The preimage of $A_\infty (Q_0)$ for the local Hardy-Littlewood maximal operator | ||
| International Journal of Nonlinear Analysis and Applications | ||
| مقاله 35، دوره 13، شماره 2، مهر 2022، صفحه 379-386 اصل مقاله (382.31 K) | ||
| نوع مقاله: Research Paper | ||
| شناسه دیجیتال (DOI): 10.22075/ijnaa.2020.22106.2327 | ||
| نویسنده | ||
| Alvaro Corvalan* | ||
| Instituto del Desarrollo Humano, Universidad Nacional de General Sarmiento, Facultad de Ingenieriacuteia y Ciencias Agrarias, Pontificia Universidad Catoacutelica, Argentina | ||
| چکیده | ||
| 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. | ||
| کلیدواژهها | ||
| Maximal Operators؛ $A_{\infty }$ classes؛ Weigths؛ Rearrangements | ||
| مراجع | ||
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