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