Generalized Cauchy and Poisson Integrals and Distributional Boundary Values

Source abstract

Let C be an open connected cone, and let $O(C)$ denote its convex envelope. Cauchy and Poisson integrals of distributions in $\mathcal{D}'_{Lp}$, $1 < p \leqq 2$, corresponding to tubular radial domains $T^{O(C)} = \mathbb{R}^n + iO(C)$ are defined; and properties of these integrals are obtained. The boundary values of these integrals are obtained in the distributional sense on the distinguished boundary of $T^{O(C)}$ Functions which are analytic and have a specified growth condition in $T^{O(C)}$ are related to the Cauchy and Poisson integrals of their distributional boundary values. The results concerning these functions extend some well-known theorems concerning the Hardy $H^p (T^{O(C)} )$-spaces to our distributional setting. Further, functions which are analytic in disconnected tubular cones are considered; and in particular conditions are obtained under which such a function has an analytic extension to the convex envelope of the tubular cone.