Weighted boundedness for the maximal operator associated with matrices

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 $M_{\alpha ,A^{-1}}$.

14 pages