Cauchy and Poisson Integral of the Convolutor in Beurling Ultradistributions of -Growth

Let be a regular cone in and let be a tubular radial domain. Let be the convolutor in Beurling ultradistributions of -growth corresponding to . We define the Cauchy and Poisson integral of and show that the Cauchy integral of？？ is analytic in and satisfies a growth property. We represent？？ as the boundary value of a finite sum of suitable analytic functions in tubes by means of the Cauchy integral representation of . Also we show that the Poisson integral of corresponding to attains as boundary value in the distributional sense. 1. Introduction Let be a regular cone in and let denote its convex envelope. In [1] (or [2]) Carmichael defined the Cauchy and Poisson integrals for Schwartz distributions , , corresponding to tubular domain . Carmichael obtained the boundary values of these integrals in the distributional sense on the boundary of and found the relation between analytic functions with a specific growth condition in and the Cauchy and Poisson integrals of their distributional boundary values. In [3] Pilipovi？ defined ultradistributions of -growth, where , , is a certain sequence of positive numbers, and studied the Cauchy and Poisson integrals for elements of in the case that the Cauchy and Poisson kernel functions are defined corresponding to the first quadrant ; , in . Pilipovi？ showed that elements in are boundary values of suitable analytic functions with a certain -norm condition by means of the Cauchy integral representation and an analytic function with a certain -norm condition determines, as a boundary value, an element from . In [4] Carmichael et al. defined ultradistributions of Beurling type of -growth and of Roumieu type of -growth, both of which generalize the Schwartz distributions , and studied the Cauchy and Poisson integrals for elements of both and for corresponding to the arbitrary tubes where is an open connected cone in of which the quadrants are special cases. They showed that the Cauchy integral for elements of both and for corresponding to is shown to be analytic in , to satisfy a growth property and to obtain an ultradistribution boundary value, which leads to an analytic representation for the ultradistributions. They also showed that the Poisson integrals for elements of both and for corresponding to are shown to have an ultradistribution boundary value. We can find the works of the Cauchy and Poisson integrals for ultradistributions of compact support in [5] and for various kinds of distributions in [2]. In the meantime, Betancor et al. [6] introduced the spaces of Beurling ultradistributions of -growth, , of which