In this paper we extend results on Fredholmness of singular integral operators with piecewise continuous coefficients in reflexive rearrangement-invariant spaces \(X(\Gamma)\) with nontrivial Boyd indices \(\alpha_X,\beta_X\) [K98] to the weighted case. Suppose a weight \(w\) belongs to the Muckenhoupt classes \(A_{\frac{1}{\alpha_X}}(\Gamma)\) and \(A_{\frac{1}{\beta_X}}(\Gamma)\). We prove that these conditions guarantee the boundedness of the Cauchy singular integral operator \(S\) in the weighted rearrangement-invariant space \(X(\Gamma,w)\). Under a ``disintegration condition'' we construct a symbol calculus for the Banach algebra generated by singular integral operators with matrix-valued piecewise continuous coefficients and get a formula for the index of an arbitrary operator from this algebra. We give nontrivial examples of spaces, for which this ``disintegration condition'' is satisfied. One of such spaces is a Lebesgue space with a general Muckenhoupt weight over an arbitrary Carleson curve.
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.
We establish a Fredholm criterion for an arbitrary operator in the Banach algebra of singular integral operators
with piecewise continuous coefficients on Nakano spaces (generalized Lebesgue spaces with variable exponent) with Khvedelidze weights over Carleson curves with logarithmic whirl points.
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