On Sharp Aperture-Weighted Estimates for Square Functions

Abstract

Let $S_{\alpha ,\psi }(f)$ be the square function defined by means of the cone in ${\mathbb R}^{n+1}_{+}$ of aperture $\alpha$, and a standard kernel $\psi$. Let $[w]_{A_p}$ denote the $A_p$ characteristic of the weight $w$. We show that for any $1<p<\infty$ and $\alpha \ge 1$,

$$\begin{aligned} \Vert S_{\alpha ,\psi }\Vert _{L^p(w)}\lesssim \alpha ^n[w]_{A_p}^{\max \left( \frac{1}{2},\frac{1}{p-1}\right) }. \end{aligned}$$

For each fixed $\alpha$ the dependence on $[w]_{A_p}$ is sharp. Also, on all class $A_p$ the result is sharp in $\alpha$. Previously this estimate was proved in the case $\alpha =1$ using the intrinsic square function. However, that approach does not allow to get the above estimate with sharp dependence on $\alpha$. Hence we give a different proof suitable for all $\alpha \ge 1$ and avoiding the notion of the intrinsic square function.