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

Shunchao Long

Published 2021-03-16Version 1

$BMO$ 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 $0<q<p$.