L-bounds for spherical maximal operators on Z.
We prove analogue statements of the spherical maximal theorem of E. M. Stein, for the lattice points Z. We decompose the discrete spherical measures as an integral of Gaussian kernels s(x) = e. By using Minkowski's integral inequality it is enough to prove L-bounds for the corresponding convolution operators. The proof is then based on L-estimates by analysing the Fourier transforms ^s(ξ), which can be handled by making use of the circle method for exponential sums. As a corollary one obtains some...