Singulars integrals, quasiconformal mappings and Sobolev spaces

We propose to study some aspects of the potential theory of the signed Riesz kernels R\sub d\nosub (x)=xIxI\super 1+d\nosuper. Specificaclly, we would like to deal with the problem, raised by David and Semmes, of obtained rectifiability from the L\super 2\nosuper-boundedness of the operators associated to the Riesz kernels on Ahlfors regular sets. A question suggested by the above problem consists in describing the homogeneous smooth Calderon-Zygmund operators in R\super n\nosuper for which the L\super 2\nosuper norm of the maximal singular integral cn be estimated by a constant times the L\super 2\nosuper norm of the singular integral itself. In quasi-conformal mapping theory we propose to study general distortion inequaliteis involving Hausdorff content and sharp removability problems for bounded quasi-regular mappings. In Sobolev space theory we intend to consider characterizations of the Sobolev space in metric-measure spaces by means of a new quadratic multi-scale function