Abstract

We prove mixed inequalities for commutators of Calderón–Zygmund operators (CZO) with multilinear symbols. Concretely, let $$m\in {\mathbb {N}}$$ and $${\mathbf {b}}=(b_1,b_2,\ldots , b_m)$$ be a vectorial symbol such that each component $$b_i\in \mathrm {Osc}_{\mathrm {exp}\, L^{r_i}}$$ , with $$r_i\ge 1$$ . If $$u\in A_1$$ and $$v\in A_\infty (u)$$ we prove that the inequality $$\begin{aligned} uv\left( \left\{ x\in {\mathbb {R}}^n: \frac{|T_{\mathbf {b}}(fv)(x)|}{v(x)}>t\right\} \right) \le C\int _{{\mathbb {R}}^n}\Phi \left( \Vert {\mathbf {b}}\Vert \frac{|f(x)|}{t}\right) u(x)v(x)\,dx \end{aligned}$$ 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.

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