Hörmander type theorem for multilinear pseudo-differential operators

We establish a Hörmander type theorem for the multilinear pseudo-differential operators, which is also a generalization of the results in [32] 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-S_ρ,δ^m(R^d), 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 (Hörmander 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-S_ρ,δ^m(R^d), 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-S_ρ,δ^m(R^d).