Coulhon–Duong conjecture on the Riesz transform for complete manifolds

Let $M$ be a complete Riemannian manifold, and let $\mathcal{L}$ be its non-negative Laplace–Beltrami operator. The Riesz transform is the operator $\nabla\mathcal{L}^{-1/2}$, initially defined on suitable functions.

Coulhon–Duong conjecture. The Riesz transform is bounded on $L^p(M)$ for every $1<p<2$.

This conjecture asks whether the $L^p$ boundedness known under volume doubling and Gaussian heat-kernel upper bounds holds on every complete manifold. The cited context establishes the result under additional geometric and analytic assumptions, while the unrestricted statement remains open in the source.