On a modification of the Poisson integral operator

Dariusz Partyka

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 \(\gamma\).

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

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References


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DOI: http://dx.doi.org/10.2478/v10062-011-0019-0
Date of publication: 2016-07-27 21:54:11
Date of submission: 2016-07-27 13:37:57


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