# Hardy Spaces Associated to Non-Negative Self-Adjoint Operators Satisfying Davies-Gaffney Estimates

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*Steve Hofmann; Guozhen Lu; Dorina Mitrea; Marius Mitrea; Lixin Yan*

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

# Table of Contents

## Hardy Spaces Associated to Non-Negative Self-Adjoint Operators Satisfying Davies-Gaffney Estimates

- Chapter 1. Introduction 18 free
- Chapter 2. Notation and preliminaries 512 free
- Chapter 3. Davies-Gaffney estimates 1320
- Chapter 4. The decomposition into atoms 1724
- Chapter 5. Relations between atoms and molecules 3138
- Chapter 6. BMOL,M(X): Duality with Hardy spaces 4148
- Chapter 7. Hardy spaces and Gaussian estimates 4552
- Chapter 8. Hardy spaces associated to Schrödinger operators 5360
- Chapter 9. Further properties of Hardy spaces associated to operators 6572
- Bibliography 7582