On sharp aperture-weighted estimates for square functions

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