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

Abstract

Let $M$ be a smooth Riemannian manifold that is the union of a compact part and a finite number of Euclidean ends, $\mathbb{R}^n {\setminus} B(0,R)$ 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 $1\lt p \lt n$ 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 $1\lt p\leq 2$ and unbounded for $p \gt n$; the result is new for $2\lt p\leq n$. 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 an $n$-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