Oleksiy Karlovych

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 $X(\Gamma)$ 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.

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