发表机构
Soochow University; National Yang Ming Chiao Tung University; South China University of Technology（苏州大学; 国立阳明交通大学; 华南理工大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文解决了Dirichlet级数Hardy空间的局部嵌入问题，通过频率分解和对偶方法证明嵌入性质恰在p≥2时成立，将阈值从偶数整数改进为p=2。
AI 中文摘要
我们解决了Dirichlet级数Hardy空间的局部嵌入问题，这是一个无维数迹问题，询问全局$\mathscr{H}^p$范数是否控制临界线$\operatorname{Re}s=1/2$上的局部$L^p$质量。更精确地说，对于每个$2<p<\infty$，存在常数$C_p<\infty$，使得每个Dirichlet多项式$P$满足$$ \sup_{\theta\in\mathbb{R}}\int_{\theta}^{\theta+1}\left|P\left(\frac12+it\right)\right|^p\\,\mathrm{d}t\le C_p\left\lVert P\right\rVert_{\mathscr{H}^p}^{p}, $$ 其中$C_p$与$P$所依赖的素数变量的数量和选择无关。在本工作之前，该嵌入在$p=2$时已知，并且通过取整数幂，在偶数指数$p=2k$时已知；此前曾猜想这些情况穷尽了大于2的有限正指数情形。结合已知的$0<p<2$时失效的结果，我们的定理给出了尖锐的有限指数分类：局部嵌入性质恰好对$p\ge2$成立。因此真正的阈值是$p=2$，而非偶数整数性。证明转向对偶指数$q=p/(p-1)\in(1,2)$，在此处精确的频率分解分离出单个共振Euler乘积项。对共享素数因子进行协方差保持的替换，将所得的矩估计归结为对数相关高斯场，而临界分支随机游走界提供了所需的多尺度衰减。有限循环平方函数估计组装共振尺度，Hardy商对偶性将所得的向量值界转化为临界线迹。对于$1\le p<\infty$，已知的等价性在若干经典问题中给出相同的尖锐阈值，包括共形不变半平面嵌入、反向局部Carleson测度转移以及所有特征零Gordon--Hedenmalm复合算子的有界性。
英文摘要
We solve the local embedding problem for Hardy spaces of Dirichlet series, which is a dimension-free trace problem asking whether the global $\mathscr{H}^p$-norm controls local $L^p$-mass on the critical line $\operatorname{Re}s=1/2$. More precisely, for every $2<p<\infty$, there exists a constant $C_p<\infty$ such that every Dirichlet polynomial $P$ satisfies $$ \sup_{θ\in\mathbb{R}}\int_θ^{θ+1}\left|P\left(\frac12+it\right)\right|^p\,\mathrm{d}t\le C_p\left\lVert P\right\rVert_{\mathscr{H}^p}^{p}, $$ 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 $p\ge2$. Thus, the true threshold is $p=2$, rather than even integrality. The proof passes to the dual exponent $q=p/(p-1)\in(1,2)$, 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 $1\le p<\infty$, 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.
CommentsExpanded version, with additional proof details and navigational material. 121 pages
