L-P-BOUNDS FOR SPHERICAL MAXIMAL OPERATORS ON Z(N)

We prove analogue statements of the spherical maximal theorem of E. M. Stein, for the lattice points Z(n). We decompose the discrete spheric al ''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 opera tors. The proof is then based on L-2-estimates by analysing the Fourie r transforms <(s)over cap (t,epsilon)>(xi), which can be handled by ma king 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.