On the Hardy--Littlewood majorant problem for arithmetic sets
On the Hardy--Littlewood majorant problem for arithmetic sets
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关于算术集的 Hardy--Littlewood 大问题
DOI：
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发表时间：
2015
期刊：
影响因子：
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通讯作者：
B. Trojan
中科院分区：
文献类型：
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作者：
B. Krause;Mariusz Mirek;B. Trojan
The aim of this paper is to exhibit a wide class of sparse deterministic sets, $mathbf B subseteq mathbb{N}$, so that
[ limsup_{N o infty} N^{-1}|mathbf B cap [1,N]|= 0, ]
for which the Hardy--Littlewood majorant property holds:
[ sup_{|a_n|le 1} Big| sum_{ninmathbf Bcap[1, N]} a_n e^{2 pi i n xi}Big |_{L^p(mathbb{T}, {mathrm d} xi)} leq mathbf{C}_p Big| sum_{ninmathbf Bcap[1, N]} e^{2 pi i n xi} Big|_{L^p(mathbb{T},
{mathrm d} xi)}, ] where $p geq p_{mathbf{B}}$ is sufficiently large, the implicit constant $mathbf{C}_p$ is independent of $N$, and the supremum is taken over all complex sequences $ (a_n : n in mathbb{N})$ such that $|a_n| leq 1$.
The aim of this paper is to exhibit a wide class of sparse deterministic sets, $mathbf B subseteq mathbb{N}$, so that
[ limsup_{N o infty} N^{-1}|mathbf B cap [1,N]|= 0, ]
for which the Hardy--Littlewood majorant property holds:
[ sup_{|a_n|le 1} Big| sum_{ninmathbf Bcap[1, N]} a_n e^{2 pi i n xi}Big |_{L^p(mathbb{T}, {mathrm d} xi)} leq mathbf{C}_p Big| sum_{ninmathbf Bcap[1, N]} e^{2 pi i n xi} Big|_{L^p(mathbb{T},
{mathrm d} xi)}, ] where $p geq p_{mathbf{B}}$ is sufficiently large, the implicit constant $mathbf{C}_p$ is independent of $N$, and the supremum is taken over all complex sequences $ (a_n : n in mathbb{N})$ such that $|a_n| leq 1$.