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

Abstract

We prove that if \(\tau \) is a large positive number, then the atomic Goldberg-type space \({\mathfrak {h}}^1(N)\) and the space \({\mathfrak {h}}_{{\mathscr {R}}_\tau }^1(N)\) of all integrable functions on N of which local Riesz transform \({\mathscr {R}}_\tau \) 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 \(N\times (0,\delta )\) for \(\delta >0\) small enough.