Sharp weighted norm inequalities for Littlewood-Paley operators and singular integrals

Abstract

We prove sharp $L^p(w)$ norm inequalities for the intrinsic square function (introduced recently by M. Wilson) in terms of the $A_p$ characteristic of $w$ for all $1<p<\infty$. This implies the same sharp inequalities for the classical Lusin area integral $S(f)$, the Littlewood-Paley $g$-function, and their continuous analogs $S_{\psi}$ and $g_{\psi}$. Also, as a corollary, we obtain sharp weighted inequalities for any convolution Calder\'on-Zygmund operator for all $1<p\le 3/2$ and $3\le p<\infty$, and for its maximal truncations for $3\le p<\infty$