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 \cite{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 $n$-$\mathcal{S}^m_{ρ, δ}(\mathbb{R}^d)$, $0 \le δ\le ρ\le1$, $0 \le δ<1$ for some $m \le 0$. 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 $L^2$-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 $n$-$\mathcal{S}^{m}_{ρ,δ}(\mathbb{R}^{d})$, $0 \le ρ\le 1$, $0 \le δ<1$, $m \le 0$. Moreover, it can be pointed out that our results can be applied to wider classes of symbols which do not belong to the traditional symbol classes $n$-$\mathcal{S}^{m}_{ρ,δ}(\mathbb{R}^{d})$.