Mixed inequalities for commutators with multilinear symbol

Abstract

We prove mixed inequalities for commutators of Calderón-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 $b_i\in {\mathrm{Osc}}_{\exp L^{r_i}}$, with $r_i\ge 1$. If $u\in A_1$ and $v\in A_{\infty }(u)$ we prove that the inequality \[uv\left(\left\{x\in \mathbb{R}^n: \frac{|T_\mathbf{b}(fv)(x)|}{v(x)}>t\right\}\right)\leq C\int_{\mathbb{R}^n}Φ\left(\|\mathbf{b}\|\frac{|f(x)|}{t}\right)u(x)v(x)\,dx\] 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.

30 pages