LAPLACE TRANSFORMS FOR ANALYTIC FUNCTIONS IN TUBULAR DOMAINS

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References 11

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doi:10.1007/s10473-021-0610-6

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 $K \subset B,$

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

$A^p(B)$ 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

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

$F$ of

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

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

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

CLC Number:

42B30

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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