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Besov spaces with non-doubling measures


Authors: Donggao Deng, Yongsheng Han and Dachun Yang
Journal: Trans. Amer. Math. Soc. 358 (2006), 2965-3001
MSC (2000): Primary 42B35; Secondary 46E35, 42B25, 47B06, 46B10, 43A99
DOI: https://doi.org/10.1090/S0002-9947-05-03787-6
Published electronically: June 10, 2005
MathSciNet review: 2216255
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Abstract: Suppose that $\mu $ is a Radon measure on ${\mathbb R}^d,$ which may be non-doubling. The only condition on $\mu$ is the growth condition, namely, there is a constant $C_0>0$ such that for all $x\in {\rm {\,supp\,}}(\mu)$ and $r>0,$

\begin{displaymath}\mu (B(x, r))\le C_0r^n,\end{displaymath}

where $0<n\leq d.$ In this paper, the authors establish a theory of Besov spaces $\dot B^s_{pq}(\mu)$ for $1\le p, q\le\infty$ and $\vert s\vert<\theta$, where $\theta>0$ is a real number which depends on the non-doubling measure $\mu$, $C_0$, $n$ and $d$. The method used to define these spaces is new even for the classical case. As applications, the lifting properties of these spaces by using the Riesz potential operators and the dual spaces are obtained.


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Additional Information

Donggao Deng
Affiliation: Department of Mathematics, Zhongshan University, Guangzhou 510275, People’s Republic of China
Email: stsdd@zsu.edu.cn

Yongsheng Han
Affiliation: Department of Mathematics, Auburn University, Alabama 36849-5310
Email: hanyong@mail.auburn.edu

Dachun Yang
Affiliation: School of Mathematical Sciences, Beijing Normal University, Beijing 100875, People’s Republic of China
Email: dcyang@bnu.edu.cn

DOI: https://doi.org/10.1090/S0002-9947-05-03787-6
Keywords: Non-doubling measure, Besov space, Calder\'on-type reproducing formula, approximation to the identity, Riesz potential, lifting property, dual space
Received by editor(s): June 17, 2003
Received by editor(s) in revised form: May 16, 2004
Published electronically: June 10, 2005
Additional Notes: The first author’s research was supported by NNSF (No. 10171111) of China
The second author’s research was supported by NNSF (No. 10271015) of China
The third (corresponding) author’s research was supported by NNSF (No. 10271015) and RFDP (No. 20020027004) of China
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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