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 Calderón–Zygmund inequalities and the \(L^p\) $L^p$ -positivity preserving property, i.e., \( u\in L^p\ \& \ (-\Delta + 1)u\ge 0 \Rightarrow u\ge 0\) $u\in L^p\&(-\mathrm{\Delta }+1)u\ge 0\Rightarrow u\ge 0$ . 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 \(L^p\) $L^p$ -positivity preserving property, \(p\in [1,+\infty ]\) $p\in [1,+\infty ]$ , on manifolds with subquadratic negative part of the Ricci curvature. This generalizes an idea of B. Güneysu and gives a new proof of a well-known condition for the stochastic completeness due to P. Hsu.