Riesz transform for $1 \leq p \le 2$ without Gaussian heat kernel bound

Li Chen (1) , Thierry Coulhon (2) , Joseph Feneuil (3) , Emmanuel Russ (4)

1 ICMAT - Instituto de Ciencias Matematicas
2 PSL - Université Paris Sciences et Lettres
3 Department of Mathematics [Minneapolis]
4 IF - Institut Fourier

We study the $L^p$ boundedness of 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 $L^p$ 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 $p\in (1,+\infty)$ for which respectively the Riesz transform is $L^p$ -bounded and the reverse inequality holds on $L^p$ on such manifolds and graphs. This picture is strikingly different from the Euclidean one.

Domaines: Analyse classique [math.CA]; Analyse fonctionnelle [math.FA]

Li Chen, Thierry Coulhon, Joseph Feneuil, Emmanuel Russ. Riesz transform for $1 \leq p \le 2$ without Gaussian heat kernel bound. Journal of Geometric Analysis, 2017, 27 (2), pp.1489-1514. ⟨hal-01221508⟩