Weighted estimates of commutators for $0<p<\infty$

Abstract: 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 $0<p\leq 1$.