On the Unweighted Lp Boundedness of a Weight-Adapted Dyadic Square Function

Abstract

We study the boundedness of a weight-adapted dyadic square function S_w on the unweighted Lebesgue spaces L^P (R). The operator is obtained from the classical dyadic square function by inserting the normalized local weight factor w (x)/<w>_I where I ranges over dyadic intervals and < w> I denotes the average of w over I. This factor is natural in dyadic weighted harmonic analysis and is closely related to the symbols appearing in t-Haar multipliers. We prove a sharp dichotomy between the ranges 1 < p ≤ 2 and p > 2. For 1 < p ≤ 2, the operator S w is bounded on L ^P (R) for every locally integrable weight w > 0 almost everywhere. For p > 2, we show that S_W is bounded on L^P (R) if and only if w ∈ RH_s^D
s>p/2 where RH_s^D denotes the dyadic reverse Hölder class.