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arXiv · 2609.23503

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 $L^p(M)$ for $1<p\leq2$, 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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Rui Chen, Renjin Jiang, Bo Li, Hong-Quan Li. 2026-09-20. The Coulhon--Duong conjecture for the Riesz transform on complete Riemannian manifolds. https://arxiv.org/abs/2609.23503

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