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Critical points of essential norms of singular integral operators in weighted spaces

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Abstract

We show that for any simple piecewise Ljapunov contour Γ there exists a power weight ρ such that the essential norm |S Γ| in the spaceL 2(Γ, ρ) does not depend on the angles of the contour and it is given by formula (2). All such weights are described. For the union Γ=Γ1∪Γ2 of two simple piecewise Lyapunov curves we prove that the essential norm |S Γ| inL 2(Γ) is minimal if both Γ1 and Γ2 are smooth in some neighborhoods of the common points. It is the case when the norm |S Γ| in the spaceL 2(Γ) as well as inL 2(Γ, ρ) does not depend on the values of the angles and it can be calculated by formula (5).

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Authors and Affiliations

  1. Dept. of Math. and CS, Bar-Ilan University, 52900, Ramat-Gan, Israel

    Naum Krupnik & Yafim Spigel

  2. Holon Center for Technological Education, P.O.B. 305, 58102, Holon, Israel

    Naum Krupnik & Yafim Spigel

Authors
  1. Naum Krupnik
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  2. Yafim Spigel
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Krupnik, N., Spigel, Y. Critical points of essential norms of singular integral operators in weighted spaces. Integr equ oper theory 33, 211–220 (1999). https://doi.org/10.1007/BF01233964

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  • DOI: https://doi.org/10.1007/BF01233964

MSC 1991