On a modification of the Poisson integral operator

Dariusz Partyka

Annales UMCS, Mathematica (2011)

  • Volume: 65, Issue: 2, page 121-137
  • ISSN: 2083-7402

Abstract

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Given a quasisymmetric automorphism γ of the unit circle T we define and study a modification Pγ of the classical Poisson integral operator in the case of the unit disk D. The modification is done by means of the generalized Fourier coefficients of γ. For a Lebesgue's integrable complex-valued function f on T, Pγ[f] is a complex-valued harmonic function in D and it coincides with the classical Poisson integral of f provided γ is the identity mapping on T. Our considerations are motivated by the problem of spectral values and eigenvalues of a Jordan curve. As an application we establish a relationship between the operator Pγ, the maximal dilatation of a regular quasiconformal Teichmüller extension of γ to D and the smallest positive eigenvalue of γ.

How to cite

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Dariusz Partyka. "On a modification of the Poisson integral operator." Annales UMCS, Mathematica 65.2 (2011): 121-137. <http://eudml.org/doc/267964>.

@article{DariuszPartyka2011,
abstract = {Given a quasisymmetric automorphism γ of the unit circle T we define and study a modification Pγ of the classical Poisson integral operator in the case of the unit disk D. The modification is done by means of the generalized Fourier coefficients of γ. For a Lebesgue's integrable complex-valued function f on T, Pγ[f] is a complex-valued harmonic function in D and it coincides with the classical Poisson integral of f provided γ is the identity mapping on T. Our considerations are motivated by the problem of spectral values and eigenvalues of a Jordan curve. As an application we establish a relationship between the operator Pγ, the maximal dilatation of a regular quasiconformal Teichmüller extension of γ to D and the smallest positive eigenvalue of γ.},
author = {Dariusz Partyka},
journal = {Annales UMCS, Mathematica},
keywords = {Dirichlet integral; eigenvalue of a Jordan curve; eigenvalue of a quasisymmetric automorphism; extremal quasiconformal mapping; Fourier coefficient; harmonic conjugation operator; harmonic function; Neumann-Poincaré kernel; Poisson integral; quasiconformal mapping; quasisymmetric automorphism; Teichmüller mapping; welding homeomorphism; quasisymmetric homeomorphism},
language = {eng},
number = {2},
pages = {121-137},
title = {On a modification of the Poisson integral operator},
url = {http://eudml.org/doc/267964},
volume = {65},
year = {2011},
}

TY - JOUR
AU - Dariusz Partyka
TI - On a modification of the Poisson integral operator
JO - Annales UMCS, Mathematica
PY - 2011
VL - 65
IS - 2
SP - 121
EP - 137
AB - Given a quasisymmetric automorphism γ of the unit circle T we define and study a modification Pγ of the classical Poisson integral operator in the case of the unit disk D. The modification is done by means of the generalized Fourier coefficients of γ. For a Lebesgue's integrable complex-valued function f on T, Pγ[f] is a complex-valued harmonic function in D and it coincides with the classical Poisson integral of f provided γ is the identity mapping on T. Our considerations are motivated by the problem of spectral values and eigenvalues of a Jordan curve. As an application we establish a relationship between the operator Pγ, the maximal dilatation of a regular quasiconformal Teichmüller extension of γ to D and the smallest positive eigenvalue of γ.
LA - eng
KW - Dirichlet integral; eigenvalue of a Jordan curve; eigenvalue of a quasisymmetric automorphism; extremal quasiconformal mapping; Fourier coefficient; harmonic conjugation operator; harmonic function; Neumann-Poincaré kernel; Poisson integral; quasiconformal mapping; quasisymmetric automorphism; Teichmüller mapping; welding homeomorphism; quasisymmetric homeomorphism
UR - http://eudml.org/doc/267964
ER -

References

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