Borderline weighted estimates for commutators of singular integrals

In this paper we establish the following estimate $$w\left(\left\{ x\in\mathbb{R}^{n}\,:\,\left|[b,T]f(x)\right| > \lambda\right\} \right)\leq \frac{c_{T}}{\varepsilon^{2}}\int_{\mathbb{R}^{n}}\Phi\left(\|b\|_{BMO}\frac{|f(x)|}{\lambda}\right)M_{L(\log L)^{1+\varepsilon}}w(x)dx$$ where $w\geq0, \, 0<\varepsilon<1$ and $\Phi(t)=t(t+\log^+(t))$. This inequality relies upon the following sharp $L^p$ estimate $$\|[b,T]f\|_{L^{p}(w)}\leq c_{T}\left(p'\right)^{2}p^{2}\left(\frac{p-1}{\delta}\right)^{\frac{1}{p'}} \|b\|_{BMO} \, \|f \|_{L^{p}(M_{L(\log L)^{2p-1+\delta}}w)}$$where $1<p<\infty, w\geq0 \text{ and } 0<\delta<1.$ As a consequence we recover the following estimate $$w\left(\{x\in\mathbb{R}^{n}\,:\,\left|[b,T]f(x)\right| >\lambda\}\right)\leq c_T\,[w]_{A_{\infty}}\left(1+\log^{+}[w]_{A_{\infty}}\right)^{2}\int_{\mathbb{R}^{n}} \Phi\left(\|b\|_{BMO}\frac{|f(x)|}{\lambda}\right)Mw(x)dx$$ We also obtain the analogue estimates for symbol-multilinear commutators for a wider class of symbols.