Approximation, regularity and positivity preservation on Riemannian manifolds

The paper focuses on the L p -Positivity Preservation property ( L p -PP for short) on a Riemannian manifold ( M , g ) . It states that any L p function u with 1 < p < + ∞ , which solves ( − Δ + 1 ) u ≥ 0 on M in the sense of distributions must be non-negative. Our main result is that the L p -PP holds if (the possibly incomplete) M has a finite number of ends with respect to some compact domain, each of which is q -parabolic for some, possibly different, values 2 p / ( p − 1 ) < q ≤ + ∞ . When p = 2 , since ∞ -parabolicity coincides with geodesic completeness, our result settles in the affirmative a conjecture by M. Braverman, O. Milatovic and M. Shubin in 2002. On the other hand, we also show that the L p -PP is stable by removing from a complete manifold a possibly singular set with Hausdorff co-dimension strictly larger than 2 p / ( p − 1 ) or with a uniform Minkowski-type upper estimate of order 2 p / ( p − 1 ) . The threshold value 2 p / ( p − 1 ) is sharp as we show that when the Hausdorff co-dimension of the removed set is strictly smaller, then the L p -PP fails. This gives a rather complete picture. The tools developed to carry out our investigations include smooth monotonic approximation and consequent regularity results for subharmonic distributions, a manifold version of the Brezis–Kato inequality, Liouville-type theorems in low regularity, removable singularities results for L p -subharmonic distributions and a Frostman-type lemma. Since the seminal works by T. Kato, the L p -PP has been linked to the spectral theory of Schrödinger operators with singular potentials Δ − V . Here we present some applications of the main results of this paper to the case where V ∈ L l o c p , addressing the essential self-adjointness of the operator when p = 2 and whether or not C c ∞ ( M ) is an operator core for Δ − V in L p . • L p -Positivity preservation property. • Removable singularities in the L p property, and relation to completeness of the manifold. • Applications to spectral theory.