HARDY SPACES ON RIEMANNIAN MANIFOLDS WITH QUADRATIC CURVATURE DECAY

Let (M, g) be a complete Riemannian manifold. Assume that the Ricci curvature of M has quadratic decay and that the volume growth is strictly faster than quadratic. We establish that the Hardy spaces of exact 1-differential forms on M, introduced by Auscher et al. (J. Geom. Anal. 18:1 (2008), 192-248), coincide with the closure in Lp of 7Z(d) boolean AND Lp(A1T*M) when v/(v - 1) < p < v, where v > 2 is related to the volume growth. Throughout, 7Z(d) denotes the range of d as an unbounded operator from L2 to L2(A1T*M). This result applies, in particular, when M has a finite number of Euclidean ends.

Automated summary

In this paper, we study the Hardy spaces on complete Riemannian manifolds and establish the equivalence between the Hardy spaces of exact 1-differential forms and the closure in Lp. This result holds when the Ricci curvature has quadratic decay and the volume growth is strictly faster than quadratic, and it applies particularly to manifolds with a finite number of Euclidean ends.