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 $f\in L(\mathbb{R}^n)$.
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
.
DOI: 10.1007/s10473-021-0610-6
[1] Stein E M, Weiss G. Introduction to Fourier Analysis on Euclidean Spaces. Princeton University Press, 1971
[2] Deng G T. Complex Analysis (in Chinese). Beijing Normal Universty Press, 2010
[3] Deng G T, Qian T. Rational approximation of functions in Hardy spaces. Complex Anal Oper Theory, 2016, 10:903-920
[4] Li H C, Deng G T, Qian T. Fourier spectrum characterizations of Hp space on tubes over cones for 1≤ p ≤ ∞. Complex Anal Oper Theory, 2017:1-26
[5] Li H C. The Theory of Hardy Spaces on Tube Domains[D]. University of Macau, 2015
[6] Deng G T, Huang Y, Qian T. Paley-Wiener-type theorem for analytic functions in tubular domains. Journal of Mathematical Analysis and Applications, 2019, 480(1):123367
[7] Folland G B. Real Analysis:Modern Techniques and Their Applications. John Wiley & Sons, Inc, Canada, 1999
[8] Faraut J, Korányi A. Analysis on Symmetric Cones. Oxford:Clarendon Press, 1994
[9] Rudin W. Real and Complex Analysis. Third ed. New York:McGraw-Hill Book Co, 1987
[10] Rudin W. Function Analysis. Second edition. New York:McGraw-Hill Book Co, 1991
[11] Zhu K. Analysis on Fock Space. Graduate Texts in Math, vol 263. New York:Springer, 2012
