A Discrete Carleson Theorem Along the Primes with a Restricted Supremum

2016/04/29 by Cladek, Laura, Henriot, Kevin, Krause, Ben +2
#Classical Analysis and ODEs (math.CA) #FOS: Mathematics

paper · doi:10.48550/arxiv.1604.08695

Abstract

Consider the discrete maximal function acting on finitely supported functions on the integers, C_Λf(n) := sup_{λ∈ Λ} | ∑_{p ∈ ± ℙ} f(n-p) log |p| \frace^{2πi λp}p |, where ± ℙ := \ ± p : p is a prime \, and Λ⊂ [0,1]. We give sufficient conditions on Λ, met by (finite unions of) lacunary sets, for this to be a bounded sublinear operator on ℓ^p(ℤ) for (3)/(2) < p < 4.

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