Algebras of Singular Integral Operators with PQC Coefficients on Weighted Lebesgue Spaces

Abstract

Let \(\mathcal {B}_{p,w}\) be the Banach algebra of all bounded linear operators on the weighted Lebesgue space \(L^p(\mathbb T,w)\) with p ∈ (1, ∞) and a Muckenhoupt weight \(w\in A_p(\mathbb T)\) which is locally equivalent at open neighborhoods ut of points \(t\in \mathbb T\) to weights Wt for which the functions \(\tau \mapsto (\tau -t)(\ln W_t)'(\tau )\) are quasicontinuous on ut, and let PQC be the C∗-algebra of all piecewise quasicontinuous functions on \(\mathbb T\). The Banach algebra $$\displaystyle \mathfrak {A}_{p,w}=\operatorname {alg}\{aI,S_{\mathbb T}: a\in PQC\}\subset \mathcal {B}_{p,w} $$ generated by all multiplication operators aI by functions a ∈ PQC and by the Cauchy singular integral operator \(S_{\mathbb T}\) is studied. A Fredholm symbol calculus for the algebra \(\mathfrak {A}_{p,w}\) is constructed and a Fredholm criterion for the operators \(A\in \mathfrak {A}_{p,w}\) in terms of their Fredholm symbols is established by applying the Allan-Douglas local principle, the two idempotents theorem and a localization of Muckenhoupt weights Wt to power weights by using quasicontinuous functions and Mellin pseudodifferential operators with non-regular symbols.