2019/09/06 by Rui Han, Ben Krause, Han, Rui +5 · 1 citation
Mathematics · #Analytic Number Theory Research #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Limits and Structures in Graph Theory #Mathematical Approximation and Integration
paper · pdf · doi:10.48550/arxiv.1909.02883
openalex publication_date 2019/09/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
Consider averages along the prime integers \mathbb P given by AN f (x) = N -1 ∑ p ∈ \mathbb P : p≤ N (log p) f (x-p). These averages satisfy a uniform scale-free ℓ p-improving estimate. For all 1< p < 2, there is a constant Cp so that for all integer N and functions f supported on [0,N], there holds N -1/p' ‖ AN f‖ℓp' ≤ Cp N - 1/p ‖ f‖ℓp. The maximal function A∗ f =supN | AN f | satisfies (p,p) sparse bounds for all 1< p < 2. The latter are the natural variants of the scale-free bounds. As a corollary, A∗ is bounded on ℓ p (w), for all weights w in the Muckenhoupt Ap class. No prior weighted inequalities for A∗ were known.