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, 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 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