Commutators of Riesz potential in the vanishing generalized weighted Morrey spaces with variable exponent

Abstract

Let $\Omega \subset {\mathbb{R}}^n$ be an unbounded open set. We consider the generalized weighted Morrey spaces ${\mathcal{M}}_{\omega }^{p(\cdot ),\phi }(\Omega )$ and the vanishing generalized weighted Morrey spaces $V{\mathcal{M}}_{\omega }^{p(\cdot ),\phi }(\Omega )$ with variable exponent $p(x)$ and a general function $\phi (x,r)$ defining the Morrey-type norm. The main result of this paper are the boundedness of Riesz potential and its commutators on the spaces ${\mathcal{M}}_{\omega }^{p(\cdot ),\phi }(\Omega )$ and $V{\mathcal{M}}_{\omega }^{p(\cdot ),\phi }(\Omega )$. This result generalizes several existing results for Riesz potential and its commutators on Morrey type spaces. Especially, it gives a unified result for generalized Morrey spaces and variable Morrey spaces which currently gained a lot of attentions from researchers in theory of function spaces.

23 pages