Mellin Pseudodifferential Operator Techniques in the Theory of Singular Integral Operators on Carleson Curves
We will present an approach to the calculation of the local and essential spectra of singular integral operators (SIOs) acting in -spaces with Muckenhoupt weights on a class of Carleson curves which is based on the Mellin pseudodifferential operators technique. From the 70's the local representation of SIOs acting on -spaces with power weights on piece-wise Lyapunov curves are well-known as to be Mellin convolutions. Such representation together with the local principle have allowed to construct a complete Fredholm theory for SIOs with piece-wise continuous coefficients acting on -spaces with power weights on piece-wise Lyapunov curves. As an extension of this approach we will show that SIOs with discontinuous coefficients acting on -spaces with Muckenhoupt weights on a class of Carleson curves have local representations as general Mellin pseudodifferential operators. By means of the limit operators method we obtain the complete description of the local spectra of SIOs which leads then to the description of the essential spectra of SIOs. Also, we are going to discuss some applications of this method to SIOs on Carleson curves acting in Hölder spaces with general weights.
