Hardy spaces on Riemannian manifolds with quadratic curvature decay

TL;DR

This paper proves that Hardy spaces of exact 1-forms on certain Riemannian manifolds with quadratic curvature decay and specific volume growth conditions coincide with a particular Lp-closure, establishing an optimal p-range.

Contribution

It establishes the equivalence of Hardy spaces and Lp-closures for exact 1-forms on manifolds with quadratic curvature decay, extending previous results to a broader class of manifolds.

Findings

Hardy spaces coincide with Lp-closures for 1-forms on the manifolds studied.

The p-range for the equivalence is proven to be optimal.

Results apply to manifolds with finite Euclidean ends.

Abstract

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 in [4], coincide with the closure in L p of R(d) $\cap$ L p ($\Lambda$ 1 T * M) when 1 < p < $\nu$, where $\nu$ > 2 is related to the volume growth. The range of p is optimal. This result applies, in particular, when M has a finite number of Euclidean ends.