Improved ℓp -Boundedness for Integral k -Spherical Maximal Functions
Copyright policy )Improved ℓp -Boundedness for Integral k -Spherical Maximal Functions
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.
18 pages. Published in Discrete Analysis Journal on 29 May 2018
- Towson University United States
- Towson University
- Towson University
- University of Bristol United Kingdom
- Department of Mathematics
11, 42, Mathematics - Number Theory, Maximal functions, Littlewood-Paley theory, Singular and oscillatory integrals (Calderón-Zygmund, etc.), 510, Mathematics - Classical Analysis and ODEs, QA1-939, Classical Analysis and ODEs (math.CA), FOS: Mathematics, Number Theory (math.NT), Mathematics
11, 42, Mathematics - Number Theory, Maximal functions, Littlewood-Paley theory, Singular and oscillatory integrals (Calderón-Zygmund, etc.), 510, Mathematics - Classical Analysis and ODEs, QA1-939, Classical Analysis and ODEs (math.CA), FOS: Mathematics, Number Theory (math.NT), Mathematics
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