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
Dachun Yang
  School of Mathematical Sciences
  Beijing Normal University
  Beijing 100875
  People's Republic of China

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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