{
  "id": "1507.02550",
  "version": "v1",
  "published": "2015-07-08T14:57:05.000Z",
  "updated": "2015-07-08T14:57:05.000Z",
  "title": "Sharp Poincaré-Hardy and Poincaré-Rellich inequalities on the hyperbolic space",
  "authors": [
    "Elvise Berchio",
    "Debdip Ganguly",
    "Gabriele Grillo"
  ],
  "categories": [
    "math.CA",
    "math.AP"
  ],
  "abstract": "We study Hardy-type inequalities associated to the quadratic form of the shifted Laplacian $-\\Delta_{\\mathbb H^N}-(N-1)^2/4$ on the hyperbolic space ${\\mathbb H}^N$, $(N-1)^2/4$ being, as it is well-known, the bottom of the $L^2$-spectrum of $-\\Delta_{\\mathbb H^N}$. We find the optimal constant in the resulting Poincar\\'e-Hardy inequality, which includes a further remainder term which makes it sharp also locally. A related inequality under suitable curvature assumption on more general manifolds is also shown. Similarly, we prove Rellich-type inequalities associated with the shifted Laplacian, in which at least one of the constant involved is again sharp.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-07-08T14:57:05.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "hyperbolic space",
      "poincaré-rellich inequalities",
      "sharp poincaré-hardy",
      "study hardy-type inequalities",
      "shifted laplacian"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150702550B"
    }
  }
}