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.
Highlights
Our interest lies in proving p(Zd)-bounds for the integral k-spherical maximal functions when k ≥ 3
These maximal functions are defined in terms of their associated averages, which we describe
This proposition suffices to establish the p-boundedness of the dyadic maximal functions AΛ, but falls just short of what is needed for an quick proof of Theorem 1
Summary
Our interest lies in proving p(Zd)-bounds for the integral k-spherical maximal functions when k ≥ 3. It is instructive to compare d0∗(k) to the known bounds for the function G(k) in the theory of Waring’s problem (defined as the least value of d for which the asymptotic formula (1.2) holds). In [7], this sort of improvement - which used the recent resolution of the Vinogradov mean values theorems [18, 3] - was limited to maximal functions over sufficiently sparse sequences. We follow the paradigm in [11] and strengthen the connection to Waring’s problem as initiated in [7] by using a lemma from [1]; we use recent work on Waring’s problem to obtain improved bounds.
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