Some Functional Properties on Cartan-Hadamard Manifolds of Very Negative Curvature

In this paper, we consider Cartan-Hadamard manifolds (i.e., simply connected, complete, of non-positive sectional curvature) whose negative Ricci curvature grows polynomially at infinity. We show that a number of functional properties, which typically hold on manifolds of bounded curvature, remain true in this setting. These include the characterization of Sobolev spaces on manifolds, the so-called Calder & oacute;n-Zygmund inequalities and the L-p-positivity preserving property, i.e., u is an element of L-p & (-Delta+1)u >= 0 double right arrow u >= 0. The main tool is a new class of first- and second-order Hardy-type inequalities on Cartan-Hadamard manifolds with a polynomial upper bound on the curvature. In the last part of the manuscript we prove the L-p-positivity preserving property, p is an element of[1,+infinity], on manifolds with subquadratic negative part of the Ricci curvature. This generalizes an idea of B. G & uuml;neysu and gives a new proof of a well-known condition for the stochastic completeness due to P. Hsu.