Riesz transform and Lp-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, Rn∖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 Lp(M)→Lp(M;T*M) for 1<p<n and unbounded for p≥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≤2 and unbounded for p>n; the result is new for 2<p≤n. We also give some heat kernel estimates on such manifolds. We then consider the implications of boundedness of the Riesz transform in Lp 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(rn) upper bound on the volume growth of geodesic balls. We show that boundedness of the Riesz transform on Lp for some p>2 implies a Hodge–de Rham interpretation of the Lp-cohomology in degree 1 and that the map from L2- to Lp-cohomology in this degree is injective