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

Naum Krupnik; Yafim Spigel

doi:10.1007/BF01233964

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

Naum Krupnik, Yafim Spigel

Department of Mathematics

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2 Scopus citations

Abstract

We show that for any simple piecewise Ljapunov contour Γ there exists a power weight ρ such that the essential norm |SΓ| in the space L₂(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 |SΓ| in L₂(Γ) is minimal if both Γ₁ and Γ₂ are smooth in some neighborhoods of the common points. It is the case when the norm |SΓ| in the space L₂(Γ) as well as in Z₂(Γ, p) does not depend on the values of the angles and it can be calculated by formula (5).

Original language	English
Pages (from-to)	211-220
Number of pages	10
Journal	Integral Equations and Operator Theory
Volume	33
Issue number	2
DOIs	https://doi.org/10.1007/BF01233964
State	Published - 1999

Bibliographical note

Funding Information:
1 This research was partially supported by the Israel Science Foundation founded by the Israel Academy of Sciences and Humarrities.

Funding

1 This research was partially supported by the Israel Science Foundation founded by the Israel Academy of Sciences and Humarrities.

Funders	Funder number
Israel Academy of Sciences and Humarrities
Israel Science Foundation

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10.1007/BF01233964

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title = "Critical points of essential norms of singular integral operators in weighted spaces",

abstract = "We show that for any simple piecewise Ljapunov contour Γ there exists a power weight ρ such that the essential norm |SΓ| in the space L2(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 |SΓ| in L2(Γ) is minimal if both Γ1 and Γ2 are smooth in some neighborhoods of the common points. It is the case when the norm |SΓ| in the space L2(Γ) as well as in Z2(Γ, p) does not depend on the values of the angles and it can be calculated by formula (5).",

author = "Naum Krupnik and Yafim Spigel",

note = "Funding Information: 1 This research was partially supported by the Israel Science Foundation founded by the Israel Academy of Sciences and Humarrities.",

year = "1999",

doi = "10.1007/BF01233964",

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AU - Spigel, Yafim

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AB - We show that for any simple piecewise Ljapunov contour Γ there exists a power weight ρ such that the essential norm |SΓ| in the space L2(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 |SΓ| in L2(Γ) is minimal if both Γ1 and Γ2 are smooth in some neighborhoods of the common points. It is the case when the norm |SΓ| in the space L2(Γ) as well as in Z2(Γ, p) does not depend on the values of the angles and it can be calculated by formula (5).

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

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