We prove estimates, , on the Hilbert transform along a one variable vector field acting on functions with frequency support in an annulus. Estimates when were proved by Lacey and Li. This paper also contains key technical ingredients for a companion paper with Christoph Thiele in which 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.
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Michael Bateman, Single annulus estimates for Hilbert transforms along vector fields. Rev. Mat. Iberoam. 29 (2013), no. 3, pp. 1021–1069DOI 10.4171/RMI/748