The Hörmander multiplier theorem for \(n\)-linear operators. (English) Zbl 1475.42020
Summary: In this paper, we study the Hörmander multiplier theorem for multilinear operators. We generalize the result of N. Tomita [J. Funct. Anal. 259, No. 8, 2028–2044 (2010; Zbl 1201.42005)] to wider target spaces and extend that of L. Grafakos and H. Van Nguyen [Monatsh. Math. 190, No. 4, 735–753 (2019; Zbl 1426.42012)] to multilinear operators. We indeed give two different proofs: The first proof is based on the results of L. Grafakos et al. [Can. J. Math. 65, No. 2, 299–330 (2013; Zbl 1275.42016); J. Math. Soc. Japan 69, No. 2, 529–562 (2017; Zbl 1372.42007)], L. Grafakos and H. van Nguyen [Colloq. Math. 144, No. 1, 1–30 (2016; Zbl 1339.42012); Monatsh. Math. 190, No. 4, 735–753 (2019; Zbl 1426.42012)], and A. Miyachi and N. Tomita [Rev. Mat. Iberoam. 29, No. 2, 495–530 (2013; Zbl 1275.42017)] and for the second one we provide a new and original approach, inspired by C. Muscalu et al. [Acta Math. 193, No. 2, 269–296 (2004; Zbl 1087.42016)]. We also give an application and discuss the sharpness of the result.
MSC:
| 42B15 | Multipliers for harmonic analysis in several variables |
| 42B20 | Singular and oscillatory integrals (Calderón-Zygmund, etc.) |
| 42B25 | Maximal functions, Littlewood-Paley theory |
Citations:
Zbl 1201.42005; Zbl 1426.42012; Zbl 1275.42016; Zbl 1372.42007; Zbl 1339.42012; Zbl 1275.42017; Zbl 1087.42016References:
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