$S'$-convolvability with the Poisson kernel in the Euclidean case and the product domain case
Tom 156 / 2003
Studia Mathematica 156 (2003), 143-163
MSC: 46F10, 46F05, 46F12.
DOI: 10.4064/sm156-2-5
Streszczenie
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 ${\mathbb R}_{+}^{n+1}$.
