Summary: In this paper we establish the following estimate:
\[ \begin{split} \omega (\{ x \in \mathbb{R}^n:| [b,T]f(x)| > \lambda \}) \\ \leq \frac{c_T}{\varepsilon^2}\int_{\mathbb{R}^n} \Phi \left( \| b \|_{BMO}\frac{| f(x)|}{\lambda} \right)M_{L( \log L)^{1 + \varepsilon}} \omega (x)dx \end{split} \]
where \(\omega \geq 0\), \(0 < \varepsilon < 1\) and \(\Phi(t)=t(1 + \log^{+}(t))\). This inequality relies upon the following sharp \(L^p\) estimate:
\[ \| [b,T]f\|_{L^p(\omega)} \leq c_T(p')^2p^2\left ( \frac{p - 1}{\delta} \right)^{\frac{1}{p'}}\| b \|_{BMO}\| f \|_{L^p\left (M_{L( \log_L)^{2p - 1 + \delta^{\omega}}}\right )} \]
where \(1<p< \infty\), \(\omega \geq 0\) and \(0 < \delta < 1\). As a consequence we recover the following estimate essentially contained in [C. Ortiz-Caraballo, Indiana Univ. Math. J. 60, No. 6, 2107–2130 (2011; Zbl 1261.42023)]:
\[ \begin{split} \omega (\{ x \in \mathbb{R}^n:| [b,T]f(x)| > \lambda \}) \\ \leq c_T[ \omega ]_{A_\infty}(1 + \log^+[ \omega ]_{A_{\infty}})^2\int_{\mathbb{R}^n} \Phi \left( \| b \|_{BMO}\frac{| f(x)|}{\lambda} \right)M \omega (x)dx.\end{split} \]
We also obtain the analogue estimates for symbol-multilinear commutators for a wider class of symbols.