Boundedness of operators on certain power-weighted Morrey spaces beyond the Muckenhoupt weights

TL;DR

This paper extends the boundedness results of operators on Morrey spaces with power weights beyond classical Muckenhoupt weights, providing new estimates and necessary conditions in a very general setting.

Contribution

It introduces a broader class of weights for operator boundedness on Morrey spaces and establishes new estimates, including for weak type spaces, with general applicability.

Findings

Boundedness of operators on Morrey spaces with extended power weights.

New estimates for weak type Morrey spaces, including the Hardy-Littlewood maximal operator.

Necessity of certain A_q weight conditions for boundedness.

Abstract

We prove that for operators satistying weighted inequalities with $A_p$ weights the boundedness on a certain class of Morrey spaces holds with weights of the form $|x|^\alpha w(x)$ for $w\in A_p$. In the case of power weights the shift with respect to the range of Muckenhoupt weights was observed by N.~Samko for the Hilbert transform, by H.~Tanaka for the Hardy-Littlewood maximal operator, and by S.~Nakamura and Y.~Sawano for Calder\'on-Zygmund operators and others. We extend the class of weights and establish the results in a very general setting, with applications to many operators. For weak type Morrey spaces, we obtain new estimates even for the Hardy-Littlewood maximal operator. Moreover, we prove the necessity of certain $A_q$ condition.