Weak and strong type estimates for the multilinear Littlewood-Paley operators

Abstract

Let $S_{\alpha }$ be the multilinear square function defined on the cone with aperture $\alpha \ge 1$. In this paper, we investigate several kinds of weighted norm inequalities for $S_{\alpha }$. We first obtain a sharp weighted estimate in terms of aperture $\alpha$ and $\vec{w}\in A_{\vec{p}}$. By means of some pointwise estimates, we also establish two-weight inequalities including bump and entropy bump estimates, and Fefferman-Stein inequalities with arbitrary weights. Beyond that, we consider the mixed weak type estimates corresponding Sawyer's conjecture, for which a Coifman-Fefferman inequality with the precise $A_{\infty }$ norm is proved. Finally, we present the local decay estimates using the extrapolation techniques and dyadic analysis respectively. All the conclusions aforementioned hold for the Littlewood-Paley $g_{\lambda }^{\ast }$ function. Some results are new even in the linear case.

35pages, Some new results have been added