L^p-parabolicity of Riemannian manifolds

Abstract

We describe the L^p-parabolicity of a Riemannian manifold in terms of a unified theory of nonlinear L^p-capacities, for the whole range 1≤ p≤ ∞. For example, the L¹-parabolicity is equivalent to the usual notion of parabolicity/recurrence, while L²-parabolicity is equivalent to biparabolicity. For any 1≤ p≤ ∞, the L^p-parabolicity turns out to be equivalent to the L^q-Liouville property for positive superharmonic functions, where p and q are Hölder conjugate exponents. We also provide a new capacitary characterization of the L¹-Liouville property. Finally we obtain an almost optimal volume growth condition for the L^p-parabolicity for 1<p≤ 2 as well as a sharp volume condition for all 1<p<∞ for manifolds of non-negative Ricci curvature and for model manifolds.

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BibTeX

@misc{grigoryan2026lpparabolicity,
  title = {{$L^p$-parabolicity of Riemannian manifolds}},
  author = {Alexander Grigor'yan and Leandro F. Pessoa and Alberto G. Setti},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.27302},
  eprint = {2610.27302},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.27302},
}

RIS

TY  - UNPB
TI  - L^p-parabolicity of Riemannian manifolds
AU  - Alexander Grigor'yan
AU  - Leandro F. Pessoa
AU  - Alberto G. Setti
PY  - 2026
DA  - 2026/10/19
PB  - arXiv
DO  - 10.48550/arxiv.2610.27302
UR  - https://arxiv.org/abs/2610.27302
AB  - We describe the L^p-parabolicity of a Riemannian manifold in terms of a unified theory of nonlinear L^p-capacities, for the whole range 1≤ p≤ ∞. For example, the L^1-parabolicity is equivalent to the usual notion of parabolicity/recurrence, while L^2-parabolicity is equivalent to biparabolicity. For any 1≤ p≤ ∞, the L^p-parabolicity turns out to be equivalent to the L^q-Liouville property for positive superharmonic functions, where p and q are Hölder conjugate exponents. We also provide a new capacitary characterization of the L^1-Liouville property. Finally we obtain an almost optimal volume growth condition for the L^p-parabolicity for 1<p≤ 2 as well as a sharp volume condition for all 1<p<∞ for manifolds of non-negative Ricci curvature and for model manifolds.
ER  -

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