Improved ℓp-Boundedness for Integral k-Spherical Maximal Functions

Kevin Hughes

doi:10.19086/da.3675

Improved ℓp-Boundedness for Integral k-Spherical Maximal Functions

Kevin Hughes

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Abstract

We improve the range of ℓp(Zd)-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.

Original language	English
Article number	10.19086/da.3675
Pages (from-to)	1-30
Number of pages	30
Journal	Discrete Analysis
DOIs	https://doi.org/10.19086/da.3675
Publication status	Published - 29 May 2018

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AB - We improve the range of ℓp(Zd)-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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Improved ℓp-Boundedness for Integral k-Spherical Maximal Functions

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