Uniform estimates for paraproducts and related multilinear multipliers. (English) Zbl 1183.42012

Bernicot, Frédéric

Rev. Mat. Iberoam. 25, No. 3, 1055-1088 (2009).

Let \(\mathcal{S}(\mathbb{R}^{dn})\) be the Schwartz space on \(\mathbb{R}^{dn}\). A \(n\)-linear multiplier \(T\) is given by its symbol \(\sigma\in \mathcal{S}(\mathbb{R}^{dn})\), with the formula: \[ T(f_1,\dots, f_n)(x):=\int_{\mathbb{R}^{dn}} e^{ix.(\xi_1+\dots +\xi_n)}\sigma(\xi)\prod_{i=1}^{n}\widehat{f_i}(\xi_i)d\xi. \] In the paper under review some continuous extendedness results for the operator \(T\) are proved. The author uses the ideas of the Calderón-Zygmund theory with a few improvements.

Reviewer: Vitaly Vladimirovich Volchkov (Donetsk)

MSC:

42B15	Multipliers for harmonic analysis in several variables
42B20	Singular and oscillatory integrals (Calderón-Zygmund, etc.)
42B25	Maximal functions, Littlewood-Paley theory

Keywords:

paraproducts; uniform estimate; multilinear operators; Littlewood-Paley theory; Calderón-Zygmund decomposition

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