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We consider extremal polynomials with respect to a Sobolev-type $p$-norm,with $1<p<\infty$ and measures supported on compact subsets of the real line.For a wide class of such extremal polynomials with respect to mutually singularmeasures (i.e. supported on disjoint subsets of the real line), it is provedthat their critical points are simple and contained in the interior of theconvex hull of the support of the measures involved and the asymptotic criticalpoint distribution is studied. We also find the $n$th root asymptotic behaviorof the corresponding sequence of Sobolev extremal polynomials and theirderivatives.

Sobolev extremal polynomials with respect to mutually singular measures