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

Let S_α be the multilinear square function defined on the cone with aperture α≥ 1. In this paper, we investigate several kinds of weighted norm inequalities for S_α. We first obtain a sharp weighted estimate in terms of aperture α and w→ ∈ A_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_∞ 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 gλ∗ function. Some results are new even in the linear case.