Improved $\ell^p$-Boundedness for Integral $k$-Spherical Maximal Functions
Classical Analysis and ODEs
2018-05-31 v2 Number Theory
Abstract
We improve the range of -boundedness of the integral -spherical maximal functions introduced by Magyar. The previously best known bounds for the full -spherical maximal function require the dimension to grow at least cubicly with the degree . Combining ideas from our prior work with recent advances in the theory of Weyl sums by Bourgain, Demeter, and Guth and by Wooley, we reduce this cubic bound to a quadratic one. As an application, we deduce improved bounds in the ergodic Waring--Goldbach problem.
Cite
@article{arxiv.1707.08667,
title = {Improved $\ell^p$-Boundedness for Integral $k$-Spherical Maximal Functions},
author = {Theresa C. Anderson and Brian Cook and Kevin Hughes and Angel Kumchev},
journal= {arXiv preprint arXiv:1707.08667},
year = {2018}
}
Comments
18 pages. Published in Discrete Analysis Journal on 29 May 2018