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

Research paper by **Paul A. Hagelstein, Ioannis Parissis**

Indexed on: **21 Jan '15**Published on: **21 Jan '15**Published in: **Mathematics - Classical Analysis and ODEs**

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#### 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 \[ \mathsf{C}_w (\alpha) := \sup_{E:\, 0
< w(E) < \infty}w(E)^{-1}w\big(\big\{x \in \mathbb{R}^n:\, \mathsf{M}_w \chi_E
(x) > \alpha\big\}\big). \] In this paper, we show that the Solyanik estimate
\[ \lim_{\alpha \rightarrow 1^-}\mathsf{C}_w(\alpha) = 1 \] 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 $w$: \[ \mathsf C ^w
(\alpha) := \sup_{E:\, 0 < w(E) < \infty} w(E)^{-1} w\big(\big\{x \in \mathbb
R^n:\, \mathsf{M} \chi_E (x) > \alpha\big\}\big). \] 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 $A_p$.