12 research outputs found
On a modification of the Poisson integral operator
Given a quasisymmetric automorphism of the unit circle we define and study a modification of the classical Poisson integral operator in the case of the unit disk . The modification is done by means of the generalized Fourier coefficients of . For a Lebesgue’s integrable complexvalued function on , is a complex-valued harmonic function in and it coincides with the classical Poisson integral of provided is the identity mapping on . 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 , the maximal dilatation of a regular quasiconformal Teichmuller extension of to and the smallest positive eigenvalue of