Mixed inequalities for commutators with multilinear symbol

TL;DR

This paper establishes mixed inequalities for commutators of Calderón-Zygmund operators with multilinear symbols, extending to less regular kernels and deriving boundedness results in weighted Lebesgue spaces.

Contribution

It introduces new mixed inequalities for commutators with multilinear symbols and extends results to operators with less regular kernels, including boundedness in weighted spaces.

Findings

Proved mixed inequalities for commutators with multilinear symbols.

Extended results to convolution-type operators with less regular kernels.

Established $L^p(w)$-boundedness for these operators with weights.

Abstract

We prove mixed inequalities for commutators of Calder\'on-Zygmund operators (CZO) with multilinear symbols. Concretely, let $m\in\mathbb{N}$ and $\mathbf{b}=(b_1,b_2,\dots, b_m)$ be a vectorial symbol such that each component $b_i\in \mathrm{Osc}_{\mathrm{exp}\, L^{r_i}}$, with $r_i\geq 1$. If $u\in A_1$ and $v\in A_\infty(u)$ we prove that the inequality \[uv\left(\left\{x\in \mathbb{R}^n: \frac{|T_\mathbf{b}(fv)(x)|}{v(x)}>t\right\}\right)\leq C\int_{\mathbb{R}^n}\Phi\left(\|\mathbf{b}\|\frac{|f(x)|}{t}\right)u(x)v(x)\,dx\] holds for every $t>0$, where $\Phi(t)=t(1+\log^+t)^r$, with $1/r=\sum_{i=1}^m 1/r_i$. We also consider operators of convolution type with kernels satisfying less regularity properties than CZO. In this setting, we give a Coifman type inequality for the associated commutators with multilinear symbol. This result allows us to deduce the $L^p(w)$-boundedness of these operators when $1<p<\infty$ and $w\in A_p$. As a consequence, we can obtain the desired mixed inequality in this context.