UC Irvine

We prove the following superexponential distribution inequality: For any integrable g on [0, 1)d with zero average, and any λ > 0, (Formula Presented) 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 2d in the denominator cannot be replaced by C2d with 0 < C < 1 independent of d when d → ∞. For d = 1 this is a classical result of Chang, Wilson, and Wolff; however, in the case of d > 1 they work with special square function S∞, and their result does not imply the estimates for the classical square function. Using a good λ inequalities technique, we then obtain unweighted and weighted Lp 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 L2 lower bounds for S obtained by Domelevo et al.