Cauchy and poisson intergral of ultredistributions

Abstract

Let C be an open convex cone in such that [Cbar] does not contain any entire straight line and let be the corresponding rube domain. Generalized Cauchy and Poisson integrals of ultradistributions of Beurling type and of Roumieu type both of which generalize the Schwartz distributions are defined and studied for appropriate values of s The Caucthy integral is shown to be holomorphic in T to satisfy a growth property, and to obtain an ultradistribution boundary value. Which leads to a holomorphic decomposition dresult for the ultradistributions. The Poission integral is shown to have an ultradistribution boundary value. The analysis of this paper lays the foundation and is necessary for future research concerning a study of holomorphic functions in tubes which are characterixed by either pointwise or norm growths, their boundary values, their recovery in terms of generalized integrals, and other associated properties.