Borderline weighted estimates for commutators of singular integrals

Abstract

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

31 pages. Final version, accepted for publication in Israel J. Math

References in corpus (3)

• Sharp Reverse Hölder property for A_\infty weights on spaces of homogeneous type
• Borderline Weak Type Estimates for Singular Integrals and Square Functions
• A Bellman function counterexample to the $A_1$ conjecture: the blow-up of the weak norm estimates of weighted singular operators

Cited by in corpus (4)

• A class of multilinear bounded oscillation operators on measure spaces and applications
• Three observations on commutators of Singular Integral Operators with BMO functions
• Weighted estimates for bilinear fractional integral operators and their commutators
• Sparse dominations and weighted variation inequalities for singular integrals and commutators