On a modification of the Poisson integral operator

Partyka, Dariusz

Annales Universitatis Mariae Curie-Skłodowska. Mathematica , Tome 65 (2011) no. 2.

Résumé

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.

Détail

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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