Karlovich, Alexei Yu., and Ilya M. Spitkovsky. "
The Cauchy singular integral operator on weighted variable Lebesgue spaces."
Concrete Operators, Spectral Theory, Operators in Harmonic Analysis and Approximation. Operator Theory: Advances and Applications, 236. Eds. Manuel Cepedello Boiso, Håkan Hedenmalm, Marinus A. Kaashoek, Alfonso Montes Rodríguez, and Sergei Treil. Basel: Birkhäuser, 2014. 275-291.
AbstractLet \(p:\mathbb{R}\to(1,\infty)\) be a globally log-Hölder continuous variable exponent and \(w:\mathbb{R}\to[0,\infty]\) be a weight. We prove that the Cauchy singular integral operator \(S\) is bounded on the weighted variable Lebesgue space \(L^{p(\cdot)}(\mathbb{R},w)=\{f:fw\in L^{p(\cdot)}(\mathbb{R})\}\) if and only if the weight \(w\) satisfies $$ \sup_{-\infty < a < b < \infty} \frac{1}{b-a} \|w\chi_{(a,b)}\|_{p(\cdot)} \|w^{-1}\chi_{(a,b)}\|_{p'(\cdot)}<\infty \quad (1/p(x)+1/p'(x)=1). $$
Karlovich, Alexei Yu., and Yuri I. Karlovich. "
Compactness of commutators arising in the Fredholm theory of singular integral operators with shifts."
Factorization, Singular Operators and Related Problems. Eds. Stefan Samko, Amarino Lebre, and António Ferreira dos Santos. Dordrecht: Kluwer Academic Publishers, 2003. 111-129.
AbstractThe paper is devoted to the compactness of the commutators \(aS_\Gamma - S_\Gamma aI\) and \(W_\alpha S_\Gamma - S_\Gamma W_\alpha\), where \(S_\Gamma\) is the Cauchy singular integral operator, \(a\) is a bounded measurable function, \(W_\alpha\) is the shift operator given by \(W_\alpha f = f\circ\alpha\), and \(\alpha\) is a bi-Lipschitz homeomorphism (shift). The cases of the unit circle and the unit interval are considered. We prove that these commutators are compact on rearrangement-invariant spaces with nontrivial Boyd indices if and only if the function a or, respectively, the derivative of the shift a has vanishing mean oscillation.