Pseudodifferential operators with analytic symbols (Q1363482)

Under study is the action of pseudodifferential operators, with symbols \(\hat k(\xi)\) analytic in the strip \(|\Im \xi|\leq\rho\), in the scale \(V^s(\rho)\) of Banach spaces of distributions on the real axis. Each element of \(V^s(\rho)\) is a sum of distributions periodic at both infinities with period \(L\) and belonging to the class \(H^s(0,L)\), which are fused together by an exponentially decreasing function of the class \(E^s(\rho)\). The basic properties of such operators are described. As an application, the author considers the boundary value problem for harmonic functions in a strip with discontinuous boundary data. Properties of eigenfunctions are described and necessary and sufficient conditions are found for the existence of a solution.