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Single annulus estimates for the variation-norm Hilbert transforms along Lipschitz vector fields

Author: Shaoming Guo
Journal: Proc. Amer. Math. Soc. 145 (2017), 601-615
MSC (2010): Primary 42B20, 42B25
Published electronically: October 24, 2016
MathSciNet review: 3577864
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Abstract: Let $ v$ be a planar Lipschitz vector field. We prove that the $ r$-th variation-norm Hilbert transform along $ v$ (defined as in (1.8)), composed with a standard Littlewood-Paley projection operator $ P_k$, is bounded from $ L^2$ to $ L^{2, \infty }$, and from $ L^p$ to itself for all $ p>2$. Here $ r>2$ and the operator norm is independent of $ k\in \mathbb{Z}$. This generalises Lacey and Li's result (2006) for the case of the Hilbert transform. However, their result only assumes measurability for vector fields. In contrast to that, we need to assume vector fields to be Lipschitz.

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Shaoming Guo
Affiliation: Institute of Mathematics, University of Bonn, Endenicher Allee 60, 53115, Bonn, Germany
Address at time of publication: 831 E. Third Street, Bloomington, Indiana 47405

Received by editor(s): November 13, 2015
Published electronically: October 24, 2016
Communicated by: Alexander Iosevich
Article copyright: © Copyright 2016 American Mathematical Society