Riesz transform and $L^p$ cohomology for manifolds with Euclidean ends

Details Riesz transform and $L^p$ cohomology for manifolds with Euclidean ends Riesz transform and $L^p$ cohomology for manifolds with Euclidean ends

2004-11-30

arxiv.org/abs/math/0411648v1

Mathematics

Analysis of PDEs

Scientific paper

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.

Affiliated with

Also associated with

No associations

Rating

Riesz transform and $L^p$ cohomology for manifolds with Euclidean ends does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Riesz transform and $L^p$ cohomology for manifolds with Euclidean ends, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Riesz transform and $L^p$ cohomology for manifolds with Euclidean ends will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-14149