  In this paper, the authors characterize, in terms of pointwise inequalities,
the classical Besov spaces $\dot B^s_{p,\,q}$ and Triebel-Lizorkin spaces $\dot
F^s_{p,\,q}$ for all $s\in(0,\,1)$ and $p,\,q\in(n/(n+s),\,\infty],$ both in
${\mathbb R}^n$ and in the metric measure spaces enjoying the doubling and
reverse doubling properties. Applying this characterization, the authors prove
that quasiconformal mappings preserve $\dot F^s_{n/s,\,q}$ on $\rn$ for all
$s\in(0,\,1)$ and $q\in(n/(n+s),\,\infty]$. A metric measure space version of
the above morphism property is also established.
