Maximal functions of the multilinear pseudo-differentials operators

Abstract

In this paper, we consider the maximal multilinear pseudo-differential operator with symbols $\sigma \in B{S}_{1,\delta }^0(R^n\times R^{mn})$, and establish the $L^p$ estimate with a sharp bound $O\left(\sqrt{\log (N+2)}\right)$. Our work improves the work of Chen, Dai and Lu [5] by extending the symbol σ from the Hörmander class $B{S}_{1,0}^0$ to $B{S}_{1,\delta }^0$ with $0\le \delta <1$. The main tools such as localizing the maximal pseudo-differential operators and the time-frequency analysis in [5] may not accommodate symbols $\sigma \in B{S}_{1,\delta }^0(R^n\times R^{mn})$ with $0<\delta <1$. In this work, we handle this difficulty by applying the inhomogeneous Littlewood-Paley-Stein decomposition to the space variable and using Taylor's expansion to track the size of those decomposed pieces. Then together with the ideas of using martingales, some related pointwise estimates and the good-λ inequality as in [16], [19], we will be able to obtain the boundedness with the optimal bound.