-bounds for spherical maximal operators on

  • Akos Magyar

    California Institute of Technology, Pasadena, USA


We prove analogue statements of the spherical maximal theorem of E.M. Stein for the lattice points . We decompose the discrete spherical measures as an integral of Gaussian kernels . By using Minkowski's integral inequality it is enough to prove -bounds for the corresponding convolution operators. The proof is then based on 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.

Cite this article

Akos Magyar, -bounds for spherical maximal operators on . Rev. Mat. Iberoam. 13 (1997), no. 2, pp. 307–317

DOI 10.4171/RMI/222