Multilinear Marcinkiewicz-Zygmund inequalities

TL;DR

This paper extends classical Marcinkiewicz-Zygmund inequalities to the multilinear setting, providing bounds for multilinear operators and applications to weighted vector-valued inequalities for Calderón-Zygmund operators.

Contribution

It introduces multilinear Marcinkiewicz-Zygmund inequalities and computes optimal constants in some cases, advancing the understanding of multilinear operator bounds.

Findings

Established multilinear inequalities with explicit bounds

Calculated best constants for certain cases

Applied results to weighted inequalities for Calderón-Zygmund operators

Abstract

We extend to the multilinear setting classical inequalities of Marcinkiewicz and Zygmund on $\ell^r$-valued extensions of linear operators. We show that for certain $1 \leq p, q_1, \dots, q_m, r \leq \infty$, there is a constant $C\geq 0$ such that for every bounded multilinear operator $T\colon L^{q_1}(\mu_1) \times \cdots \times L^{q_m}(\mu_m) \to L^p(\nu)$ and functions $\{f_{k_1}^1\}_{k_1=1}^{n_1} \subset L^{q_1}(\mu_1), \dots, \{f_{k_m}^m\}_{k_m=1}^{n_m} \subset L^{q_m}(\mu_m)$, the following inequality holds \begin{equation}\label{MZ ineq abstract} (1) \quad \quad \left\Vert \left(\sum_{k_1, \dots, k_m} |T(f_{k_1}^1, \dots, f_{k_m}^m)|^r\right)^{1/r} \right\Vert_{L^p(\nu)} \leq C \|T\| \prod_{i=1}^m \left\| \left(\sum_{k_i=1}^{n_i} |f_{k_i}^i|^r\right)^{1/r} \right\|_{L^{q_i}(\mu_i)}. \end{equation} In some cases we also calculate the best constant $C\geq 0$ satisfying the previous inequality. We apply these results to obtain weighted vector-valued inequalities for multilinear Calder\'on-Zygmund operators.