In this paper we establish the following estimate:
$$\omega \left( {\left\{ {x \in {\mathbb{R}^n}:\left| {\left {b,T} \right]f\left( x \right)} \right| > \lambda } \right\}} \right) \leqslant \frac{{{c_T}}}{{{\varepsilon ^2}}}\int_{{\mathbb{R}^n}} {\Phi \left( {{{\left\| b \right\|}_{BMO}}\frac{{\left| {f\left( x \right)} \right|}}{\lambda }} \right){M_{L{{\left( {\log L} \right)}^{1 + \varepsilon }}}}} \omega \left( x \right)dx$$
where
ω ≥ 0, 0 <
ε < 1 and Φ(
t) =
t(1 + log
+(
t)). This inequality relies upon the following sharp
L p estimate:
$${\left\| {\left {b,T} \right]f} \right\|_{{L^p}\left( \omega \right)}} \leqslant {c_T}{\left( {p'} \right)^2}{p^2}{\left( {\frac{{p - 1}}{\delta }} \right)^{\frac{1}{{p'}}}}{\left\| b \right\|_{BMO}}{\left\| f \right\|_{{L^p}\left( {{M_{L{{\left( {{{\log }_L}} \right)}^{2p - 1 + {\delta ^\omega }}}}}} \right)}}$$
where 1 <
p < ∞,
ω ≥ 0 and 0 <
δ < 1. As a consequence we recover the following estimate essentially contained in 18]:
$$\omega \left( {\left\{ {x \in {\mathbb{R}^n}:\left| {\left {b,T} \right]f\left( x \right)} \right| > \lambda } \right\}} \right) \leqslant {c_T}{\left \omega \right]_{{A_\infty }}}{\left( {1 + {{\log }^ + }{{\left \omega \right]}_{{A_\infty }}}} \right)^2}\int_{{\mathbb{R}^n}} {\Phi \left( {{{\left\| b \right\|}_{BMO}}\frac{{\left| {f\left( x \right)} \right|}}{\lambda }} \right)M} \omega \left( x \right)dx.$$
We also obtain the analogue estimates for symbol-multilinear commutators for a wider class of symbols.