Local Riesz transform and local Hardy spaces on Riemannian manifolds with bounded geometry

We prove that if $τ$ is a large positive number, then the atomic Goldberg-type space $\mathfrak{h}^1(N)$ and the space $\mathfrak{h}_{\mathcal R_τ}^1(N)$ of all integrable functions on $N$ whose local Riesz transform $\mathcal R_τ$ is integrable are the same space on any complete noncompact Riemannian manifold $N$ with Ricci curvature bounded from below and positive injectivity radius. We also relate $\mathfrak{h}^1(N)$ to a space of harmonic functions on the slice $N\times (0,δ)$ for $δ>0$ small enough.

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