Sobolev extremal polynomials with respect to mutually singular measures

We consider extremal polynomials with respect to a Sobolev-type $p$-norm, with $1<p<\mathrm{\infty }$ and measures supported on compact subsets of the real line. For a wide class of such extremal polynomials with respect to mutually singular measures (i.e. supported on disjoint subsets of the real line), it is proved that their critical points are simple and contained in the interior of the convex hull of the support of the measures involved and the asymptotic critical point distribution is studied. We also find the $n$th root asymptotic behavior of the corresponding sequence of Sobolev extremal polynomials and their derivatives.