A Wiener–Wintner theorem for the Hilbert transform

Michael Lacey School of Mathematics, Georgia Institute of Technology Erin Terwilleger Department of Mathematics, U-3009, University of Connecticut

TBD mathscidoc:1701.333136

Arkiv for Matematik, 46, (2), 315-336, 2005.12
We prove the following extension of the Wiener–Wintner theorem and the Carleson theorem on pointwise convergence of Fourier series: For all measure-preserving flows ($X$,μ,$T$_{$t$}) and$f$∈$L$^{$p$}($X$,μ), there is a set$X$_{$f$}⊂$X$of probability one, so that for all$x$∈$X$_{$f$}, $$\lim_{s\downarrow0}\int_{s<|t|<1/s}e^{i\theta t} f(\textup{T}_tx)\,\frac{dt}t\quad\text{exists for all}\ \theta.$$ The proof is by way of establishing an appropriate oscillation inequality which is itself an extension of Carleson’s theorem.
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@inproceedings{michael2005a,
  title={A Wiener–Wintner theorem for the Hilbert transform},
  author={Michael Lacey, and Erin Terwilleger},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203623551380945},
  booktitle={Arkiv for Matematik},
  volume={46},
  number={2},
  pages={315-336},
  year={2005},
}
Michael Lacey, and Erin Terwilleger. A Wiener–Wintner theorem for the Hilbert transform. 2005. Vol. 46. In Arkiv for Matematik. pp.315-336. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203623551380945.
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