The first author acknowledges financial support from the Spanish Ministry of Economy andCompetitiveness, through the “Severo Ochoa Programme for Centres of Excellence in R&D”(SEV-2015-0554). A part of this research was accomplished when the first author workedat Sun Yat-sen University. He is grateful to Professor Lixin Yan and the Department ofMathematics for their hospitality. The second author is supported by NSFC (Nos. 11671039,11871101) and NSFC-DFG (No. 11761131002). The third author is supported partly bygrant-in-Aid for Scientific Research (C) Nr. 15K04942, Japan Society for the Promotion ofScience

In this paper, we investigate the boundedness of the multilinear pseudo-differential operator T. First, we establish the local exponential decay estimates for T. In terms of the corresponding commutators T, we obtain the local subexponential decay estimates. Secondly, we derive the weighted mixed weak type inequality for T, which parallels Sawyer's conjecture for Calderón-Zygmund operators and covers the endpoint weighted inequalities. Last but not least, we present the sharp weighted estimates for T and T. It is worth mentioning that our results are totally new even in the linear case.

Peer reviewed