An isomorphism theorem for linear evolution problems on the line
The author considers the problem \[ {du\over dt}- Au= f,\tag{1} \] where \(X\) is a complex Banach space, \(u: \mathbb{R}\to X\), \(A\) is a closed linear operator on \(X\) with dense domain and \(f\in L^p(\mathbb{R}, X)\) for \(p\in (1,\infty)\). Weis' \(L^p\) multiplier theorem [\textit{L. Weis}, Math. Ann. 319, 735--758 (2001; Zbl 0989.47025)] is used to prove existence and uniqueness of solutions of (1) under the assumptions that \(X\) has ``unconditionality of martingale differences'', the spectrum of \(A\) contains no purely imaginary numbers, plus an assumption on the resolvent of \(A\) which involves the Rademacher bound. Rademacher boundedness is a key concept for the Weis multiplier theorem. Applications are discussed in section 5, including Hamiltonian systems. The concepts mentioned above which are probably not familiar to all readers are discussed in detail. The author makes very detailed connections with the literature and includes 62 references.
scientific article; zbMATH DE number 2057849