Multilinear pseudo-differential operators on the multidimensional torus

In the talk, we discuss some results on the boundedness of the toroidal linear pseudo-differential operators of the form
$$T_\sigma (f; x) := \sum_{\xi\in\mathbb{Z}^m} \sigma(x, \xi) \widehat{f}(\xi) e^{2\pi i \xi x}$$
and multilinear pseudo-differential operators of the form
$$T_\sigma (f_1, \ldots, f_n; x) := \sum_{(\xi_1, \ldots, \xi_n)\in\mathbb{Z}^{nm}} \sigma(x, \xi_1, \ldots, \xi_n) \widehat{f}_1(\xi_1)\times \cdots \times\widehat{f}_n(\xi_n) e^{2\pi i (\xi_1 + \cdots + \xi_n) x}$$
with the symbols from the Hörmander class and from its multilinear analog, respectively, as operators from the Nikol'skii–Besov or Lizorkin–Triebel function spaces and from the tensor product of such spaces into other similar spaces.