On sharp aperture-weighted estimates for square functions

Let $S_{\a,\psi}(f)$ be the square function defined by means of the cone in ${\mathbb R}^{n+1}_{+}$ of aperture $\a$, 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 $\a\ge 1$, $$\|S_{\a,\psi}\|_{L^p(w)}\lesssim \a^n[w]_{A_p}^{\max(1/2,\frac{1}{p-1})}.$$ For each fixed $\a$ the dependence on $[w]_{A_p}$ is sharp. Also, on all class $A_p$ the result is sharp in $\a$. Previously this estimate was proved in the case $\a=1$ using the intrinsic square function. However, that approach does not allow to get the above estimate with sharp dependence on $\a$. Hence we give a different proof suitable for all $\a\ge 1$ and avoiding the notion of the intrinsic square function.