$S'$-convolvability with the Poisson kernel in the Euclidean case and the product domain case

Josefina Alvarez ¹   ; Martha Guzmán-Partida ²   ; Urszula Skórnik ³

¹ Department of Mathematics New Mexico State University Las Cruces, NM 88003, U.S.A.
² Departamento de Matemáticas Universidad de Sonora Hermosillo, Sonora 83000, México
³ Warsaw University of Agriculture Nowoursynowska 166 02-787 Warszawa, Poland

Studia Mathematica, Tome 156 (2003) no. 2, pp. 143-163

Résumé

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 ${\mathbb R}_{+}^{n+1}$.

Détail

DOI : 10.4064/sm156-2-5

Keywords: obtain real variable complex variable formulas integral integrable distribution n dimensional these formulas involve specific versions cauchy kernel poisson kernel namely euclidean version product domain version interpret real variable formulas integrals prime convolutions characterize those tempered distribution prime convolvable poisson kernel euclidean product domain application results prove every integrable distribution mathbb has harmonic extension upper half space mathbb