Superexponential estimates and weighted lower bounds for the square function

Paata Ivanisvili, Sergei Treil
2019 Transactions of the American Mathematical Society  
We prove the following superexponential distribution inequality: for any integrable g on [0, 1) d with zero average, and any λ > 0 , where S(g) denotes the classical dyadic square function in [0, 1) d . The estimate is sharp when dimension d tends to infinity in the sense that the constant 2 d in the denominator cannot be replaced by C2 d with 0 < C < 1 independent of d when d → ∞. For d = 1 this is a classical result of Chang-Wilson-Wolff [4]; however, in the case d > 1 they work with special
more » ... work with special square function S∞, and their result does not imply the estimates for the classical square function. Using good λ inequalities technique we then obtain unweighted and weighted L p lower bounds for S; to get the corresponding good λ inequalities we need to modify the classical construction. We also show how to obtain our superexponential distribution inequality (although with worse constants) from the weighted L 2 lower bounds for S, obtained in [5] .
doi:10.1090/tran/7795 fatcat:7ckkehzj5vb6dkep7jlzp2pfoy