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

TL;DR

This paper extends the Muckenhoupt-Wheeden theorem to generalized weighted Morrey spaces, establishing conditions under which Riesz potentials and fractional maximal functions are norm-equivalent, with implications for harmonic analysis.

Contribution

It provides new conditions on functions and weights ensuring norm equivalence in generalized weighted Morrey spaces, extending classical theorems to broader contexts.

Findings

Established norm equivalence conditions for Riesz potential and fractional maximal function

Extended Muckenhoupt-Wheeden theorem to generalized weighted Morrey spaces

Identified weight class $A_{ abla}$ for norm equivalence

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.