On a modification of the Poisson integral operator

Dariusz Partyka

Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica (2011)

  • Volume: 65, Issue: 2
  • ISSN: 0365-1029

Abstract

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Given a quasisymmetric automorphism γ of the unit circle 𝕋 we define and study a modification P γ 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 complexvalued function f on 𝕋 , P γ [ 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 P γ , the maximal dilatation of a regular quasiconformal Teichmuller 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 Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica 65.2 (2011): null. <http://eudml.org/doc/289763>.

@article{DariuszPartyka2011,
abstract = {Given a quasisymmetric automorphism $\gamma $ of the unit circle $\mathbb \{T\}$ we define and study a modification $P_\{\gamma \}$ of the classical Poisson integral operator in the case of the unit disk $\mathbb \{D\}$. The modification is done by means of the generalized Fourier coefficients of $\gamma $. For a Lebesgue’s integrable complexvalued function $f$ on $\mathbb \{T\}$, $P_\{\gamma \}[f]$ is a complex-valued harmonic function in $\mathbb \{D\}$ and it coincides with the classical Poisson integral of $f$ provided $\gamma $ is the identity mapping on $\mathbb \{T\}$. 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 $P_\{\gamma \}$, the maximal dilatation of a regular quasiconformal Teichmuller extension of $\gamma $ to $\mathbb \{D\}$ and the smallest positive eigenvalue of $\gamma $.},
author = {Dariusz Partyka},
journal = {Annales Universitatis Mariae Curie-Skłodowska, sectio A – 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-Poincare kernel; Poisson integral},
language = {eng},
number = {2},
pages = {null},
title = {On a modification of the Poisson integral operator},
url = {http://eudml.org/doc/289763},
volume = {65},
year = {2011},
}

TY - JOUR
AU - Dariusz Partyka
TI - On a modification of the Poisson integral operator
JO - Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica
PY - 2011
VL - 65
IS - 2
SP - null
AB - Given a quasisymmetric automorphism $\gamma $ of the unit circle $\mathbb {T}$ we define and study a modification $P_{\gamma }$ of the classical Poisson integral operator in the case of the unit disk $\mathbb {D}$. The modification is done by means of the generalized Fourier coefficients of $\gamma $. For a Lebesgue’s integrable complexvalued function $f$ on $\mathbb {T}$, $P_{\gamma }[f]$ is a complex-valued harmonic function in $\mathbb {D}$ and it coincides with the classical Poisson integral of $f$ provided $\gamma $ is the identity mapping on $\mathbb {T}$. 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 $P_{\gamma }$, the maximal dilatation of a regular quasiconformal Teichmuller extension of $\gamma $ to $\mathbb {D}$ and the smallest positive eigenvalue of $\gamma $.
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-Poincare kernel; Poisson integral
UR - http://eudml.org/doc/289763
ER -

References

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