p>2时Dirichlet Carleson嵌入id: D_{p-1}^p → L^p(μ)的完整刻画
A Full Characterization of the Dirichlet Carleson embedding $\operatorname{id}: \mathcal D_{p-1}^p \to L^p(μ)$ for $p>2$
- Auburn University（奥本大学）
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中文总结 AI 辅助
本文对p>2时Dirichlet Carleson嵌入id: D_{p-1}^p → L^p(μ)有界的有限正Borel测度μ给出完整刻画，解决相关长期问题，构造测度表明两个能量条件不同，关键是将Dirichlet嵌入约化为二进Whitney嵌入。
中文摘要 AI 辅助
本文中，我们对单位圆盘𝔻上使得嵌入id: 𝒟_{p-1}^p → L^p(μ)（其中p>2）有界的有限正Borel测度μ给出了完整刻画。更准确地说，对于单位圆周𝕋上的任意二进系统𝒟，我们证明该嵌入有界当且仅当𝒞_{p,𝒟}(μ) + _{p,𝒟}(μ) < ∞，其中𝒞_{p,𝒟}(μ)和_{p,𝒟}(μ)分别表示μ的填充能量和Haar能量。这一结果解决了Dirichlet型空间理论中一个长期存在的刻画问题，该问题源于Wu在1999年提出的猜想，对应于Arcozzi、Rochberg和Sawyer在2002年的工作中未覆盖的端点情形，并且在Girela与Peláez2006年的工作以及Galanopoulos、Girela与Peláez2011年的工作之后仍然悬而未决。我们还构造了有限测度，表明这两个能量条件确实是不同的。证明中的关键步骤是将Dirichlet嵌入约化为二进Whitney嵌入，这使得我们可以将树上的加权Hardy不等式与概率论证相结合。
英文摘要
In this paper, we obtain a full characterization of the finite positive Borel measures $μ$ on $\mathbb D$ for which the embedding $$ \operatorname{id}:\mathcal D_{p-1}^p\longrightarrow L^p(μ),\qquad p>2, $$ is bounded. More precisely, for any dyadic system $\mathcal D$ on $\mathbb T$, we prove that this embedding is bounded if and only if $$ \mathcal C_{p,\mathcal D}(μ)+\mathcal H_{p,\mathcal D}(μ)<\infty, $$ where $\mathcal C_{p,\mathcal D}(μ)$ and $\mathcal H_{p,\mathcal D}(μ)$ denote the packing energy and the Haar energy of $μ$, respectively. This resolves a longstanding characterization problem in the theory of Dirichlet-type spaces that arose from Wu's 1999 conjecture, corresponds to the endpoint case not covered by the work of Arcozzi, Rochberg, and Sawyer in 2002, and remained open after the works of Girela and Peláez in 2006 and Galanopoulos, Girela, and Peláez in 2011. We also construct finite measures showing that the two energy conditions are genuinely distinct. The key ingredient in the proof is a reduction of the Dirichlet embedding to a dyadic Whitney embedding, which allows us to combine weighted Hardy inequalities on trees with probabilistic arguments.
