On a modification of the Poisson integral operator

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 \(\gamma\).