Boundedness properties in a family of weighted Morrey spaces with emphasis on power weights

TL;DR

This paper introduces a unified framework for weighted Morrey spaces, providing sharp estimates for operators like the Hardy-Littlewood maximal operator, especially with power weights, and employs extrapolation techniques for broad applicability.

Contribution

It defines a new scale of weighted Morrey spaces encompassing existing variants, enabling unified weighted estimates and sharp results for key operators.

Findings

Weighted estimates for operators with power weights of the form |x|^α w(x).

Sharp results for the Hardy-Littlewood maximal operator in these spaces.

Generalized extrapolation techniques for broad operator applicability.

Abstract

We define a scale of weighted Morrey spaces which contains different weighted versions appearing in the literature. This allows us to obtain weighted estimates for operators in a unified way. In general, we obtain results for weights of the form $|x|^\alpha w(x)$ with $w\in A_p$ and nonnegative $\alpha$. We study particularly some properties of power-weighted spaces and in the case of the Hardy-Littlewood maximal operator our results for such spaces are sharp. By using extrapolation techniques the results are given in abstract form in such a way that they are automatically obtained for many operators.