  In this article, we begin a systematic study of the boundedness and the
nuclearity properties of multilinear periodic pseudo-differential operators and
multilinear discrete pseudo-differential operators on $L^p$-spaces. First, we
prove analogues of known multilinear Fourier multipliers theorems (proved by
Coifman and Meyer, Grafakos, Tomita, Torres, Kenig, Stein, Fujita, Tao, etc.)
in the context of periodic and discrete multilinear pseudo-differential
operators. For this, we use the periodic analysis of pseudo-differential
operators developed by Ruzhansky and Turunen. Later, we investigate the
$s$-nuclearity, $0<s \leq 1,$ of periodic and discrete pseudo-differential
operators. To accomplish this, we classify those $s$-nuclear multilinear
integral operators on arbitrary Lebesgue spaces defined on $\sigma$-finite
measures spaces. We also study similar properties for periodic Fourier integral
operators. Finally, we present some applications of our study to deduce the
periodic Kato-Ponce inequality and to examine the $s$-nuclearity of multilinear
Bessel potentials as well as the $s$-nuclearity of periodic Fourier integral
operators admitting suitable types of singularities.
