Lp-boundedness of Multilinear Pseudo-differential Operators on Zn and Tn

The aim of this paper is to introduce and study multilinear pseudo-differential operators on Z(n) and T-n = (R-n/2 pi Z(n)) the n-torus. More precisely, we give sufficient conditions and sometimes necessary conditions for L-p-boundedness of these classes of operators. L-2-boundedness results for multilinear pseudo-differential operators on Z(n) and T-n with L-2-symbols are stated. The proofs of these results are based on elementary estimates on the multilinear Rihaczek transforms for functions in L-2(Z(n)) respectively L-2(T-n) which are also introduced. We study the weak continuity of multilinear operators on the m-fold product of Lebesgue spaces L-pj (Z(n)), j = 1,..., m and the link with the continuity of multilinear pseudo-differential operators on Z(n). Necessary and sufficient conditions for multilinear pseudo-differential operators on Z(n) or T-n to be a Hilbert-Schmidt operators are also given. We give a necessary condition for a multilinear pseudo-differential operators on Z(n) to be compact. A sufficient condition for compactness is also given.