On the Hardy--Littlewood Majorant Problem for Arithmetic Sets

TY - JOUR

T1 - On the Hardy--Littlewood Majorant Problem for Arithmetic Sets

AU - Krause, Ben

AU - Mirek, Mariusz

AU - Trojan, Bartosz

PY - 2016/7/1

Y1 - 2016/7/1

N2 - The aim of this paper is to exhibit a wide class of sparse deterministic sets, B⊆N, so that lim supN→∞N−1|B∩[1,N]|=0,for which the Hardy–Littlewood majorant property holds: sup|an|≤1⁡‖∑n∈B∩[1,N]ane2πinξ‖Lp(T,dξ)≤Cp‖∑n∈B∩[1,N]e2πinξ‖Lp(T,dξ), where p≥pB is sufficiently large, the implicit constant Cp is independent of N, and the supremum is taken over all complex sequences (an:n∈N) such that |an|≤1.

AB - The aim of this paper is to exhibit a wide class of sparse deterministic sets, B⊆N, so that lim supN→∞N−1|B∩[1,N]|=0,for which the Hardy–Littlewood majorant property holds: sup|an|≤1⁡‖∑n∈B∩[1,N]ane2πinξ‖Lp(T,dξ)≤Cp‖∑n∈B∩[1,N]e2πinξ‖Lp(T,dξ), where p≥pB is sufficiently large, the implicit constant Cp is independent of N, and the supremum is taken over all complex sequences (an:n∈N) such that |an|≤1.

U2 - 10.1016/j.jfa.2016.04.008

DO - 10.1016/j.jfa.2016.04.008

M3 - Article

SN - 0022-1236

VL - 271

SP - 164

EP - 181

JO - JOURNAL OF FUNCTIONAL ANALYSIS

JF - JOURNAL OF FUNCTIONAL ANALYSIS

IS - 1

ER -