Cauchy and Poisson kernels for tubes

Abstract

Let C be a regular cone in ℝn. The Cauchy and Poisson kernel functions, denoted K(z - t), z ∈ ℝn + iC, t ∈ ℝn, and Q(z; t), z ∈ ℝn + iC, t ∈ ℝn, respectively, are defined and are shown to be elements in the ultradifferential spaces D(*, Ls) for 1 < s ≤ ∞ and 1 ≤ s ≤ ∞, respectively, as functions of t ∈ ℝn for z ∈ ℝn+iC arbitrary but fixed. Cauchy and Poisson integrals of ultradistributions in D′(*, Ls) for the corresponding values of s are defined and properties of them are indicated. D(*, Ls) and D′(*, Ls) are defined through the use of special sequences {Mp} of positive real numbers, and the spaces D′(*, Ls) generalize the Schwartz spaces D′Ls.