H\"ormander type theorem for multilinear Pseudo-differential operators

We establish a H\"{o}rmander type theorem for the multilinear pseudo-differential operators, which is also a generalization of the results in {MR4322619} to symbols depending on the spatial variable. Most known results for multilinear pseudo-differential operators were obtained by assuming their symbols satisfy pointwise derivative estimates(Mihlin-type condition), that is, their symbols belong to some symbol classes n-Sᵐρ, δ(Rᵈ), 0 ≤ δ ≤ ρ ≤1, 0 ≤ δ<1 for some m ≤ 0. In this paper, we shall consider multilinear pseudo-differential operators whose symbols have limited smoothness described in terms of function space and not in a pointwise form(H\"ormander type condition). Our conditions for symbols are weaker than the Mihlin-type conditions in two senses: the one is that we only assume the first-order derivative conditions in the spatial variable and lower-order derivative conditions in the frequency variable, and the other is that we make use of L²-average condition rather than pointwise derivative conditions for the symbols. As an application, we obtain some mapping properties for the multilinear pseudo-differential operators associated with symbols belonging to the classes n-Sᵐρ,δ(Rᵈ), 0 ≤ ρ ≤ 1, 0 ≤ δ<1, m ≤ 0. Moreover, it can be pointed out that our results are applied to wider classes of symbols which do not belong to the traditional symbol classes n-Sᵐρ,δ(Rᵈ). 2010 Mathematics Subject Classification. Primary 42B20, Secondary 42B15.