Description of the Space of Riesz Potentials of Functions in a Grand Lebesgue Space on~$\mathbb{R}^n$

S. M. Umarkhadzhiev

Matematičeskie zametki, Tome 104 (2018) no. 3, pp. 467-480

Résumé

The Riesz potentials $I^\alpha f$, $0\alpha\infty$, are considered in the framework of a grand Lebesgue space $L^{p),\theta}_a$, $1$, $\theta>0$, on $\mathbb{R}^n$ with grandizers $a\in L^1(\mathbb{R}^n)$, which are understood in the case $\alpha\ge n/p$ in terms of distributions on test functions in the Lizorkin space. The images under $I^\alpha$ of functions in a subspace of the grand space which satisfy the so-called vanishing condition is studied. Under certain assumptions on the grandizer, this image is described in terms of the convergence of truncated hypersingular integrals of order $\alpha$ in this subspace.

Détail

Keywords: Riesz potential, space of Riesz potentials, hypersingular integral, grandizer, Lizorkin space of test functions, identity approximation.
Mots-clés : grand Lebesgue space