On Sharp Aperture-Weighted Estimates for Square Functions

Let S_α, ψ (f) be the square function defined by means of the cone in ℝⁿ⁺¹₊ of aperture α, and a standard kernel ψ. Let [w]_Ap denote the A_p characteristic of the weight w. We show that for any 1<p<∞ and α ≥ 1, (Formula Presented) For each fixed α the dependence on [w]_Ap is sharp. Also, on all class A_p 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.