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 $\mathcal{M}^{p,\varphi_1}_{\omega_1}(\mathbb{R}^n)$ to the generalized weighted Morrey spaces $\mathcal{M}^{q,\varphi_2}_{\omega_2}(\mathbb{R}^n)$, where $0<\alpha <n$, $1<p<\frac{n}{\alpha},$ $\frac 1p-\frac 1q=\frac \alpha {n}$, $(\omega_1, \omega_2)\in A_{p,q}(\mathbb{R}^n)$, $\varphi_1$, $\varphi_2$ are generalized functions and $b\in BMO(\mathbb{R}^n)$. Furthermore, we give some applications of our results.