Topics in Analysis: Interpolation and Approximation

Let X be the Sobolev space of functions on R^n whose derivatives up to order m belong to L^p. For a subset E of R^n, let X(E) denote the space of restrictions to E of functions in X. If p>n there exists a bounded linear operator T:X(E)->X such that Tf=f on E (i.e. T is an "extension operator"). The corresponding result for p<n is unknown. The course is an attempt to find a proof.