Weighted Solyanik estimates for the strong maximal function

Paul Hagelstein, Ioannis Parissis
2018 Publicacions matemàtiques  
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
more » ... e exponent 1/(cn[w]_A_∞ ^*) can not be improved in terms of [w]_A_∞ ^*. As corollaries, we obtain a sharp reverse Hölder inequality for strong Muckenhoupt weights in R^n as well as a quantitative imbedding of A_∞^* into A_p^*. We also consider the strong maximal operator on R^n associated with the weight w and denoted by M_ S ^w. In this case the corresponding sharp Tauberian constant C_ S ^w is defined by C_ S ^w α) := _E⊂ R^n 0α}). We show that there exists some constant c_w,n>0 depending only on w and the dimension n such that C_ S ^w (α)-1 ≲_w,n (1-α)^c_w,n as α→ 1^- whenever w∈ A_∞ ^* is a strong Muckenhoupt weight.
doi:10.5565/publmat6211807 fatcat:q7p7r2xutzd6hfrgu7s6vtjx7m