Oleksiy Karlovych

Citation:

Karlovich, Alexei Yu. "Algebras of singular integral operators on rearrangement-invariant spaces and Nikolski ideals." The New York Journal of Mathematics. 8 (2002): 215-234.

Abstract:

We construct a presymbol for the Banach algebra $\operatorname{alg}(\Omega, S)$ generated by the Cauchy singular integral operator $S$ and the operators of multiplication by functions in a Banach subalgebra $\Omega$ of $L^\infty$. This presymbol is a homomorphism $\operatorname{alg}(\Omega,S)\to\Omega\oplus\Omega$ whose kernel coincides with the commutator ideal of $\operatorname{alg}(\Omega,S)$. In terms of the presymbol, necessary conditions for Fredholmness of an operator in $\operatorname{alg}(\Omega,S)$ are proved. All operators are considered on reflexive rearrangement-invariant spaces with nontrivial Boyd indices over the unit circle.

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