\(L^p\)-bounds for spherical maximal operators on \(\mathbb Z^n\)
type: Item
instance of: scholarly article
title: \(L^p\)-bounds for spherical maximal operators on \(\mathbb Z^n\)
zbMATH Open document ID: 0893.42011
DOI: 10.4171/RMI/222
author: Ákos Magyar
published in: Revista Matemática Iberoamericana
publication date: 1998-04-01T00:00:00Z
full work available at URL: https://eudml.org/doc/39542
full work available at URL: Magyar.pdf
review text: Summary: We prove analogue statements of the spherical maximal theorem of E. M. Stein, for the lattice points \(\mathbb Z^n\). We decompose the discrete spherical ``measures'' as an integral of Gaussian kernels \(s_{t,\varepsilon}(x)= e^{2\pi i| x|^2(t+ i\varepsilon)}\). By using Minkowski's integral inequality it is enough to prove \(L^p\)-bounds for the corresponding convolution operators. The proof is then based on \(L^2\)-estimates by analyzing the Fourier transforms \(\hat s_{t,\varepsilon}(\xi)\), which can be handled by making use of the ``circle'' method for exponential sums. As a corollary one obtains some regularity of the distribution of lattice points on small spherical caps.
Mathematics Subject Classification ID: 42B25
Mathematics Subject Classification ID: 11P21
Mathematics Subject Classification ID: 11L07
zbMATH DE Number: 1136584
zbMATH Keywords: spherical maximal operators
zbMATH Keywords: \(L^p\)-bounds
zbMATH Keywords: Fourier transforms
zbMATH Keywords: distribution of lattice points on small spherical caps
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1136584