Mixed inequalities for commutators with multilinear symbol

2021/08/20 by Fabio Berra, Berra, Fabio, Marilina Carena +3 · 1 citation
Mathematics · #Advanced Harmonic Analysis Research #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Nonlinear Partial Differential Equations #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.2108.09202

openalex publication_date 2021/08/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We prove mixed inequalities for commutators of Calderón-Zygmund operators (CZO) with multilinear symbols. Concretely, let m∈ℕ and b=(b₁,b₂,…, b_m) be a vectorial symbol such that each component b_i∈ Osc_{exp L^ri}, with r_i≥ 1. If u∈ A₁ and v∈ A_∞(u) we prove that the inequality uv(\x∈ ℝⁿ: (|T_b(fv)(x)|)/(v(x))gt;t\)≤ C∫_ℝⁿΦ(‖b‖(|f(x)|)/(t))u(x)v(x) dx holds for every t>0, where Φ(t)=t(1+log⁺t)^r, with 1/r=∑_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

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