Some new inhomogeneous Triebel–Lizorkin spaces on metric measure spaces and their various characterizations
Volume 167 / 2005
Abstract
Let $(X,\varrho,\mu)_{d,\theta}$ be a space of homogeneous type, i.e. $X$ is a set, $\varrho$ is a quasi-metric on $X$ with the property that there are constants $\theta\in (0,1]$ and $C_0>0$ such that for all $x, x', y\in X$, $$ |\varrho(x,y)-\varrho(x',y)|\le C_0\varrho(x,x')^\theta[\varrho(x,y) +\varrho(x',y)]^{1-\theta}, $$ and $\mu$ is a nonnegative Borel regular measure on $X$ such that for some $d>0$ and all $x\in X$, $$ \mu(\{y\in X: \varrho(x,y)< r\})\sim r^d. $$ Let $\varepsilon\in (0,\theta]$, $|s|< \varepsilon$ and $ \max\{d/(d+\varepsilon),d/(d+s+\varepsilon)\} < q \le \infty. $ The author introduces new inhomogeneous Triebel–Lizorkin spaces ${F^s_{\infty q}(X)}$ and establishes their frame characterizations by first establishing a Plancherel–Pólya-type inequality related to the norm $\|\cdot\|_{F^s_{\infty q}(X)}$, which completes the theory of function spaces on spaces of homogeneous type. Moreover, the author establishes the connection between the space ${F^s_{\infty q}(X)}$ and the homogeneous Triebel–Lizorkin space ${\dot F^s_{\infty q}(X)}$. In particular, he proves that $\mathop{\rm bmo}\nolimits(X)$ coincides with $F^0_{\infty2}(X)$.
