A differentiation theorem for uniform measures

2013/08/15 by Marc Carnovale, Carnovale, Marc
Mathematics · #11B25 #26A24 #26A99 #28A15 #28A78 #42A32 #42A38 #42A45 #42B25 #Advanced Harmonic Analysis Research #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Limits and Structures in Graph Theory #Mathematical Dynamics and Fractals #math.CA #msc:11B25 #msc:26A24 #msc:26A99 #msc:28A15 #msc:28A78 #msc:42A32 #msc:42A38 #msc:42A45 #msc:42B25

paper · pdf · doi:10.48550/arxiv.1308.3479

arxiv created 2013/08/15 · openalex publication_date 2013/08/15 · arxiv updated 2013/08/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Using the notion of higher-order Fourier dimension introduced in \citeM2 (which was a sort of psuedorandomness condition stemming from the Gowers norms of Additive Combinatorics), we prove a maximal theorem and corresponding differentiation theorem for singular measures on \R^d, d=1,2,.... This extends results begun by Hardy and Littlewood for balls in \R^d and continued by Stein \citestein for spheres in \R^{d≥ 3} and Bourgain for circles in \R², first considered for more general spaces in \citerubio, and shown to hold for some singular subsets of the reals for the first time in \citeLabaDiff. Notably, unlike the more delicate of the previous results on differentiation such as \citeBourgain and \citeLabaDiff, the assumption of higher-order Fourier dimension subsumes all of the geometric or combinatorial input necessary for one to obtain our theorem, and suggests a new approach to some problems in Harmonic Analysis.

Related

• Higher-order Fourier dimension and frequency decompositions
• Singular sets for harmonic measure on locally flat domains with locally\n finite surface measure
• Harmonic measure is absolutely continuous with respect to the Hausdorff measure on all low-dimensional uniformly rectifiable sets
• Harmonic measure on sets of codimension larger than one
• Long progressions in sets of fractional dimension