AN IMPROVED COMPACT EMBEDDING THEOREM FOR DEGENERATE SOBOLEV SPACES

This short note investigates the compact embedding of degenerate matrix weighted Sobolev spaces into weighted Lebesgue spaces. The Sobolev spaces explored are defined as the abstract completion of Lipschitz functions in a bounded domain Omega with respect to the norm:parallel to f parallel to(QH1,p(v,mu;Omega)) = parallel to f parallel to(Lvp(Omega)) + parallel to del f parallel to(LQP(mu;Omega))where the weight v is comparable to a power of the pointwise operator norm of the matrix valued function Q = Q(x) in Omega. Following our main theorem, we give an explicit application where degeneracy is controlled through an ellipticity condition of the formw(x)vertical bar xi vertical bar(p) <= (xi center dot Q(x)xi)(p/2) <= tau(x)vertical bar xi vertical bar(p)for a pair of p-admissible weights. We also give explicit examples demonstrating the sharpness of our hypotheses.