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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.79.

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Duality of Hardy and BMO spaces associated with operators with heat kernel bounds
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by Xuan Thinh Duong and Lixin Yan HTML | PDF
J. Amer. Math. Soc. 18 (2005), 943-973 Request permission

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.
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Additional 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