Hui CAO. LAPLACE TRANSFORMS FOR ANALYTIC FUNCTIONS IN TUBULAR DOMAINS[J]. Acta mathematica scientia,Series B, 2021, 41(6): 1938-1948.

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