Weighted boundedness for the maximal operator associated with matrices

TL;DR

This paper investigates the boundedness of a matrix-associated maximal operator on weighted Lebesgue spaces, providing new characterizations, examples distinguishing weight classes, and extending results to fractional operators.

Contribution

It introduces new boundedness results for matrix-related maximal operators, characterizes specific matrix cases, and extends findings to fractional maximal operators.

Findings

Characterization of boundedness for $M_{A^{-1}}$ on $L^p(w)$.

Examples showing differences between $ ext{A}_{A,p}$ and $ ext{A}_p$ weight classes.

Extension of results to fractional maximal operators $M_{eta, A^{-1}}$.

Abstract

In this paper we study the boundedness on $L^p(w)$ of the maximal operator $M_{A^{-1}}$, defined by $M_{A^{-1}}f(x)=Mf(A^{-1}x)$, that is, the maximal of Hardy-Littlewood composed with a invertible matrix $A$. We present two different results of boundedness and provide a characterization for a particular case of matrices. The main novelty lies in examples illustrating the difference between the class of weights with a matrix, $\mathcal{A}_{A,p}$, and the classical Muckenhoupt weight class, $\mathcal{A}_{p}$. Finally, we extend these results to the fractional framework, considering the fractional maximal operator $M_{\alpha, A^{-1}}$.