A variation norm Carleson theorem

Abstract

We strengthen the Carleson-Hunt theorem by proving Lp 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. © European Mathematical Society 2012.