Improved lp-Boundedness for Integral k-Spherical Maximal Functions

We improve the range of l(p)(Z(d))-boundedness of the integral k-spherical maximal functions introduced by Magyar. The previously best known bounds for the full k-spherical maximal function require the dimension d to grow at least cubically with the degree k. Combining ideas from our prior work with recent advances in the theory ofWeyl sums by Bourgain, Demeter, and Guth and by Wooley, we reduce this cubic bound to a quadratic one. As an application, we improve upon bounds in the authors' previous work [1] on the ergodic Waring-Goldbach problem, which is the analogous problem of l(p)(Z(d))-boundedness of the k-spherical maximal functions whose coordinates are restricted to prime values rather than integer values.