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Duality of Hardy and BMO spaces associated with operators with heat kernel bounds

Authors: Xuan Thinh Duong and Lixin Yan
Journal: J. Amer. Math. Soc. 18 (2005), 943-973
MSC (2000): Primary 42B30, 42B35, 47F05
Published electronically: July 12, 2005
MathSciNet review: 2163867
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Abstract: Let $L$ be the infinitesimal generator of an analytic semigroup on $L^2({\mathbb R}^n)$ with suitable upper bounds on its heat kernels. Auscher, Duong, and McIntosh defined a Hardy space $H_L^1$ by means of an area integral function associated with the operator $L$. By using a variant of the maximal function associated with the semigroup $\{e^{-tL}\}_{t\geq 0}$, a space ${\rm BMO}_L$ of functions of BMO type was defined by Duong and Yan and it generalizes the classical BMO space. In this paper, we show that if $L$ has a bounded holomorphic functional calculus on $L^2({\mathbb R}^n)$, then the dual space of $H_L^1 $ is ${\rm BMO}_{L^{\ast}} $ where $L^{\ast}$ is the adjoint operator of $L$. We then obtain a characterization of the space ${\rm BMO}_L $ in terms of the Carleson measure. We also discuss the dimensions of the kernel spaces ${\mathcal K}_L$ of BMO$_{ L} $ when $L$ is a second-order elliptic operator of divergence form and when $L$ is a Schrödinger operator, and study the inclusion between the classical BMO space and ${\rm BMO}_L$ spaces associated with operators.

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Additional Information

Xuan Thinh Duong
Affiliation: Department of Mathematics, Macquarie University, NSW 2109, Australia

Lixin Yan
Affiliation: Department of Mathematics, Macquarie University, NSW 2109, Australia and Department of Mathematics, Zhongshan University, Guangzhou, 510275, People’s Republic of China

Keywords: Hardy space, BMO, semigroup, holomorphic functional calculi, tent space, Carleson measure, second-order elliptic operator, Schr\"odinger operator.
Received by editor(s): August 3, 2004
Published electronically: July 12, 2005
Additional Notes: Both authors are supported by a grant from the Australia Research Council. The second author is also supported by NNSF of China (Grant No. 10371134) and the Foundation of Advanced Research Center, Zhongshan University
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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