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Algebras of singular integral operators with PC coefficients in rearrangement-invariant spaces with Muckenhoupt weights

Citation:
Karlovich, Alexei Yu. "Algebras of singular integral operators with PC coefficients in rearrangement-invariant spaces with Muckenhoupt weights." Journal of Operator Theory. 47 (2002): 303-323.

Abstract:

In this paper we extend results on Fredholmness of singular integral operators with piecewise continuous coefficients in reflexive rearrangement-invariant spaces X(Γ) with nontrivial Boyd indices αX,βX [K98] to the weighted case. Suppose a weight w belongs to the Muckenhoupt classes A1αX(Γ) and A1βX(Γ). We prove that these conditions guarantee the boundedness of the Cauchy singular integral operator S in the weighted rearrangement-invariant space X(Γ,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.

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