The Local Embedding Problem for Hardy Spaces of Dirichlet Series
Abstract
We solve the local embedding problem for Hardy spaces of Dirichlet series, which is a dimension-free trace problem asking whether the global ℋp-norm controls local Lp-mass on the critical line Res=1/2. More precisely, for every 2<p<∞, there exists a constant Cp<∞ such that every Dirichlet polynomial P satisfies supθ∈ℝ∫θθ+1|P(1/2+it)|p dt≤ CplVert PrVertℋpp, with Cp independent of the number and choice of prime variables on which P depends. Before the present work, the embedding was known at p=2 and, by taking integer powers, at the even exponents p=2k; it had been conjectured that these exhaust the finite positive cases above $2$. Together with the known failure for 0<p<2, our theorem gives the sharp finite-exponent classification: the local embedding property holds exactly for p≥2. Thus, the true threshold is p=2, rather than even integrality. The proof passes to the dual exponent q=p/(p−1)∈(1,2), where an exact frequency decomposition isolates a single resonant Euler-product term. A covariance-preserving replacement of the shared prime factors reduces the resulting fractional-moment estimate to a log-correlated Gaussian field, and a critical branching-random-walk bound supplies the required multiscale decay. A finite-cyclic square-function estimate assembles the resonant scales, and Hardy-quotient duality converts the resulting vector-valued bound into the critical-line trace. For 1≤ p<∞, known equivalences give the same sharp threshold in several classical problems, including the conformally invariant half-plane embedding, the reverse local Carleson-measure transfer, and boundedness of all characteristic-zero Gordon--Hedenmalm composition operators.
Related
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BibTeX
@misc{chen2026local,
title = {{The Local Embedding Problem for Hardy Spaces of Dirichlet Series}},
author = {Bonan Chen and Xiang Fang and Feng Guo and Shengzhao Hou and Yizhou Shao and Qi Zhou},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.11560},
eprint = {2609.11560},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.11560},
}
RIS
TY - UNPB
TI - The Local Embedding Problem for Hardy Spaces of Dirichlet Series
AU - Bonan Chen
AU - Xiang Fang
AU - Feng Guo
AU - Shengzhao Hou
AU - Yizhou Shao
AU - Qi Zhou
PY - 2026
DA - 2026/09/13
PB - arXiv
DO - 10.48550/arxiv.2609.11560
UR - https://arxiv.org/abs/2609.11560
AB - We solve the local embedding problem for Hardy spaces of Dirichlet series, which is a dimension-free trace problem asking whether the global ℋ^p-norm controls local L^p-mass on the critical line Res=1/2. More precisely, for every 2<p<∞, there exists a constant C_p<∞ such that every Dirichlet polynomial P satisfies sup_θ∈ℝ∫_θ^θ+1|P(1/2+it)|^p dt≤ C_plVert PrVert_ℋ^p^p, with C_p independent of the number and choice of prime variables on which P depends. Before the present work, the embedding was known at p=2 and, by taking integer powers, at the even exponents p=2k; it had been conjectured that these exhaust the finite positive cases above $2$. Together with the known failure for 0<p<2, our theorem gives the sharp finite-exponent classification: the local embedding property holds exactly for p≥2. Thus, the true threshold is p=2, rather than even integrality. The proof passes to the dual exponent q=p/(p−1)∈(1,2), where an exact frequency decomposition isolates a single resonant Euler-product term. A covariance-preserving replacement of the shared prime factors reduces the resulting fractional-moment estimate to a log-correlated Gaussian field, and a critical branching-random-walk bound supplies the required multiscale decay. A finite-cyclic square-function estimate assembles the resonant scales, and Hardy-quotient duality converts the resulting vector-valued bound into the critical-line trace. For 1≤ p<∞, known equivalences give the same sharp threshold in several classical problems, including the conformally invariant half-plane embedding, the reverse local Carleson-measure transfer, and boundedness of all characteristic-zero Gordon--Hedenmalm composition operators.
ER -