Riesz transform and Lp-cohomology for manifolds with Euclidean ends

Let M be a smooth Riemannian manifold that is the union of a compact part and a finite number of Euclidean ends, R-n\B(0, R) for some R > 0, each of which carries the standard metric. Our main result is that the Riesz transform oil M is bounded from L-p(M) -> L-p(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 L-p for some p > 2 for a more general class of manifolds. Assume that M is all n-dimensional complete manifold satisfying the Nash inequality and with an O(r(n)) upper bound oil 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 neap from L-2- to L-p-cohomology in this degree is injective.