A variation norm Carleson theorem

TL;DR

This paper enhances the Carleson-Hunt theorem by establishing $L^p$ bounds for the $r$-variation of Fourier partial sums, with implications for nonlinear Fourier analysis and ergodic theory.

Contribution

It introduces new $L^p$ estimates for the $r$-variation of Fourier partial sum operators, strengthening the classical theorem.

Findings

Proves $L^p$ bounds for $r$-variation of Fourier sums for $p > ext{max}ig brace r', 2 ig brace$.

Includes applications to nonlinear Fourier transforms.

Provides transference and variation norm Menshov-Paley-Zygmund theorems.

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 $p>\max\{r',2\}$. Four appendices are concerned with transference, a variation norm Menshov-Paley-Zygmund theorem, and applications to nonlinear Fourier transforms and ergodic theory.