Duality of Hardy and BMO spaces associated with operators with heat kernel bounds
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- by Xuan Thinh Duong and Lixin Yan
- J. Amer. Math. Soc. 18 (2005), 943-973
- DOI: https://doi.org/10.1090/S0894-0347-05-00496-0
- Published electronically: July 12, 2005
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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 $\textrm {BMO}_L$ 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 $\textrm {BMO}_{L^{\ast }}$ where $L^{\ast }$ is the adjoint operator of $L$. We then obtain a characterization of the space $\textrm {BMO}_L$ 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 $\textrm {BMO}_L$ spaces associated with operators.References
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Bibliographic Information
- Xuan Thinh Duong
- Affiliation: Department of Mathematics, Macquarie University, NSW 2109, Australia
- MR Author ID: 271083
- Email: duong@ics.mq.edu.au
- Lixin Yan
- Affiliation: Department of Mathematics, Macquarie University, NSW 2109, Australia and Department of Mathematics, Zhongshan University, Guangzhou, 510275, People’s Republic of China
- MR Author ID: 618148
- Email: lixin@ics.mq.edu.au, mcsylx@zsu.edu.cn
- 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
- © Copyright 2005
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: J. Amer. Math. Soc. 18 (2005), 943-973
- MSC (2000): Primary 42B30, 42B35, 47F05
- DOI: https://doi.org/10.1090/S0894-0347-05-00496-0
- MathSciNet review: 2163867