Riesz transform and $L^p$ cohomology for manifolds with Euclidean ends
| dc.creator | Carron, Gilles | |
| dc.creator | Coulhon, Thierry | |
| dc.creator | Hassell, Andrew | |
| dc.date | 2004-11-30 | |
| dc.date.accessioned | 2026-07-07T05:14:48Z | |
| dc.date.available | 2026-07-07T05:14:48Z | |
| dc.description | Let $M$ be a smooth Riemannian manifold which is the union of a compact part and a finite number of Euclidean ends, $\RR^n \setminus B(0,R)$ for some $R > 0$, each of which carries the standard metric. Our main result is that the Riesz transform on $M$ is bounded from $L^p(M) \to L^p(M; T^*M)$ for $1 < p < n$ and unbounded for $p \geq n$ if there is more than one end. It follows from known results that in such a case the Riesz transform on $M$ is bounded for $1 < p \leq 2$ and unbounded for $p > n$; the result is new for $2 < p \leq n$. We also give some heat kernel estimates on such manifolds. We then consider the implications of boundedness of the Riesz transform in $L^p$ for some $p > 2$ for a more general class of manifolds. Assume that $M$ is a $n$-dimensional complete manifold satisfying the Nash inequality and with an $O(r^n)$ upper bound on the volume growth of geodesic balls. We show that boundedness of the Riesz transform on $L^p$ for some $p > 2$ implies a Hodge-de Rham interpretation of the $L^p$ cohomology in degree 1, and that the map from $L^2$ to $L^p$ cohomology in this degree is injective. | |
| dc.identifier | https://arxiv.org/abs/math/0411648 | |
| dc.identifier | http://arxiv.org/abs/math/0411648 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/73419 | |
| dc.subject | Analysis of PDEs | |
| dc.subject | Differential Geometry | |
| dc.subject | Functional Analysis | |
| dc.subject | 58J37, 58J35, 42B20 | |
| dc.title | Riesz transform and $L^p$ cohomology for manifolds with Euclidean ends | |
| dc.type | text |