Resumen de Mixed inequalities for commutators with multilinear symbol

Fabio Berra, Marilina Carena, Gladis Pradolini

• 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.