  In this paper, we consider periodic boundary value problems for differential
equations whose coefficients are trigonometric polynomials. We construct the
spaces of generalized functions, where such problems have solutions. In
particular, the solvability space of a periodic analogue of the Mizohata
equation is constructed. We build also a periodic analogue and a generalization
of the construction of the nonstandard analysis, where infinitely smalls are
not only functions, but also functional spaces.
  To show that not all constructions on the torus lead to a simplification in
compare with the plane, we consider a periodic analogue of the hypoelliptic
differential operator and show that its number-theoretic properties are
significant. In particular, it turns out that if a polynomial with integer
coefficients is irreducible in the rational field, then the corresponding
differential operator is hypoelliptic on the torus.
