Riesz transform and L^p-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, 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 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 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 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 L²- to L^p-cohomology in this degree is injective.