Hardy Spaces and bmo on Manifolds with Bounded Geometry

Abstract

We develop the theory of the “local” Hardy space $\mathfrak{h}^{1}(M)$ and John-Nirenberg space $\mathop{\mathrm{bmo}}(M)$ when M is a Riemannian manifold with bounded geometry, building on the classic work of Fefferman-Stein and subsequent material, particularly of Goldberg and Ionescu. Results include $\mathfrak{h}^{1}$ – $\mathop{\mathrm{bmo}}$ duality, L ^p estimates on an appropriate variant of the sharp maximal function, $\mathfrak{h}^{1}$ and bmo-Sobolev spaces, and action of a natural class of pseudodifferential operators, including a natural class of functions of the Laplace operator, in a setting that unifies these results with results on L ^p-Sobolev spaces. We apply results on these topics to some interpolation theorems, motivated in part by the search for dispersive estimates for wave equations.