Local Riesz Transform and Local Hardy Spaces on Riemannian Manifolds with Bounded Geometry

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