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PreprintCC BY 4.0arXiv

The Coulhon--Duong conjecture for the Riesz transform on complete Riemannian manifolds

  • Rui Chen
  • Renjin Jiang
  • Bo Li
  • Hong-Quan Li
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Abstract

Let M be a complete, non-compact Riemannian manifold. We prove that its Riesz transform is of weak type (1,1), with constant $2$ for real-valued functions. Consequently, it is bounded on Lp(M) for 1<p≤2, with constants depending only on p, which proves the Coulhon--Duong conjecture. The proof uses an obstacle decomposition for positive self-adjoint operators with sub-Markovian semigroups. Applying this decomposition to the shifted square-root Laplacian and using locality of the Sobolev differential yields the endpoint estimate without geometric or heat kernel assumptions. We also obtain the corresponding result for Dirichlet spaces admitting a local Hilbertian differential calculus.

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Cite

BibTeXRIS
BibTeX
@misc{chen2026coulhonduong,
  title = {{The Coulhon--Duong conjecture for the Riesz transform on complete Riemannian manifolds}},
  author = {Rui Chen and Renjin Jiang and Bo Li and Hong-Quan Li},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.23503},
  eprint = {2609.23503},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.23503},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - The Coulhon--Duong conjecture for the Riesz transform on complete Riemannian manifolds
AU  - Rui Chen
AU  - Renjin Jiang
AU  - Bo Li
AU  - Hong-Quan Li
PY  - 2026
DA  - 2026/09/20
PB  - arXiv
DO  - 10.48550/arxiv.2609.23503
UR  - https://arxiv.org/abs/2609.23503
AB  - Let M be a complete, non-compact Riemannian manifold. We prove that its Riesz transform is of weak type (1,1), with constant $2$ for real-valued functions. Consequently, it is bounded on L^p(M) for 1<p≤2, with constants depending only on p, which proves the Coulhon--Duong conjecture. The proof uses an obstacle decomposition for positive self-adjoint operators with sub-Markovian semigroups. Applying this decomposition to the shifted square-root Laplacian and using locality of the Sobolev differential yields the endpoint estimate without geometric or heat kernel assumptions. We also obtain the corresponding result for Dirichlet spaces admitting a local Hilbertian differential calculus.
ER  -