On a modification of the Poisson integral operator

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 Teichmüller extension of γ to D and the smallest positive eigenvalue of γ. Introduction. A number of important problems in the potential theory of the complex plane C can be reduced to a linear integral equation of Fredholm type with the Neumann–Poincaré kernel k or its transposition. This kernel is assigned to a rectifiable and sufficiently smooth Jordan curve Γ ⊂ C by the formula (0.1) k(ζ, z) := − 1 π ∂ ∂~nζ log |ζ − z| , ζ, z ∈ Γ, ζ 6= z, 2000 Mathematics Subject Classification. Primary 30C62, 30C75.