An extension of Muchenhoupt-Wheeden theorem to generalized weighted (central) Morrey spaces

Abstract

In this paper we find the condition on function $\omega$ and weight $v$ which ensures the equivalency of norms of the Riesz potential and the fractional maximal function in generalized weighted Morrey spaces ${\mathcal M}_{p,\omega}({\mathbb R}^n,v)$ and generalized weighted central Morrey spaces $\dot{\mathcal M}_{p,\omega}({\mathbb R}^n,v)$, when $v$ belongs to Muckenhoupt $A_{\infty}$-class.