J. Aust. Math. Soc.  80 (2006), 229-262
Some new Besov and Triebel-Lizorkin spaces associated with para-accretive functions on spaces of homogeneous type

Dongguo Deng
  Department of Mathematics
  Zhongshan University
  Guangzhou 510275
  People's Republic of China
  stsdd@zsu.edu.cn
and
Dachun Yang
  School of Mathematical Sciences
  Beijing Normal University
  Beijing 100875
  People's Republic of China
  dcyang@bnu.edu.cn


Abstract
Let $(X,\rho,\mu)_{d,\theta}$ be a space of homogeneous type with $d>0$ and $\theta\in (0,1]$, $b$ be a para-accretive function, $\epsilon\in (0,\theta]$, $|s|<\epsilon$, and $a_0\in (0,1)$ be some constant depending on $d$, $\epsilon$ and $s$. The authors introduce the Besov space $b\dot B^s_{pq}(X)$ with $a_0<p\le\infty$ and $0<q\le\infty$, and the Triebel-Lizorkin space $b\dot F^s_{pq}(X)$ with $a_0<p<\infty$ and $a_0<q\le\infty$ by first establishing a Plancherel-Polya-type inequality. Moreover, the authors establish the frame and the Littlewood-Paley function characterizations of these spaces. Furthermore, the authors introduce the new Besov space $b^{-1}\dot B^s_{pq}(X)$ and the Triebel-Lizorkin space $b^{-1}\dot F^s_{pq}(X)$. The relations among these spaces and the known Hardy-type spaces are presented. As applications, the authors also establish some real interpolation theorems, embedding theorems, $Tb$ theorems, and the lifting property by introducing some new Riesz operators of these spaces.
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