A variation norm Carleson theorem 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.