A differentiation theorem for uniform measures

Publication date: 15 August 2013

Abstract: Using the notion of higher-order Fourier dimension introduced in cite{M2} (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 cite{stein} for spheres in

R^{d g e q 3}

and Bourgain for circles in

R^{2}

, first considered for more general spaces in cite{rubio}, and shown to hold for some singular subsets of the reals for the first time in cite{LabaDiff}. Notably, unlike the more delicate of the previous results on differentiation such as cite{Bourgain} and cite{LabaDiff}, 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.

Mathematics Subject Classification ID

Maximal functions, Littlewood-Paley theory (42B25) Differentiation (real functions of one variable): general theory, generalized derivatives, mean value theorems (26A24) Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type (42A38) Multipliers in one variable harmonic analysis (42A45) Arithmetic progressions (11B25) Hausdorff and packing measures (28A78) Functions of one variable (26A99) Abstract differentiation theory, differentiation of set functions (28A15) Trigonometric series of special types (positive coefficients, monotonic coefficients, etc.) (42A32)

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