A variation norm Carleson theorem in higher dimensions

TL;DR

This paper extends the Carleson-Hunt theorem by establishing variation norm bounds for polygonal Fourier partial sums in higher dimensions, combining Fefferman's approach with recent OSTTW results.

Contribution

It introduces higher-dimensional variation norm bounds for Fourier partial sums using Fefferman's method and OSTTW's $L^p$ estimates.

Findings

Established variation norm bounds for polygonal Fourier partial sums in higher dimensions.

Unified Fefferman's approach with OSTTW results to extend Carleson-Hunt theorem.

Provided a new perspective on convergence of Fourier series in higher dimensions.

Abstract

The celebrated Carleson-Hunt theorem gives pointwise almost everywhere convergence for the Fourier series of a function in $L^p(\mathbb T)$. R. Oberlin, A. Seeger, T. Tao, C. Thiele and J. Wright (OSTTW) strengthened this theorem by proving $L^p$ estimates for the $r$-variation of the partial sum operators for Fourier series. Also, C. Fefferman gave an extension of the theorem in higher dimensions by proving the maximal function bound for polygonal Fourier partial sums of functions in $L^p(\mathbb T^d)$. In this brief note, we observe that C. Fefferman's argument can be used, together with the OSTTW result, to establish variation norm bounds for the polygonal Fourier partial sums in higher dimensions.