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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 \({\mathbf {b}}=(b_1,b_2,\ldots , b_m)\) 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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Correspondence to Fabio Berra.

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The authors were supported by CONICET, UNL and ANPCyT.

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Berra, F., Carena, M. & Pradolini, G. Mixed inequalities for commutators with multilinear symbol. Collect. Math. (2022). https://doi.org/10.1007/s13348-022-00367-4

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  • DOI: https://doi.org/10.1007/s13348-022-00367-4

Keywords

  • Multilinear symbol
  • Commutators
  • Young functions
  • Muckenhoupt weights

Mathematics Subject Classification

  • 42B20
  • 42B25