Local spectra and index of singular integral operators with piecewise continuous coefficients on composed curves

C. J. Bishop, A. Böttcher, Yu I. Karlovich, Ilya Spitkovsky

Research output: Contribution to journalArticle

Abstract

We establish a symbol calculus for deciding whether singular integral operators with piecewise continuous coefficients are Fredholm on the Lebesgue space Lp(Γ, w) where 1 < p < ∞, Γ is a composed Carleson curve, and w is a Muckenhoupt weight in the class Ap(Γ). We also provide index formulas for the operators in the closed algebra of singular integral operators with piecewise continuous matrix-valued coefficients. Our main theorem is based upon three pillars: on the identification of the local spectrum of the Cauchy singular integral operator at the endpoints of simple Carleson arcs, on an appropriate "N projections theorem", and on results of geometric function theory pertaining to the problem of extending Carleson curves and Muckenhoupt weights.

Original languageEnglish (US)
Pages (from-to)5-83
Number of pages79
JournalMathematische Nachrichten
Volume206
DOIs
StatePublished - Jan 1 1999

Fingerprint

Local Spectrum
Piecewise continuous
Singular Integral Operator
Muckenhoupt Weights
Curve
Coefficient
Cauchy Integral
Lebesgue Space
Theorem
Calculus
Arc of a curve
Projection
Closed
Algebra
Operator

Keywords

  • Ahlfors-David curve
  • Carleson condition
  • Geometric function theory
  • Idempotent
  • Index
  • Muckenhoupt condition
  • Singular integral operator
  • Spectral theory

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

Local spectra and index of singular integral operators with piecewise continuous coefficients on composed curves. / Bishop, C. J.; Böttcher, A.; Karlovich, Yu I.; Spitkovsky, Ilya.

In: Mathematische Nachrichten, Vol. 206, 01.01.1999, p. 5-83.

Research output: Contribution to journalArticle

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