Mixed inequalities for commutators with multilinear symbol

Abstract

We prove mixed inequalities for commutators of Calder´on-Zygmund operators (CZO) with multilinear symbols. Concretely, let m ∈ N and b = (b1, b2, . . . , bm) be a vectorial symbol such that each component bi ∈ Oscexp Lri , with ri ≥ 1. If u ∈ A1 and v ∈ A∞(u) we prove that the inequality uv x ∈ R n : |Tb(fv)(x)| v(x) > t ≤ C Z Rn Φ kbk |f(x)| t u(x)v(x) dx holds for every t > 0, where Φ(t) = t(1 + log+ t) r , with 1/r = Pm i=1 1/ri. 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 < ∞ and w ∈ Ap. As a consequence, we can obtain the desired mixed inequality in this context.