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

Document Type : Research Paper

Author

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_{\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


[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 of weight functions, Studia Math. 94 (1989), 245–25
Volume 13, Issue 2
July 2022
Pages 379-386
  • Receive Date: 15 October 2020
  • Revise Date: 16 December 2020
  • Accept Date: 26 December 2020