A Wiener–Wintner theorem for the Hilbert transform

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 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.