Existence and Regularity of Periodic Solutions for a Class of Partial Differential Operators

We give a complete answer to the questions concerning existence and regularity of periodic solutions to a class of linear partial differential operators. The results depend on Diophantine conditions and also on a control on the sign of the imaginary part of the symbol, which is related to the Nirenberg–Treves condition (P). This control is based on the following aspects: the linear dependence of the imaginary part of the coefficients, the connectedness of certain sublevel sets, and the parity of the order of certain derivatives. For certain operators, the results are also influenced by the order of vanishing of the coefficients.

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