Weighted Solyanik estimates for the strong maximal function

TL;DR

This paper establishes sharp weighted estimates for the strong maximal function, characterizing strong Muckenhoupt weights via Tauberian constants and deriving related inequalities and embeddings.

Contribution

It provides a precise quantification of the limit behavior of weighted Tauberian constants for the strong maximal operator, linking them to strong Muckenhoupt weights and deriving new inequalities.

Findings

Limit of Tauberian constants characterizes $A_$ weights.

Sharp estimate for the approach of constants to 1 as lpha;

New reverse Hf6lder inequality and embedding results.

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.