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 $\gamma$.