Let \(X\) be a metric space with doubling measure, and \(L\) be a non-negative, self-adjoint operator satisfying Davies-Gaffney bounds on \(L^2(X)\). In this article the authors present a theory of Hardy and BMO spaces associated to \(L\), including an atomic (or molecular) decomposition, square function characterization, and duality of Hardy and BMO spaces. Further specializing to the case that \(L\) is a Schrödinger operator on \(\mathbb{R}^n\) with a non-negative, locally integrable potential, the authors establish additional characterizations of such Hardy spaces in terms of maximal functions. Finally, they define Hardy spaces \(H^p_L(X)\) for \(p>1\), which may or may not coincide with the space \(L^p(X)\), and show that they interpolate with \(H^1_L(X)\) spaces by the complex method.

Table of Contents

• Introduction
• Notation and preliminaries
• Davies-Gaffney estimates
• The decomposition into atoms
• Relations between atoms and molecules
• \(\mathrm{BMO}_{L,M}(X)\): Duality with Hardy spaces
• Hardy spaces and Gaussian estimates
• Hardy spaces associated to Schrödinger operators
• Further properties of Hardy spaces associated to operators
• Bibliography