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 $${\mathcal {S}}_{(\omega )}^\prime $$ .