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 \(L^p\)-positivity preserving property, i.e., \( u\in L^p\ \& \ (-\Delta + 1)u\ge 0 \Rightarrow u\ge 0\). 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 \(L^p\)-positivity preserving property, \(p\in [1,+\infty ]\), 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.

在本文中，我们考虑 Cartan-Hadamard 流形（即非正截面曲率的单连通、完备流形），其负 Ricci 曲率以多项式无穷大增长。我们证明了许多通常保持有界曲率流形的函数属性在这种情况下仍然成立。其中包括流形上 Sobolev 空间的表征、所谓的 Calderón-Zygmund 不等式和\(L^p\) -正性保持性质，即\( u\in L^p\ \& \ (-\Delta + 1)u\ge 0 \右箭头 u\ge 0\)。主要工具是 Cartan-Hadamard 流形上的一类和二阶 Hardy 型不等式，其曲率具有多项式上限。在手稿的最后一部分中，我们证明了在具有利玛窦曲率的次二次负部分的流形上的\(L^p\) -正性保持属性\(p\in [1,+\infty ]\)。这概括了 B. Güneysu 的想法，并为 P. Hsu 提出的随机完整性的众所周知的条件提供了新的证明。