Higher order distance-like functions and Sobolev spaces

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