Wiener-Wintner for Hilbert Transform

Abstract

We prove the following extension of the Wiener--Wintner Theorem in Ergodic Theor and the Carleson Theorem on pointwise convergence of Fourier series: For all measure preserving flows $ (X,\mu , T_t)$ and $ f\in L^p (X,\mu)$, there is a set $X_f\subset X $ of probability one, so that for all $x\in X_f$ we have \begin{equation*} \lim _{s\downarrow0} \int _{s<\abs t<1/s} \operatorname e ^{i \theta t} f(\operatorname T_tx)\; \frac{dt}t \qquad \text{exists for all $\theta$.} \end{equation*} The proof is by way of establishing an appropriate oscillation inequality which is itself an extension of Carleson's theorem.