\(L^p\)-bounds for spherical maximal operators on \(\mathbb Z^n\)
Summary: 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 \(s_{t,\varepsilon}(x)= e^{2\pi i| x|^2(t+ i\varepsilon)}\). 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 analyzing the Fourier transforms \(\hat s_{t,\varepsilon}(\xi)\), 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.
distribution of lattice points on small spherical caps
scientific article; zbMATH DE number 1136584
scientific article