-convolvability with the Poisson kernel in the Euclidean case and the product domain case
Volume 156 / 2003
Abstract
We obtain real-variable and complex-variable formulas for the integral of an integrable distribution in the n-dimensional case. These formulas involve specific versions of the Cauchy kernel and the Poisson kernel, namely, the Euclidean version and the product domain version. We interpret the real-variable formulas as integrals of S^{\prime }-convolutions. We characterize those tempered distribution that are S^{\prime }-convolvable with the Poisson kernel in the Euclidean case and the product domain case. As an application of our results we prove that every integrable distribution on {\mathbb R}^{n} has a harmonic extension to the upper half-space {\mathbb R}_{+}^{n+1}.