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arXiv · 2009.13814

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

Mingming Cao·Mahdi Hormozi·Gonzalo Ibañez-Firnkorn·Israel P. Rivera-Ríos·Zengyan Si·Kôzô Yabuta
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Abstract

Let $S_α$ be the multilinear square function defined on the cone with aperture $α\geq 1$. 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 $\vec{w} \in A_{\vec{p}}$. 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 $g^*_λ$ function. Some results are new even in the linear case.

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Mingming Cao, Mahdi Hormozi, Gonzalo Ibañez-Firnkorn, Israel P. Rivera-Ríos, Zengyan Si, Kôzô Yabuta. 2020-10-23. Weak and strong type estimates for the multilinear Littlewood-Paley operators. https://arxiv.org/abs/2009.13814

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