  $BMO$ commutators of some sublinear operators such as singular integral
operators and Hardy-Littelwood maximal operator are well known to be bounded
from $L_w^p$ to itself for all $1<p<\infty$ and all $w\in A_p$ (the classical
Muckenhopt class), but for these commutators, it has been an open question how
to extend the estimate to all $0<p<\infty$. In addition, there are many
classical operators, especially some maximal operators such as Carleson maximal
operator, for which the $L^p$-boundedness holds for $1<p<\infty$, but for the
$BMO$ commutators of these operators, it has also been an open question whether
there are similar unweighted or weighted estimates for each $0<p<\infty$.
  In this paper, giving a weights class $A_p^+$ with $0<p\leq \infty$ which is
an extension of $A_p$ when $1\leq p\leq \infty$ and is a refinement of $A_1$
when $0< p\leq 1$, for $BMO$ commutators of some sublinear operators, we obtain
weighted estimates from some subspaces of $L^p_w$ to $L^p_w$ (and to
themselves) for all $0<p<\infty$ and $w\in A_q^+$ with $0<q<p$. These results
are applied the commutators in above two questions. In particular, these imply
that the $BMO$ commutators of many classical operators including those
mentioned above are bounded from $H^p_w$ to $L^p_w$ and from $H^p_w$ to itself
for all $0<p\leq 1$ and $w\in A_q^+$ with $0<q<p$.
