Weak and Strong Type Estimates for the Multilinear Littlewood–Paley Operators.

Abstract

Let be the multilinear square function defined on the cone with aperture . In this paper, we investigate several kinds of weighted norm inequalities for . We first obtain a sharp weighted estimate in terms of aperture and . 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 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 function. Some results are new even in the linear case.