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Journal of Operator Theory

Volume 47, Issue 2, Spring 2002  pp. 303-323.

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

Authors Alexei Yu. Karlovich
Author institution: Department of Mathematics and Physics, South Ukrainian State Pedagogical Univ., Staroportofrankovskaya str. 26, 65020, Odessa, Ukraine

Summary:  In this paper we extend results on Fredholmness of singular integral operators with piecewise continuous coefficients in reflexive rearrange\-ment-invariant spaces $X(\Gamma)$ with nontrivial Boyd indices $\alpha_X,\beta_X$ ([22]) 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 rearrange\-ment-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.

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