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Fourier transforms of indicator functions, lattice point discrepancy, and the stability of integrals. (English) Zbl 1471.42021

Summary: We prove sharp estimates for Fourier transforms of indicator functions of bounded open sets in \({\mathbb{R}}^n\) with real analytic boundary, as well as nontrivial lattice point discrepancy results. Both are derived from estimates on Fourier transforms of hypersurface measures. Relations with maximal averages are discussed, connecting two conjectures of A. Iosevich and E. Sawyer [Adv. Math. 132, No. 1, 46–119 (1997; Zbl 0921.42015)]. We also prove a theorem concerning the stability under function perturbations of the growth rate of a real analytic function near a zero. This result is sharp in an appropriate sense. It implies a corresponding stability result for the local integrablity of negative powers of a real analytic function near a zero.

MSC:

42B10 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
42B25 Maximal functions, Littlewood-Paley theory
26E05 Real-analytic functions
11P21 Lattice points in specified regions
11K38 Irregularities of distribution, discrepancy

Citations:

Zbl 0921.42015
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References:

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