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Revista Matemática Iberoamericana


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Volume 29, Issue 3, 2013, pp. 1021–1069
DOI: 10.4171/RMI/748

Published online: 2013-08-04

Single annulus $L^p$ estimates for Hilbert transforms along vector fields

Michael Bateman[1]

(1) University of California Los Angeles, USA

We prove $L^p$ estimates, $p\in (1,\infty)$, on the Hilbert transform along a one variable vector field acting on functions with frequency support in an annulus. Estimates when $p>2$ were proved by Lacey and Li. This paper also contains key technical ingredients for a companion paper with Christoph Thiele in which $L^p$ estimates are established for the full Hilbert transform. The operators considered here are singular integral variants of maximal operators arising in the study of planar differentiation problems.

Keywords: Carleson’s theorem, time-frequency analysis, Stein’s conjecture, Zygmund’s conjecture, differentiation of vector fields, Hilbert transform in direction of a vector field

Bateman Michael: Single annulus $L^p$ estimates for Hilbert transforms along vector fields. Rev. Mat. Iberoam. 29 (2013), 1021-1069. doi: 10.4171/RMI/748