Lp-bounds for spherical maximal operators on Zn.

Akos Magyar

Revista Matemática Iberoamericana (1997)

  • Volume: 13, Issue: 2, page 307-317
  • ISSN: 0213-2230

Abstract

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We prove analogue statements of the spherical maximal theorem of E. M. Stein, for the lattice points Zn. We decompose the discrete spherical measures as an integral of Gaussian kernels st,ε(x) = e2πi|x|2(t + iε). By using Minkowski's integral inequality it is enough to prove Lp-bounds for the corresponding convolution operators. The proof is then based on L2-estimates by analysing the Fourier transforms ^st,ε(ξ), 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.

How to cite

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Magyar, Akos. "Lp-bounds for spherical maximal operators on Zn.." Revista Matemática Iberoamericana 13.2 (1997): 307-317. <http://eudml.org/doc/39542>.

@article{Magyar1997,
abstract = {We prove analogue statements of the spherical maximal theorem of E. M. Stein, for the lattice points Zn. We decompose the discrete spherical measures as an integral of Gaussian kernels st,ε(x) = e2πi|x|2(t + iε). By using Minkowski's integral inequality it is enough to prove Lp-bounds for the corresponding convolution operators. The proof is then based on L2-estimates by analysing the Fourier transforms ^st,ε(ξ), 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.},
author = {Magyar, Akos},
journal = {Revista Matemática Iberoamericana},
keywords = {Operadores integrales; Operadores maximales; Espacios LP; Integrales singulares; spherical maximal operators; -bounds; Fourier transforms; distribution of lattice points on small spherical caps},
language = {eng},
number = {2},
pages = {307-317},
title = {Lp-bounds for spherical maximal operators on Zn.},
url = {http://eudml.org/doc/39542},
volume = {13},
year = {1997},
}

TY - JOUR
AU - Magyar, Akos
TI - Lp-bounds for spherical maximal operators on Zn.
JO - Revista Matemática Iberoamericana
PY - 1997
VL - 13
IS - 2
SP - 307
EP - 317
AB - We prove analogue statements of the spherical maximal theorem of E. M. Stein, for the lattice points Zn. We decompose the discrete spherical measures as an integral of Gaussian kernels st,ε(x) = e2πi|x|2(t + iε). By using Minkowski's integral inequality it is enough to prove Lp-bounds for the corresponding convolution operators. The proof is then based on L2-estimates by analysing the Fourier transforms ^st,ε(ξ), 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.
LA - eng
KW - Operadores integrales; Operadores maximales; Espacios LP; Integrales singulares; spherical maximal operators; -bounds; Fourier transforms; distribution of lattice points on small spherical caps
UR - http://eudml.org/doc/39542
ER -

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