Dimension-Free Estimates for the Discrete Spherical Maximal Functions

Article Mathematics

Some remarks on dimension-free estimates for the discrete Hardy-Littlewood maximal functions

Dariusz Kosz et al.

Summary: This paper studies the dependencies of optimal constants in strong and weak type bounds for maximal functions over convex symmetric bodies on Double-struck capital R-d and DOUBLE-STRUCK CAPITAL Z(d). Firstly, it is shown that these optimal constants in L-p (Double-struck capital R-d) are not larger than their discrete analogues in l(p)(DOUBLE-STRUCK CAPITAL Z(d)), and equality holds for cubes when p = 1. This implies that the best constant in the weak type (1, 1) inequality for the discrete Hardy-Littlewood maximal function associated with centered cubes in DOUBLE-STRUCK CAPITAL Z(d) grows to infinity as the dimension d approaches infinity. Secondly, dimension-free estimates are proved for the l(p)(DOUBLE-STRUCK CAPITAL Z(d)) norms of the discrete Hardy-Littlewood maximal operators with restricted ranges of scales. Finally, the results are extended to the l(2)(DOUBLE-STRUCK CAPITAL Z(d)) case for maximal operators restricted to dyadic scales.

ISRAEL JOURNAL OF MATHEMATICS (2023)

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Article Mathematics

BOUNDS FOR LACUNARY MAXIMAL FUNCTIONS GIVEN BY BIRCH-MAGYAR AVERAGES

Brian Cook et al.

Summary: Positive and negative results are obtained for lacunary discrete maximal operators defined by dilations of sufficiently nonsingular hypersurfaces arising from Diophantine equations in many variables. The negative results demonstrate substantial differences from those of lacunary discrete maximal operators defined along a nonsingular hypersurface. The positive results involve improvements over bounds for the corresponding full maximal functions initially studied by Magyar.

TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY (2021)

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Article Mathematics, Applied

THE IONESCU-WAINGER MULTIPLIER THEOREM AND THE ADELES

Terence Tao

Summary: The Ionescu-Wainger multiplier theorem provides important tools for discrete harmonic analysis, with a simplified proof and explicit constants given. A related adelic sampling theorem is also established, showing that functions with Fourier transform supported on major arcs have comparable norms in different spaces.

MATHEMATIKA (2021)

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Article Mathematics