Higher order distance-like functions and Sobolev spaces

Abstract

On non-compact Riemannian manifolds, we construct distance-like functions with derivatives controlled up to some order k assuming bounds on the growth of the derivatives of the curvature up to order k−2 and on the decay of the injectivity radius. This construction extends previously known results in various directions, permitting to obtain consequences which are (in a sense) sharp. As a first main application, we give refined conditions guaranteeing the density of compactly supported smooth functions in the Sobolev space Wk,p on the manifold. Contrary to all previously known results this can be obtained also on manifolds with possibly unbounded geometry. In the particular case p=2, making use of the Weitzenböck formula for a Lichnerowicz Laplacian acting on k-covariant totally symmetric tensor fields, we can weaken the assumptions needed to obtain the density property, avoiding any condition on the highest order derivatives of the curvature. Distance-like functions are also used to obtain new disturbed Sobolev inequalities, disturbed Lp-Calderón-Zygmund inequalities and the full Omori-Yau maximum principle for the Hessian under weak assumptions.