Algebras of singular integral operators on rearrangement-invariant spaces and Nikolski ideals
Let \(\mathbb{T}\) be the unit circle and \(X({\mathbb{T}}, |d\tau|/2\pi)\) be a reflexive rearrangement-invariant space with nontrivial Boyd indices. It is considered the algebra \(\text{Alg} (\Omega, S)\) generated by the Cauchy singular integral operator \(S\) and the operator \(M_\varphi\) of multiplication by the function \(\varphi \in \Omega \subset L^\infty\). The author constructs a pre-symbol of the operators from \(\text{Alg} (\Omega, S)\) and gives necessary conditions for Fredholmness. Contents: 1. Introduction. 2. Rearrangement-invariant spaces and their indices. 3. Nikolski ideals associated with Douglas algebras. 4. The pre-symbol of the algebra \(\text{Alg} (\Omega, S)\). References.
scientific article; zbMATH DE number 1893532