, Volume 46, Issue 2, pp 315-336

A Wiener–Wintner theorem for the Hilbert transform

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Abstract

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 fL p (X,μ), there is a set X f X of probability one, so that for all xX 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.