Weighted Solyanik Estimates for the Hardy-Littlewood maximal operator and embedding of $A_{\infty }$ into $A_p$

Abstract

Let $w$ denote a weight in ${\mathbb{R}}^n$ which belongs to the Muckenhoupt class $A_{\infty }$ and let ${\mathsf{M}}_w$ denote the uncentered Hardy-Littlewood maximal operator defined with respect to the measure $w(x)dx$. The \emph{sharp Tauberian constant} of ${\mathsf{M}}_w$ with respect to $\alpha$, denoted by ${\mathsf{C}}_w(\alpha )$, is defined by \[ \mathsf{C}_w (α) := \sup_{E:\, 0 < w(E) < \infty}w(E)^{-1}w\big(\big\{x \in \mathbb{R}^n:\, \mathsf{M}_w χ_E (x) > α\big\}\big). \] In this paper, we show that the Solyanik estimate \[ \lim_{α\rightarrow 1^-}\mathsf{C}_w(α) = 1 \] holds. Following the classical theme of weighted norm inequalities we also consider the sharp Tauberian constants defined with respect to the usual uncentered Hardy-Littlewood maximal operator $\mathsf{M}$ and a weight $w$: \[ \mathsf C ^w (α) := \sup_{E:\, 0 < w(E) < \infty} w(E)^{-1} w\big(\big\{x \in \mathbb R^n:\, \mathsf{M} χ_E (x) > α\big\}\big). \] We show that we have ${\lim }_{\alpha \to 1^{-}}{\mathsf{C}}^w(\alpha )=1$ if and only if $w\in A_{\infty }$. As a corollary of our methods we obtain a quantitative embedding of $A_{\infty }$ into $A_p$.

20 pages, submitted for publication, v.2 this is the final version, numbering has changed to match the published version, one reference updated, incorporates referee's report, to appear in J. Geom. Anal

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• Sharp Reverse Hölder property for A_\infty weights on spaces of homogeneous type
• Tauberian conditions, Muckenhoupt weights, and differentiation properties of weighted bases
• Variations on the Theme of Journe's Lemma
• Sharp estimates involving $A_{\infty }$ and $LlogL$ constants, and their applications to PDE

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• Proof of an extension of E. Sawyer's conjecture about weighted mixed weak-type estimates
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• Degenerate Poincaré-Sobolev inequalities via fractional integration
• Two-weight extrapolation on function spaces and applications
• Hölder continuity of Tauberian constants associated with discrete and ergodic strong maximal operators
• A note on local Hölder continuity of weighted Tauberian functions
• Sharp inequalities for one-sided Muckenhoupt weights