The p--Bloch--Carleson measure problem for 0<p<∞: A complete characterization via Carleson tents
Abstract
We characterize the boundedness and compactness of the embedding id:ℬ ⟶ Lp(μ), 0<p<∞, where ℬ denotes the Bloch space and μ is a positive Borel measure on the unit disc. The criteria depend only on the dyadic Carleson tents. These results refine our earlier L2(μ) characterization based on a full dyadic resolution of the disc and extend it to all positive exponents. The proof combines discretization on Carleson tents, representations of Bloch functions by series of analytic polynomials, and weighted matrix estimates. We also obtain corresponding boundedness and compactness characterizations for id:𝒬p⟶ Lq(μ), 0<p≤1, 0<q<∞. These results resolve the general p--Bloch--Carleson measure problem studied by Girela, Peláez, Pérez-González and Rättyä in 2008.
Related
- Littlewood-Paley and Carleson measure characterizations of Lipschitz spaces adapted to Schrödinger operators
- Parabolic uniform rectifiability is characterized by Carleson measure estimates for bounded caloric functions
- The Lp Neumann problem for parabolic operators with coefficients satisfying small Carleson condition
- Characterization of C1,α-domains in terms of Carleson-type ε-coefficients
- A Full Characterization of the Dirichlet Carleson embedding id: 𝒟p−1p → Lp(μ) for p>2
BibTeX
@misc{hu2026pblochcarleson,
title = {{The $p$--Bloch--Carleson measure problem for $0<p<\infty$: A complete characterization via Carleson tents}},
author = {Bingyang Hu and Xiaojing Zhou},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2606.16760},
eprint = {2606.16760},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.16760},
}
RIS
TY - UNPB
TI - The p--Bloch--Carleson measure problem for 0<p<∞: A complete characterization via Carleson tents
AU - Bingyang Hu
AU - Xiaojing Zhou
PY - 2026
DA - 2026/10/26
PB - arXiv
DO - 10.48550/arxiv.2606.16760
UR - https://arxiv.org/abs/2606.16760
AB - We characterize the boundedness and compactness of the embedding id:ℬ ⟶ L^p(μ), 0<p<∞, where ℬ denotes the Bloch space and μ is a positive Borel measure on the unit disc. The criteria depend only on the dyadic Carleson tents. These results refine our earlier L^2(μ) characterization based on a full dyadic resolution of the disc and extend it to all positive exponents. The proof combines discretization on Carleson tents, representations of Bloch functions by series of analytic polynomials, and weighted matrix estimates. We also obtain corresponding boundedness and compactness characterizations for id:𝒬_p⟶ L^q(μ), 0<p≤1, 0<q<∞. These results resolve the general p--Bloch--Carleson measure problem studied by Girela, Peláez, Pérez-González and Rättyä in 2008.
ER -