Optimal extension of Fourier multiplier operators in \(L^p(G)\)
Let \(G\) be a compact abelian group. For \(1 \leq p < \infty\), in this paper the authors study the optimal domain (a Banach function space containing \(L^p(G)\)) extension question for the Fourier multiplier operator \(T_{\psi}^{(p)}\), with values in \(L^p(G)\). The recent monograph by \textit{S. Okada, W. J. Ricker} and \textit{E. A. Sánchez Pérez} [Optimal domain and integral extension of operators, acting in function spaces. Basel: Birkhäuser (2008; Zbl 1145.47027)] is a good source for background material on this topic. We now quote some results without introducing the notation. The authors identify \(L^1(m_\psi^{(p)})\) as those \(f \in L^1(G)\) for which \((\psi . (\chi_A))^\wedge \in (L^p(G))^\wedge\) for all \(A \in {\mathcal B}(G)\). They also show that \(T^{(p)}_\psi\) is upper semi-Fredholm (lower semi-Fredholm) if and only if it is Fredholm. When this happens, the operator is a finite rank perturbation of an invertible \(p\)-multiplier operator and \(L^1(m_\psi^{(p)}) = L^p(G)\).
scientific article; zbMATH DE number 5835394
scientific article