Algebras of singular integral operators on rearrangement-invariant spaces and Nikolski ideals

Abstract:

We construct a presymbol for the Banach algebra $\alg(\Omega, S)$ 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 $\alg(\Omega,S)\to\Omega\oplus\Omega$ whose kernel coincides with the commutator ideal of $\alg(\Omega,S)$. In terms of the presymbol, necessary conditions for Fredholmness of an operator in $\alg(\Omega,S)$ are proved. All operators are considered on reflexive rearrangement-invariant spaces with nontrivial Boyd indices over the unit circle.

The author is partially supported by F.C.T. (Portugal) grants POCTI 34222/MAT/2000 and PRAXIS XXI/BPD/22006/99.

Departamento de Matem\'{a}tica, Instituto Superior T\'{e}cnico, Av. Rovisco Pais, 1049--001 Lisboa, Portugal
akarlov@math.ist.utl.pt
http://www.math.ist.utl.pt/~akarlov