Estimates of Some Dyadic Operators in the Weighted Setting

Abstract

The dyadic square function and the constant Haar multiplier have been estimated linearly with the A₂ characteristic of the weight, [w]_A₂ in the weighted Lebesgue space, L²(w).  In this paper, we explore the estimation of the dyadic variable square function and the estimation of its composition with a constant Haar multiplier. This work shows that, the weight function, w in the dyadic reverse Hölder class 2, RH₂ᵈ, characterizes the boundedness of S_w and S_w∘ T_σ . More precisely, our work is concerned with the boundedness of the dyadic variable square function and the boundedness of its composition from L²(ℝ)  to L²(ℝ,w); a single weight case.