Dachun Yang (Beijing Normal University)
Besov and Triebel-Lizorkin spaces on spaces of homogeneous type
Abstract.
Let $(X,\rho,\mu)_{d,\theta}$ be a space of homogeneous type which includes metric measure spaces and some fractals, namely, $X$ is a set, $\rho$ is a quasi-metric on $X$ satisfying that there are constants $\theta\in (0,1]$ and $C_0>0$ such that for all $x,\ x',\ y\in X$, $$|\rho(x,y)-\rho(x',y)|\le C_0\rho(x,x')^\theta[\rho(x,y) +\rho(x',y)]^{1-\theta},$$ and $\mu$ is a nonnegative Borel regular measure on $X$ satisfying that for some $d>0$ and all $x\in X$, $$\mu(\{y\in X:\ r>\rho(x,y)\})\sim r^d.$$ We introduce the fractional integrals and derivatives, and prove that ${B^s_{pq}(X)}$ and ${F^s_{pq}(X)}$ have the lifting properties for $\theta>|s|$. We establish the frame decompositions of ${B^s_{pq}(X)}$ and ${F^s_{pq}(X)}$. Applying these frame characterizations, we obtain the estimates of entropy numbers of compact embeddings between ${B^s_{pq}(X)}$ or ${F^s_{pq}(X)}$ when $\infty>\mu(X)$. Parts of these results are new even when $(X,\ \rho,\ \mu)_{d,\theta}$ is just the $n$-dimensional Euclidean space, or a compact $d$-set, $\Gamma$, in ${\mathbb R}^n$, which includes various kinds of fractals. We also introduce new spaces of Lipschitz type and show that these spaces are just the Besov spaces or Triebel-Lizorkin spaces when the smooth index is less than $\theta$. Moreover, we clarify the relationships amongst these Lipschitz-type spaces, Haj\l asz-Sobolev spaces, Korevaar-Schoen-Sobolev spaces, Newtonian Sobolev space and the Cheeger-Sobolev spaces on metric-measure spaces, showing that they are the same space with equivalence of norms. Furthermore, we prove a Sobolev embedding theorem, namely that the Lipschitz-type spaces with large orders of smoothness can be embedded in Lipschitz spaces. For metric-measure spaces with heat kernel, we establish a Hardy-Littlewood-Sobolev theorem, and hence prove that Lipschitz-type spaces with small orders of smoothness can be embedded in Lebesgue spaces.

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