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SHARP WEIGHTED BOUNDS FOR GEOMETRIC MAXIMAL OPERATORS

Part of: Harmonic analysis in several variables Linear function spaces and their duals Stochastic processes

Published online by Cambridge University Press:  10 June 2016

ADAM OSȨKOWSKI
ADAM OSȨKOWSKI*
Affiliation:
Department of Mathematics, Informatics and Mechanics, University of Warsaw, Banacha 2, 02-097 Warsaw, Poland e-mail: ados@mimuw.edu.pl
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Abstract

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Let $\mathcal{M}$ and G denote, respectively, the maximal operator and the geometric maximal operator associated with the dyadic lattice on $\mathbb{R}^d$ .

  1. (i) We prove that for any 0 < p < ∞, any weight w on $\mathbb{R}^d$ and any measurable f on $\mathbb{R}^d$ , we have Fefferman–Stein-type estimate

    $$\begin{equation*}||G(f)||_{L^p(w)}\leq e^{1/p}||f||_{L^p(\mathcal{M}w)}.\end{equation*}$$
    For each p, the constant e1/p is the best possible.

  2. (ii) We show that for any weight w on $\mathbb{R}^d$ and any measurable f on $\mathbb{R}^d$ ,

    $$\begin{equation*}\int_{\mathbb{R}^d} G(f)^{1/\mathcal{M}w}w\mbox{d}x\leq e\int_{\mathbb{R}^d} |f|^{1/w}w\mbox{d}x\end{equation*}$$
    and prove that the constant e is optimal.

Actually, we establish the above estimates in a more general setting of maximal operators on probability spaces equipped with a tree-like structure.

MSC classification

Primary: 42B25: Maximal functions, Littlewood-Paley theory
Secondary: 46E30: Spaces of measurable functions ($L^p$-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) 60G42: Martingales with discrete parameter

Information

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 59 , Issue 3 , September 2017 , pp. 533 - 547
Copyright
Copyright © Glasgow Mathematical Journal Trust 2016 

References

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