JournalsrmiVol. 13, No. 2pp. 307–317

LpL^p-bounds for spherical maximal operators on Zn\mathbb Z^n

  • Akos Magyar

    California Institute of Technology, Pasadena, USA
$L^p$-bounds for spherical maximal operators on $\mathbb Z^n$ cover
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Abstract

We prove analogue statements of the spherical maximal theorem of E.M. Stein for the lattice points Zn\mathbb Z^n. We decompose the discrete spherical measures as an integral of Gaussian kernels st,ϵ(x)=e2πix2(t+iϵ)s_{t, \epsilon}(x) = e^{2\pi i|x|^2(t+i\epsilon)}. By using Minkowski's integral inequality it is enough to prove LpL^p-bounds for the corresponding convolution operators. The proof is then based on L2L^2 estimates by analysing the Fourier transforms $\hat{s}_{t,\epsilon}(\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.

Cite this article

Akos Magyar, LpL^p-bounds for spherical maximal operators on Zn\mathbb Z^n. Rev. Mat. Iberoam. 13 (1997), no. 2, pp. 307–317

DOI 10.4171/RMI/222