A variation norm Carleson theorem

Abstract

We strengthen the Carleson-Hunt theorem by proving L^p estimates for the r -variation of the partial sum operators for Fourier series and integrals, for r>\max\{p',2\} . Four appendices are concerned with transference, a variation norm Menshov-Paley-Zygmund theorem, and applications to nonlinear Fourier transforms and ergodic theory.