Borderline weighted estimates for commutators of singular integrals

TL;DR

This paper establishes new weighted estimates for commutators of singular integrals, providing sharp inequalities and extending results to multilinear commutators with broader symbol classes.

Contribution

It introduces novel weighted bounds for commutators of singular integrals, including sharp $L^p$ estimates and inequalities involving Orlicz and maximal functions, extending to multilinear cases.

Findings

Established a new weighted weak-type estimate involving Orlicz and maximal functions.

Derived sharp $L^p$ bounds for commutators with explicit dependence on $p$ and $A_ ext{infinity}$ constants.

Extended estimates to multilinear commutators with wider symbol classes.

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.