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

Abstract

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