We study the Lp boundedness of the Riesz transform as well as the reverse inequality on Riemannian manifolds and graphs under the volume doubling property and a sub-Gaussian heat kernel upper bound. We prove that the Riesz transform is then bounded on Lp for 1 < p< 2 , which shows that Gaussian estimates of the heat kernel are not a necessary condition for this. In the particular case of Vicsek manifolds and graphs, we show that the reverse inequality does not hold for 1 < p< 2. This yields a full picture of the ranges of p∈ (1 , + ∞) for which respectively the Riesz transform is Lp-bounded and the reverse inequality holds on Lp on such manifolds and graphs. This picture is strikingly different from the Euclidean one.
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- Heat kernel
- Riemannian manifolds
- Riesz transforms
- Sub-Gaussian estimates