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Hardy spaces associated to non-negative self-adjoint operators satisfying Davies-Gaffney estimates
About this Title
Steve Hofmann, Department of Mathematics, University of Missouri, Columbia, Missouri 65211, Guozhen Lu, Department of Mathematics, Wayne State University, Detroit, Michigan 48202, Dorina Mitrea, Department of Mathematics, University of Missouri, Columbia, Missouri 65211, Marius Mitrea, Department of Mathematics, University of Missouri, Columbia, Missouri 65211 and Lixin Yan, Department of Mathematics, Zhongshan University, Guangzhou, 510275, People’s Republic of China
Publication: Memoirs of the American Mathematical Society
Publication Year:
2011; Volume 214, Number 1007
ISBNs: 978-0-8218-5238-5 (print); 978-1-4704-0624-0 (online)
DOI: https://doi.org/10.1090/S0065-9266-2011-00624-6
Published electronically: March 22, 2011
Keywords: Hardy space,
non-negative self-adjoint operator,
Davies-Gaffney condition,
atom,
molecule,
BMO,
Schrödinger operators,
space of homogeneous type.
MSC: Primary 42B20, 42B25; Secondary 46B70, 47G30
Table of Contents
Chapters
- 1. Introduction
- 2. Notation and preliminaries
- 3. Davies-Gaffney estimates
- 4. The decomposition into atoms
- 5. Relations between atoms and molecules
- 6. $\textrm {BMO}_{L,M}(X)$: Duality with Hardy spaces
- 7. Hardy spaces and Gaussian estimates
- 8. Hardy spaces associated to Schrödinger operators
- 9. Further properties of Hardy spaces associated to operators
Abstract
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 we 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, we establish additional characterizations of such Hardy spaces in terms of maximal functions. Finally, we 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.- Pascal Auscher, On necessary and sufficient conditions for $L^p$-estimates of Riesz transforms associated to elliptic operators on $\Bbb R^n$ and related estimates, Mem. Amer. Math. Soc. 186 (2007), no. 871, xviii+75. MR 2292385, DOI 10.1090/memo/0871
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