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Weighted Solyanik estimates for the strong maximal function. (English) Zbl 1385.42011

Summary: 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 \[ \mathsf{C}_{\mathsf{S},w}(\alpha):=\mathop{\sup}\limits_{\mathop{E\subset\mathbb{R}^n}\limits_{0<w(E)<+\infty}}\frac1{w(E)}w(\{x\in\mathbb{R}^n:\mathsf{M}_{\mathsf{S}}(\mathbf{1}_E)(x)>\alpha\}). \] We show that \(\lim_{\alpha\to1^-}\mathsf{C}_{\mathsf{S},w}(\alpha)=1\) if and only if \(w\in A_\infty^\ast\), that is if and only if \(w\) is a strong Muckenhoupt weight. This is quantified by the estimate \(\mathsf{C}_{\mathsf{S},w}(\alpha)-1\lesssim_n(1-\alpha)^{(cn[w]_{A_\infty^\ast})^{-1}}\) as \(\alpha\to1^-\), where \(c>0\) is a numerical constant independent of \(n\); this estimate is sharp in the sense that the exponent \(1/(cn[w]_{A_\infty^\ast})\) can not be improved in terms of \([w]_{A_\infty^\ast}\). As corollaries, we obtain a sharp reverse Hölder inequality for strong Muckenhoupt weights in \(\mathbb{R}^n\) as well as a quantitative imbedding of \(A_\infty^\ast\) into \(A_p^\ast\). 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 \[ \mathsf{C}_{\mathsf{S}}^w(\alpha):=\mathop{\sup}\limits_{\mathop{E\subset\mathbb{R}^n}\limits_{0<w(E)<+\infty}}\frac1{w(E)}w(\{x\in\mathbb{R}^n:\mathsf{M}_{\mathsf{S}}^w(\mathbf{1}_E)(x)>\alpha\}). \] 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 \(\alpha\to1^-\) whenever \(w\in A_\infty^\ast\) is a strong Muckenhoupt weight.

MSC:

42B25 Maximal functions, Littlewood-Paley theory
42B35 Function spaces arising in harmonic analysis
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