Representations of generalized Hardy functions in Beurling’s tempered distributions

Abstract

Let B be a proper open subset in ${{\mathbb {R}}}^N$ and C be a regular cone in ${{\mathbb {R}}}^N$. On our previous paper of Acta Scientiarum Mathematicarum 85, 595–611 (2019), we have defined the space of generalized Hardy functions, $G_{\omega ^*,A}^p(T^B)$, $1< p \le 2,$ and $A \ge 0$, and have shown that the functions in $G_{\omega ^*,A}^p(T^B)$ have distributional boundary values in the weak topology of Beurling tempered distributions, ${\mathcal {S}}_{(\omega )}^\prime$. In this paper we show that if the distributional boundary values are convolutors in Beurling ultradistributions of $L_2$-growth, then the functions in $G_{\omega ^*,0}^p(T^C)$, $1< p \le 2,$ can be represented as Cauchy and Poisson integral of the boundary values in ${\mathcal {S}}_{(\omega )}^\prime$.