Weighted Solyanik estimates for the strong maximal function

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 Tauberian constant $\mathsf C_{\mathsf {S},w}$ associated with $\mathsf M_{\mathsf{S}}$ by \[ \mathsf C_{\mathsf{S},w}(\alpha) := \sup_{\begin{subarray}{c} E\subset \mathbb{R}^n \\ 0\lt w(E) \lt+\infty\end{subarray}}\frac{1}{w(E)}w(\{x\in\mathbb{R}^n: \mathsf M_{\mathsf{S}}( {\mathbf 1}_E)(x)>\alpha\}). \] We show that $\lim_{\alpha\to 1^-} \mathsf C_{\mathsf {S},w}(\alpha)=1$ 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 $\mathsf C_{\mathsf {S},w}(\alpha)-1\lesssim_{n} (1-\alpha)^{ (cn [w]_{A_\infty^*})^{-1}}$ as $\alpha\to 1^-$, where $c>0$ is a numerical constant independent of $n$; 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 Hölder 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 Tauberian constant $\mathsf C_{\mathsf{S}} ^w$ is defined by \[ \mathsf C _{\mathsf{S}}^w(\alpha) := \sup_{\begin{subarray}{c} E\subset \mathbb{R}^n \\ 0\lt w(E) \lt +\infty\end{subarray}}\frac{1}{w(E)}w(\{x\in\mathbb{R}^n: \mathsf M_{\mathsf{S}}^{w}({\mathbf 1}_E)(x)>\alpha\}). \] 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.