Weak and strong $A_p$-$A_{\mathrm{\infty }}$ estimates for square functions and related operators

We prove sharp weak and strong type weighted estimates for a class of dyadic operators that includes majorants of both standard singular integrals and square functions. Our main new result is the optimal bound $[w{]}_{A_p}^{1\mathrm{/}p}[w{]}_{A_{\mathrm{\infty }}}^{1\mathrm{/}2-1\mathrm{/}p}\lesssim [w{]}_{A_p}^{1\mathrm{/}2}$ for the weak type norm of square functions on $L^p(w)$ for $p>2$; previously, such a bound was only known with a logarithmic correction. By the same approach, we also recover several related results in a streamlined manner.