2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/73264One considers the class of complete non-compact Riemannian manifolds whose heat kernel satisfies Gaussian estimates from above and below. One shows that the Riesz transform is $L^p$ bounded on such a manifold, for $p$ ranging in an open interval above 2, if and only if the gradient of the heat kernel satisfies a certain $L^p$ estimate in the same interval of $p$'s.to appear in Annales de l'Ecole Normale Superieure de ParisAnalysis of PDEsClassical Analysis and ODEsDifferential Geometry58J35, 42B20text