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On the essential norm of the Cauchy singular integral operator in weighted rearrangement-invariant spaces

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Abstract

In this paper we extend necessary conditions for Fredholmness of singular integral operators with piecewise continuous coefficients in rearrangement-invariant spaces [19] to the weighted caseX(Γ,w). These conditions are formulated in terms of indices α(Q t w) and β(Q t w) of a submultiplicative functionQ t w, which is associated with local properties of the space, of the curve, and of the weight at the pointt∈Γ. Using these results we obtain a lower estimate for the essential norm |S| of the Cauchy singular integral operatorS in reflexive weighted rearrangement-invariant spacesX(Γ,w) over arbitrary Carleson curves Γ:

$$\left| S \right| \geqslant \cot \left( {\pi \lambda _\Gamma ,w/2} \right)$$

where\(\lambda _{\Gamma ,w} : = \begin{array}{*{20}c} {\inf } \\ {t \in \Gamma } \\ \end{array} \min \left\{ {\alpha \left( {Q_t w} \right),1 - \beta \left( {Q_t w} \right)} \right\}\). In some cases we give formulas for computation of α(Q t w) and β(Q t w).

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Author notes

  1. Alexei Yu. Karlovich

    Present address: Departamento de Matematica, Instituto Superior Tecnico, Av. Rovisco Pais, 1096, Lisboa, Portugal

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  1. Department of Mathematics and Physics, South Ukrainian State Pedagogical University, Staroportofrankovskaya str. 26, 65020, Odessa, Ukraine

    Alexei Yu. Karlovich

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Karlovich, A.Y. On the essential norm of the Cauchy singular integral operator in weighted rearrangement-invariant spaces. Integr equ oper theory 38, 28–50 (2000). https://doi.org/10.1007/BF01192300

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