Weighted Solyanik estimates for the strong maximal function

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.