The aim of this paper is to introduce and study multilinear pseudo-differential operators
on Zn and Tn =
(Rn/
2πZn) the
n-torus.
More precisely, we give sufficient conditions and sometimes necessary conditions for
Lp-boundedness of these
classes of operators. L2-boundedness results for multilinear
pseudo-differential operators on Zn and Tn with
L2-symbols are stated. The proofs of these
results are based on elementary estimates on the multilinear Rihaczek transforms for
functions in L2(Zn)
respectively L2(Tn)
which are also introduced.
We study the weak continuity of multilinear operators on the m-fold product of Lebesgue
spaces Lpj(Zn),
j =
1,...,m and the
link with the continuity of multilinear pseudo-differential operators on Zn.
Necessary and sufficient conditions for multilinear pseudo-differential operators on
Zn or Tn to be a
Hilbert-Schmidt operators are also given. We give a necessary condition for a multilinear
pseudo-differential operators on Zn to be compact. A sufficient
condition for compactness is also given.