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arXiv · 1707.08667

Improved $\ell^p$-Boundedness for Integral $k$-Spherical Maximal Functions

Theresa C. Anderson·Brian Cook·Kevin Hughes·Angel Kumchev
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Abstract

We improve the range of $\ell^p(\mathbb Z^d)$-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 cubicly with the degree $k$. 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.

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Theresa C. Anderson, Brian Cook, Kevin Hughes, Angel Kumchev. 2017-07-26. Improved $\ell^p$-Boundedness for Integral $k$-Spherical Maximal Functions. https://doi.org/10.19086/da.3675

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