Quasi-periodic analytic solutions of a partial differential equations with constant coefficients (Q2756113)

The author is investigating in depth, using both old and new concepts related to almost periodicity, the problem of existence of almost periodic solutions to PDEs of the form \(Q(D)u=f\), with \(Q\) a polynomial with constant coefficients. The paper contains a good deal of results concerning various spaces of almost periodic functions or distributions, among them the spaces \(S_{ap}(\Lambda)= \cap^\infty_{m=1} C^m_{ap}(\mathbb{R}^s; \Lambda)\). The \(C^m_{ap}\) spaces consist of Bohr almost periodic functions with values in \(\mathbb{R}^s\), while \(\Lambda\) is a countable subset of \(\mathbb{R}^s\) (actually, a semigroup linearly generated by a finite number of elements). The set \(\Lambda\) is containing the spectrum of each function in \(S_{ap}(\Lambda)\). The Theorem 3 of this paper establishes the equivalence of three basic properties of the operator \(Q(D)\), namely: A) The operator \(Q(D)\) has a fundamental solution in the space \(S_{ap}'(\Lambda)\), the dual of \(S_{ap}(\Lambda)\); B) The equation \(Q(D)u=f\) has a solution \(u\in S_{ap}'(\Lambda)\) for each \(f\in S_{ap}' (\Lambda)\); C) \(Q(\lambda)\neq 0\) for each \(\lambda\in\Lambda\), and there exists a positive number \(c\), such that \(|Q(\lambda) |\geq c(1+\lambda^2)^{-m}\) for \(\lambda\in\Lambda\) and some positive integer \(m\).