Let
$\Omega $ be an open set
in a metric space
$H$,
$1\le {p}_{0},p\le q<\infty $,
$a,b,\gamma \in \mathbb{R}$,
$a\ge 0$. Suppose
$\sigma ,\mu ,w$
are Borel measures. Combining results from earlier work (2009) with those
obtained in work with Wheeden (2011) and with Rodney and Wheeden
(2013), we study embedding and compact embedding theorems of sets
$\mathfrak{S}\subset {L}_{\sigma ,loc}^{1}\left(\Omega \right)\times {L}_{w}^{p}\left(\Omega \right)$ to
${L}_{\mu}^{q}\left(\Omega \right)$ (projection to the first
component) where
$\mathfrak{S}$
(abstract Sobolev space) satisfies a Poincarétype inequality,
$\sigma $ satisfies certain weak doubling
property and
$\mu $ is absolutely
continuous with respect to
$\sigma $.
In particular, when
$H={\mathbb{R}}^{n}$,
$w,\mu ,\rho $ are weights so that
$\rho $ is essentially constant on
each ball deep inside in
$\Omega \setminus F$,
and
$F$ is
a finite collection of points and hyperplanes. With the help of a simple observation,
we apply our result to the study of embedding and compact embedding of
${L}_{{\rho}^{\gamma}}^{{p}_{0}}\left(\Omega \right)\cap {E}_{w{\rho}^{b}}^{p}\left(\Omega \right)$ and weighted fractional
Sobolev spaces to
${L}_{\mu {\rho}^{a}}^{q}\left(\Omega \right)$,
where
${E}_{w{\rho}^{b}}^{p}\left(\Omega \right)$
is the space of locally integrable functions in
$\Omega $ such that their weak
derivatives are in
${L}_{w{\rho}^{b}}^{p}\left(\Omega \right)$.
In
${\mathbb{R}}^{n}$,
our assumptions are mostly sharp. Besides extending numerous results in the
literature, we also extend a result of Bourgain et al. (2002) on cubes to John
domains.
