| Abstract: |
| The classical Sobolev inequality not only holds in Euclidean space, but also on Cartan-Hadamard manifolds (with the same optimal constant), that is complete and simply connected Riemannian manifolds having nonpositive sectional curvatures at every point. On the other hand, the Poincar\`e (or spectral gap) inequality fails on Euclidean space but holds on hyperbolic space or more in general on any Cartan-Hadamard manifold with sectional curvatures bounded from above by a negative constant: this is a result due to McKean (1970). However, almost nothing seems to be known in between, namely when curvatures are negative but allowed to vanish at infinity. Here we establish some results in this direction, along with related consequences regarding smoothing effects for certain nonlinear diffusions of porous medium type. We are able to prove suitable Sobolev-type inequalities in the radial setting when curvatures are allowed to vanish with a power-type rate at infinity. Surprisingly enough, such inequalities tend to fail for nonradial functions, as counterexamples show. |
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