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

Holon Institute of Technology
Bar-Ilan University

Research output: Contribution to journal › Article › peer-review

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
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.",

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doi = "10.1007/bf01233964",

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AU - Spigel, Yafim

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N2 - 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).

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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