@article {
author = {Corvalan, Alvaro},
title = {The preimage of $A_\infty (Q_0)$ for the local Hardy-Littlewood maximal operator},
journal = {International Journal of Nonlinear Analysis and Applications},
volume = {13},
number = {2},
pages = {379-386},
year = {2022},
publisher = {Semnan University},
issn = {2008-6822},
eissn = {2008-6822},
doi = {10.22075/ijnaa.2020.22106.2327},
abstract = {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.},
keywords = {Maximal Operators,$A_{infty }$ classes,Weigths,Rearrangements},
url = {https://ijnaa.semnan.ac.ir/article_6463.html},
eprint = {https://ijnaa.semnan.ac.ir/article_6463_3b1766d0f69b3aa22af15167bd76f33d.pdf}
}