Approximation, regularity and positivity preservation on Riemannian manifolds

TL;DR

This paper establishes conditions under which the $L^{p}$-Positivity Preservation property holds on Riemannian manifolds, linking geometric features like ends and parabolicity to positivity and spectral properties of differential operators.

Contribution

It provides a comprehensive characterization of $L^{p}$-PP on manifolds with ends, including stability under removal of singular sets and applications to Schrödinger operators with singular potentials.

Findings

$L^{p}$-PP holds if manifold has ends that are $q$-parabolic with $q eq 2p/(p-1)$.

$L^{p}$-PP is stable under removal of sets with Hausdorff co-dimension > $2p/(p-1)$.

Failure of $L^{p}$-PP when the co-dimension is smaller than the threshold.

Abstract

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 $1<p<+\infty$, which solves $(-\Delta + 1)u\ge 0$ 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 $2p/(p-1) < q \leq +\infty$. When $p=2$, since $\infty$-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 $2p/(p-1)$ or with a uniform Minkowski-type upper estimate of order $2p/(p-1)$. The threshold value $2p/(p-1)$ 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 Schr\"odinger operators with singular potentials $\Delta - V$. Here we present some applications of the main results of this paper to the case where $V\in L^p_{loc}$, addressing the essential self-adjointness of the operator when $p=2$ and whether or not $C^\infty_c(M)$ is an operator core for $\Delta-V$ in $L^p$.