On Sharp Aperture-Weighted Estimates for Square Functions

Let Sα,ψ( f ) be the square function defined by means of the cone in R n+1 + of aperture α, and a standard kernel ψ . Let [w]Ap denote the Ap characteristic of the weight w. We show that for any 1 < p < ∞ and α ≥ 1, ‖Sα,ψ‖L p(w) αn[w] max ( 1 2 , 1 p−1 ) Ap . For each fixed α the dependence on [w]Ap is sharp. Also, on all class Ap the result is sharp in α. Previously this estimate was proved in the case α = 1 using the intrinsic square function. However, that approach does not allow to get the above estimate with sharp dependence on α. Hence we give a different proof suitable for all α ≥ 1 and avoiding the notion of the intrinsic square function.