A variation norm Carleson theorem in higher dimensions

2025/08/24 by Himali Dabhi, Dabhi, Himali
Mathematics · #37A46 #42B05 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #math.CA #msc:37A46 #msc:42B05

paper · pdf · doi:10.48550/arxiv.2508.17272

To appear in Colloquium Mathematicum, 18 pages

arxiv created 2026/07/30 · arxiv updated 2026/07/31

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 < ∞. 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 L²(\mathbb T^d), we establish r-variational and 2-oscillation bounds for partial Fourier sums over nested rectangles whenever r>2.

Citations

• Oscillation inequalities in ergodic theory and analysis: one-parameter and multi-parameter perspectives
• A variation norm Carleson theorem

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