On the Toroidal Quantization of Periodic Pseudo-Differential Operators

On the torus, pseudo-differential operators can be presented globally by Fourier series, without local coordinate charts. Periodization of partial differential equations leads to investigating the correspondence between toroidal and Euclidean symbols of operators. Fourier series operators—the analogues of Fourier integral operators—are introduced, and formulae for their compositions with pseudo-differential operators are presented. The L 2 boundedness of pseudo-differential and Fourier series operators follows from certain conditions on phases and amplitudes.