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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 | ||
| تاریخ دریافت: 24 مهر 1399، تاریخ بازنگری: 26 آذر 1399، تاریخ پذیرش: 06 دی 1399 | ||
| 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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[1] I.U. Asekritova, N. Krugljak, L. Maligranda and L.E. Persson, Distribution and rearrangement estimates of the maximal function and interpolation, Studia Math. 124 (1997), 107–132. [2] J. Bastero, M. Milman and F. Ruiz, Reverse H¨older inequalities and interpolation, Function spaces, interpolation spaces, and related topics, Israel Math. Conf. Proc. 13 (1999), 11–23 [3] J. Bergh and J. L¨ofstr¨om, Interpolation spaces. An introduction, Grundlehren der mathematischen Wissenschaften 223. Berlin-Heidelberg-New York, Springer-Verlag, 1976. [4] C. Bennett and R. Sharpley, Interpolation of Operators. Pure and Applied Mathematics Series, Academic Press, New York, 1988. [5] A. Corval´an, Some characterizations of the preimage of A∞ for the Hardy-Littlewood maximal operator and consequences, Real Anal. Exchange 44 (2019), no. 1, 141–166. [6] R. Coifman and C. Fefferman, Weighted norm inequalities for maximal functions and singular integrals, Studia Math. 15 (1974), 241–250. [7] R.R. Coifman and R. Rochberg, Another characterization of BMO, Proc. Amer. Math. Soc. 79 (1980), 249–254. [8] D. Cruz-Uribe SFO, Piecewise monotonic doubling measures, Rocky Mount. J. Math. 26 (1996), 1–39. [9] J. Garcia-Cuerva and J.L. Rubio de Francia, Weighted Norm Inequalities and Related Topics, North Holland, New York, 1985. [10] C. P´erez, Endpoint estimates for commutators of singular integral operators, J. Funct. Anal. 128 (1995), 163–185. [11] W. Rudin, Real and complex analysis, 3rd edition, New York, London, McGraw-Hill, 1987. [12] I. Wik, On Muckenhoupt’s classes of weight functions, Studia Math. 94 (1989), 245–25. | ||
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