{
  "id": "2301.05159",
  "version": "v1",
  "published": "2023-01-12T17:26:24.000Z",
  "updated": "2023-01-12T17:26:24.000Z",
  "title": "Approximation, regularity and positivity preservation on Riemannian manifolds",
  "authors": [
    "Stefano Pigola",
    "Daniele Valtorta",
    "Giona Veronelli"
  ],
  "comment": "29 pages. This paper and its companion [Guneysu,Pigola,Stollmann,Veronelli, Regularity of subharmonic distributions on local spaces, and a conjecture by Braverman, Milatovic, Shubin, Preprint 2022] supersede the unpublished preprint arXiv:2105.14847 by two of the authors",
  "categories": [
    "math.AP",
    "math.DG"
  ],
  "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. Its relevance, since the seminal works by T. Kato, arises from the spectral theory of Schr\\\"odinger operators with singular potentials. First, we prove 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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-01-12T17:26:24.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "58J05",
      "47B25",
      "35J10"
    ],
    "keywords": [
      "riemannian manifold",
      "subharmonic distributions",
      "uniform minkowski-type upper estimate",
      "hausdorff co-dimension strictly larger",
      "positivity preservation property"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 29,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}