Vector-valued Hilbert transforms along curves

Guixiang Hong; Honghai Liu

doi:10.1215/17358787-3589397

April 2016 Vector-valued Hilbert transforms along curves

Guixiang Hong, Honghai Liu

Banach J. Math. Anal. 10(2): 430-450 (April 2016). DOI: 10.1215/17358787-3589397

Abstract

In this paper, we show that Hilbert transforms along some curves are bounded on $L^{p}({\mathbb{R}}^{n};X)$ for some $1\lt p\lt \infty$ and some UMD spaces $X$ . In particular, we prove that Hilbert transforms along some curves are completely $L^{p}$ -bounded in the terminology from operator space theory. Moreover, we obtain the $L^{p}(\mathbb{R}^{n};X)$ -boundedness of anisotropic singular integrals by using the “method of rotations” of Calderón–Zygmund. All these results extend preexisting related ones.

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Guixiang Hong and Honghai Liu "Vector-valued Hilbert transforms along curves," Banach Journal of Mathematical Analysis 10(2), 430-450, (April 2016). https://doi.org/10.1215/17358787-3589397

Received: 9 April 2015; Accepted: 27 July 2015; Published: April 2016

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