A Full Characterization of the Dirichlet Carleson embedding id: 𝒟_p−1^p → L^p(μ) for p>2

@misc{hu2026full,
  title = {{A Full Characterization of the Dirichlet Carleson embedding $\operatornameid: \mathcal D_p-1^p \to L^p(μ)$ for $p>2$}},
  author = {Bingyang Hu and Xiaojing Zhou},
  howpublished = {arXiv},
  year = {2026},
  month = aug,
  doi = {10.48550/arxiv.2608.31134},
  eprint = {2608.31134},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.31134},
}

TY  - UNPB
TI  - A Full Characterization of the Dirichlet Carleson embedding id: 𝒟_p−1^p → L^p(μ) for p>2
AU  - Bingyang Hu
AU  - Xiaojing Zhou
PY  - 2026
DA  - 2026/08/31
PB  - arXiv
DO  - 10.48550/arxiv.2608.31134
UR  - https://arxiv.org/abs/2608.31134
AB  - In this paper, we obtain a full characterization of the finite positive Borel measures μ on 𝔻 for which the embedding id:𝒟_p−1^p⟶ L^p(μ), p>2, is bounded. More precisely, for any dyadic system 𝒟 on 𝕋, we prove that this embedding is bounded if and only if 𝒞_p,𝒟(μ)+ℋ_p,𝒟(μ)<∞, where 𝒞_p,𝒟(μ) and ℋ_p,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.
ER  -