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Weighted Solyanik estimates for the strong maximal function

2018
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Publicacions matemàtiques
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Let M_ S denote the strong maximal operator on R^n and let w be a non-negative, locally integrable function. For α∈(0,1) we define the weighted sharp Tauberian constant C_ S associated with M_ S by C_ S (α):= _E⊂ R^n 0α}). We show that _α→ 1^- C_ S (α)=1 if and only if w∈ A_∞ ^*, that is if and only if w is a strong Muckenhoupt weight. This is quantified by the estimate C_ S(α)-1≲_n (1-α)^(cn [w]_A_∞ ^*)^-1 as α→ 1^-, where c>0 is a numerical constant; this estimate is sharp in the sense that

doi:10.5565/publmat6211807
fatcat:q7p7r2xutzd6hfrgu7s6vtjx7m