TWO-WEIGHTED INEQUALITIES FOR RIESZ POTENTIAL AND ITS COMMUTATORS IN GENERALIZED WEIGHTED MORREY SPACES

Abstract

In this paper we find the conditions for the boundedness of Riesz potential $I^{\alpha }$ and its commutators from the generalized weighted Morrey spaces ${\mathcal{M}}_{{\omega }_1}^{p,{\phi }_1}({\mathbb{R}}^n)$ to the generalized weighted Morrey spaces ${\mathcal{M}}_{{\omega }_2}^{q,{\phi }_2}({\mathbb{R}}^n)$, where $0<\alpha <n$, $1<p<\frac{n}{\alpha },$ $\frac{1}{p}-\frac{1}{q}=\frac{\alpha }{n}$, $({\omega }_1,{\omega }_2)\in A_{p,q}({\mathbb{R}}^n)$, ${\phi }_1$, ${\phi }_2$ are generalized functions and $b\in BMO({\mathbb{R}}^n)$. Furthermore, we give some applications of our results.