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