Resumen de Lp bounds for spherical maximal operators on Zn

Akos Magyar

• We prove analogue statements of the spherical maximal theorem of E. M. Stein, for the lattice points Zn. We decompose the discrete spherical measures as an integral of Gaussian kernels st,e(x) = e2pi|x|2(t + ie). By using Minkowski's integral inequality it is enough to prove Lp-bounds for the corresponding convolution operators. The proof is then based on L2-estimates by analysing the Fourier transforms ^st,e(?), 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.