  We establish weighted inequalities for $BMO$ commutators of sublinear
operators for all $0<p<\infty$.
  For weights $w$ satisfying the doubling condition of order $q$ with $0<q<p$
and the reverse H\"{o}lder condition, we prove that
  $\bullet$ commutators $T_b$, which are bounded on $L^p$ with $1<p<\infty$,
are bounded from some subspaces of $L^p_w$ to $L^p_w$ and to themselves for all
$0<p<\infty$, these are applied to the commutators of singular integral
operators and Hardy-Littelwood maximal operator, et.al, which are known to fail
to be bounded from $H^1$ to $L^1$ and whose estimate has been open problems for
$0<p$ enough small;
  $\bullet$ commutators $T_b$, whose associated operators $T$ are bounded on
$L^p$ with $1<p<\infty$, are bounded from some subspaces of $L^p_w$ to $L^p_w$
and from some subspaces of $L^p_w$ to others for all $0<p<\infty$, these are
applied to the commutators of maximal operators such as singular integral
maximal operators, Carleson operator and the polynomial Carleson operator,
et.al, the estimate of these commutators has been open problems for each
$0<p<\infty$;
  $\bullet$ in particular, these imply that the commutators above are bounded
from $H^p_w$ to $L^p_w$ and to itself for all $0<p\leq 1$.
