Abstract. Let M be a smooth Riemannian manifold which 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 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 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.

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