Commutators of Riesz potential in the vanishing generalized weighted Morrey spaces with variable exponent
Let $Ω\subset \mathbb{R}^n$ be an unbounded open set. We consider the generalized weighted Morrey spaces $\mathcal{M}^{p(\cdot),φ}_ω(Ω)$ and the vanishing generalized weighted Morrey spaces $V\mathcal{M}^{p(\cdot),φ}_ω(Ω)$ with variable exponent $p(x)$ and a general function $φ(x,r)$ defining the Morrey-type norm. The main result of this paper are the boundedness of Riesz potential and its commutators on the spaces $\mathcal{M}^{p(\cdot),φ}_ω(Ω)$ and $V\mathcal{M}^{p(\cdot),φ}_ω(Ω)$. 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.