# Weighted Solyanik estimates for the strong maximal function

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

Indexed on: **13 Oct '14**Published on: **13 Oct '14**Published in: **Mathematics - Classical Analysis and ODEs**

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#### Abstract

Let $\mathsf M_{\mathsf S}$ denote the strong maximal operator on $\mathbb
R^n$ and let $w$ be a non-negative, locally integrable function. For
$\alpha\in(0,1)$ we define the weighted sharp Tauberian constant $\mathsf
C_{\mathsf S}$ associated with $\mathsf M_{\mathsf S}$ by $$ \mathsf C_{\mathsf
S} (\alpha):= \sup_{\substack {E\subset \mathbb R^n \\
0<w(E)<+\infty}}\frac{1}{w(E)}w(\{x\in\mathbb R^n:\, \mathsf M_{\mathsf
S}(\mathbf{1}_E)(x)>\alpha\}). $$ We show that $\lim_{\alpha\to 1^-} \mathsf
C_{\mathsf S} (\alpha)=1$ if and only if $w\in A_\infty ^*$, that is if and
only if $w$ is a strong Muckenhoupt weight. This is quantified by the estimate
$\mathsf C_{\mathsf S}(\alpha)-1\lesssim_{n} (1-\alpha)^{(cn [w]_{A_\infty
^*})^{-1}}$ as $\alpha\to 1^-$, where $c>0$ is a numerical constant; this
estimate is sharp in the sense that the exponent $1/(cn[w]_{A_\infty ^*})$ can
not be improved in terms of $[w]_{A_\infty ^*}$. As corollaries, we obtain a
sharp reverse H\"older inequality for strong Muckenhoupt weights in $\mathbb
R^n$ as well as a quantitative imbedding of $A_\infty^*$ into $A_{p}^*$. We
also consider the strong maximal operator on $\mathbb R^n$ associated with the
weight $w$ and denoted by $\mathsf M_{\mathsf S} ^w$. In this case the
corresponding sharp Tauberian constant $\mathsf C_{\mathsf S} ^w$ is defined by
$$ \mathsf C_{\mathsf S} ^w \alpha) := \sup_{\substack {E\subset \mathbb R^n \\
0<w(E)<+\infty}}\frac{1}{w(E)}w(\{x\in\mathbb R^n:\, \mathsf M_{\mathsf S} ^w
(\mathbf{1}_E)(x)>\alpha\}).$$ We show that there exists some constant
$c_{w,n}>0$ depending only on $w$ and the dimension $n$ such that $\mathsf
C_{\mathsf S} ^w (\alpha)-1 \lesssim_{w,n} (1-\alpha)^{c_{w,n}}$ as $\alpha\to
1^-$ whenever $w\in A_\infty ^*$ is a strong Muckenhoupt weight.