Strong weighted and restricted weak weighted estimates of the square function

TL;DR

This paper provides sharp weighted estimates for the dyadic square function, including a refined $T1$-type testing condition and a sharp restricted weak type estimate that eliminates logarithmic corrections.

Contribution

It introduces a sharper weighted estimate for the square function using $T1$-type testing conditions and establishes a sharp restricted weak type estimate for characteristic functions.

Findings

Sharp weighted estimate for the square function from $L^2(w)$ to $L^2(w)$.

Refinement of estimates using $T1$-type testing conditions.

Sharp restricted weak type estimate without logarithmic correction for characteristic functions.

Abstract

In this note we give a sharp weighted estimate for square function from $L^2(w)$ to $L^2(w)$, $w\in A_2$. This has been known. But we also give a sharpening of this weighted estimate in the spirit of $T1$-type testing conditions. Finally we show that for any weight $w\in A^d_2$ and any characteristic function of a measurable set $\|S_w\chi_E\|_{L^{2, \infty}(w^{-1})} \le C \sqrt{[w]_{A^d_2}}\, \|\chi_E\|_w$, and this estimate is sharp. So on characteristic functions of measurable sets at least, no logarithmic correction is needed for the weak type of the dyadic square function.The sharp estimate for the restricted weak type is at most $ \sqrt{[w]_{A^d_2}}$.