Document Type : Research Paper

Author

• Alvaro Corvalan

Instituto del Desarrollo Humano, Universidad Nacional de General Sarmiento, Facultad de Ingenieriacuteia y Ciencias Agrarias, Pontificia Universidad Catoacutelica, Argentina

Abstract

We describe here all those weight functions $u$ such that $Mu\in A_{\mathrm{\infty }}\left(Q\right)$ for $M$ the local Hardy-Littlewood maximal operator restricted to a cube $Q\subset {\mathbb{R}}^n$. In a recent paper it is shown that for the maximal operator in ${\mathbb{R}}^n$, $Mu\in A_{\mathrm{\infty }}$  implies that $Mu\in A_1$; here we see that the same is true for the local $M$ but this imposes a stronger condition for weights in $Q$, that is, for $M$  restricted to a finite cube $Mu\in A_{\mathrm{\infty }}$ if and only if $u\in A_{\mathrm{\infty }}$. This differs from the case in ${\mathbb{R}}^n$ where there are weights  $u$ not belonging to $A_{\mathrm{\infty }}$ such that $Mu$ is in $A_{\mathrm{\infty }}$. As an application we get a new shorter proof of a result of I. Wik. We also give a  characterization for those weights in terms the $K$-functional of Peetre.

Keywords

• Maximal Operators
• $A_{\mathrm{\infty }}$ classes
• Weigths
• Rearrangements