A variation norm Carleson theorem in higher dimensions

Himali Dabhi

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arXiv · 2508.17272

A variation norm Carleson theorem in higher dimensions

Himali Dabhi

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Abstract

In 1971, C. Fefferman established a higher dimensional extension of the celebrated Carleson--Hunt theorem which gives pointwise almost everywhere convergence of the partial Fourier sums of functions in $L^p(\mathbb T), 1 < p < \infty.$ More precisely, Fefferman proved a maximal function bound for polygonal Fourier partial sums of functions in $L^p(\mathbb T^d), p>1.$ In this note, we extend Fefferman's maximal function bound to strong $r$-variation norm bounds whenever $r>2$ as well as uniform $2$-oscillation bounds. Furthermore, for functions in $L^2(\mathbb T^d)$, we establish $r$-variational and $2$-oscillation bounds for partial Fourier sums over nested rectangles whenever $r>2$.

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Himali Dabhi. 2025-08-24. A variation norm Carleson theorem in higher dimensions. https://arxiv.org/abs/2508.17272

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