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’-convolutions. We characterize those tempered distribution that are S’-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 ℝⁿ has a harmonic extension to the upper half-space .
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm156-2-5,
author = {Josefina Alvarez and Martha Guzm\'an-Partida and Urszula Sk\'ornik},
title = {S'-convolvability with the Poisson kernel in the Euclidean case and the product domain case},
journal = {Studia Mathematica},
volume = {157},
year = {2003},
pages = {143-163},
zbl = {1023.46041},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm156-2-5}
}
Josefina Alvarez; Martha Guzmán-Partida; Urszula Skórnik. S'-convolvability with the Poisson kernel in the Euclidean case and the product domain case. Studia Mathematica, Tome 157 (2003) pp. 143-163. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm156-2-5/