Borderline weighted estimates for commutators of singular integrals (1507.08568v2)

Abstract: 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.