The preimage of $A_\infty (Q_0)$ for the local Hardy-Littlewood maximal operator

Document Type : Research Paper


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


We describe here all those weight functions $u$ such that $Mu\in A_{\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 $\mathbb{R}^{n}$, $Mu\in A_{\infty }$  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 $Mu\in A_{\infty }$ if and only if $u\in A_{\infty }$. This differs from the case in $\mathbb{R}^{n}$ where there are weights  $u$ not belonging to $A_{\infty } $ such that $Mu$ is in $A_{\infty }$. 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.


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Volume 13, Issue 2
July 2022
Pages 379-386
  • Receive Date: 15 October 2020
  • Revise Date: 16 December 2020
  • Accept Date: 26 December 2020