# Revista Matemática Iberoamericana

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**Volume 13, Issue 2, 1997, pp. 307–317**

**DOI: 10.4171/RMI/222**

$L^p$-bounds for spherical maximal operators on $\mathbb Z^n$

Akos Magyar^{[1]}(1) Mathematcis, Sloan Lab. 253-37, California Institute of Technology, 1200 E. California Blvd., CA 91125-8100, PASADENA, UNITED STATES

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 $s_{t, \epsilon}(x) = e^{2\pi i|x|^2(t+i\epsilon)}$. 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 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.

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Magyar A. $L^p$-bounds for spherical maximal operators on $\mathbb Z^n$. *Rev. Mat. Iberoamericana* 13 (1997), 307-317. doi: 10.4171/RMI/222