Endpoint estimates for the maximal function over prime numbers

2019/07/10 by Trojan, Bartosz
#37A45 (Primary) 46E30 #42B25 (Secondary) #Classical Analysis and ODEs (math.CA) #Dynamical Systems (math.DS) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1907.04753

Abstract

Given an ergodic dynamical system (X, B, μ, T), we prove that for each function f belonging to the Orlicz space L(log L)²(log log L)(X, μ), the ergodic averages (1)/(π(N)) ∑_{p ∈ ℙN} f(T^p x), converge for μ-almost all x ∈ X, where ℙ_N is the set of prime numbers not larger that N and π(N) = # ℙ_N.

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