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 \(g^*_{\lambda }\) function. Some results are new even in the linear case.