Let be a smooth Riemannian manifold that is the union of a compact part and a finite number of Euclidean ends, for some , each of which carries the standard metric. Our main result is that the Riesz transform on is bounded from for and unbounded for if there is more than one end. It follows from known results that in such a case, the Riesz transform on is bounded for and unbounded for ; the result is new for . We also give some heat kernel estimates on such manifolds.
We then consider the implications of boundedness of the Riesz transform in for some for a more general class of manifolds. Assume that is an -dimensional complete manifold satisfying the Nash inequality and with an upper bound on the volume growth of geodesic balls. We show that boundedness of the Riesz transform on for some implies a Hodge–de Rham interpretation of the -cohomology in degree and that the map from - to -cohomology in this degree is injective
"Riesz transform and -cohomology for manifolds with Euclidean ends." Duke Math. J. 133 (1) 59 - 93, 15 May 2006. https://doi.org/10.1215/S0012-7094-06-13313-6