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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