Some new inhomogeneous Triebel–Lizorkin spaces on metric measure spaces and their various characterizations

Dachun Yang¹

¹ Department of Mathematics Beijing Normal University Beijing 100875 People's Republic of China

Studia Mathematica, Tome 167 (2005) no. 1, pp. 63-98

Résumé

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 $F^0_{\infty2}(X)$.

Détail

DOI : 10.4064/sm167-1-5

Keywords: varrho theta space homogeneous type set varrho quasi metric property there constants theta varrho varrho varrho theta varrho varrho theta nonnegative borel regular measure varrho sim varepsilon theta varepsilon max varepsilon varepsilon infty author introduces inhomogeneous triebel lizorkin spaces infty establishes their frame characterizations first establishing plancherel lya type inequality related norm cdot infty which completes theory function spaces spaces homogeneous type moreover author establishes connection between space infty homogeneous triebel lizorkin space dot infty particular proves mathop bmo nolimits coincides infty