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.
@article{bwmeta1.element.doi-10_2478_v10062-011-0019-0,
author = {Dariusz Partyka},
title = {On a modification of the Poisson integral operator},
journal = {Annales UMCS, Mathematica},
volume = {65},
year = {2011},
pages = {121-137},
zbl = {1269.30029},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_v10062-011-0019-0}
}
Dariusz Partyka. On a modification of the Poisson integral operator. Annales UMCS, Mathematica, Tome 65 (2011) pp. 121-137. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_v10062-011-0019-0/
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