On a modification of the Poisson integral operator
Dariusz Partyka
Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica, Tome 65 (2011), / Harvested from The Polish Digital Mathematics Library

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 complexvalued 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 Teichmuller extension of γ to 𝔻 and the smallest positive eigenvalue of γ.

Publié le : 2011-01-01
EUDML-ID : urn:eudml:doc:289763
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Dariusz Partyka. On a modification of the Poisson integral operator. Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica, Tome 65 (2011) . http://gdmltest.u-ga.fr/item/bwmeta1.element.ojs-doi-10_17951_a_2011_65_2_121-137/

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