Absolutely summing operators on Bergman spaces over convex domains of finite type
Abstract
Let Ω⊂ℂn be a smoothly bounded convex domain of finite type. We characterize absolute summability on its Bergman spaces and estimate approximation errors in the summing norm. For 1<p<∞ and 1≤ q,r<∞, we characterize absolutely r-summing Carleson embeddings Ap(Ω)→ Lq(μ) by diagonal operators formed from normalized masses on McNeal balls. For 1<p,q<∞ and 1≤ r<∞, we obtain a corresponding big Hankel criterion using local Lq distances to holomorphic functions, with the appropriate volume weights. Both criteria give upper and lower norm estimates. The proofs combine uniform local nuclear estimates, kernel synthesis, and an analytic decomposition at two fixed scales. Local Taylor polynomials give explicit rank and error bounds for the embeddings. These bounds transfer to general Hankel operators when p=2 or q=2. Suitable atomic embeddings have the same best approximation errors as their diagonal models, up to constants. A family of Hankel operators contains a complemented copy of the diagonal ideal and has matching approximation rates for power sequences in the Hilbert case. For positive Toeplitz operators on finite-type ellipsoids, we compute exact spectral asymptotics in three regimes, including the critical logarithmic factor and the leading constants for geometric symbols. These asymptotics give optimal approximation rates in the summing norm. Applications to composition and Volterra operators include the target endpoint q=1.
Related
- Maximal Spaceability of the set of non-absolutely summing operators in quasi-Banach spaces
- Absolutely summing Toeplitz operators on weighted Fock spaces
- Quantum harmonic analysis and the Differential operators on weighted Bergman spaces of bounded symmetric domains
- Generalized Volterra Companion Operators on Bergman Spaces over Convex Domains of Finite Type
- Pseudoconvex domains in complex Euclidean space with finite-dimensional Bergman spaces
BibTeX
@misc{jianxiang2026absolutely,
title = {{Absolutely summing operators on Bergman spaces over convex domains of finite type}},
author = {Dong Jianxiang and Xu Chunxu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.23653},
eprint = {2609.23653},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.23653},
note = {CC BY-NC-SA 4.0},
}
RIS
TY - UNPB
TI - Absolutely summing operators on Bergman spaces over convex domains of finite type
AU - Dong Jianxiang
AU - Xu Chunxu
PY - 2026
DA - 2026/09/20
PB - arXiv
DO - 10.48550/arxiv.2609.23653
UR - https://arxiv.org/abs/2609.23653
AB - Let Ω⊂ℂ^n be a smoothly bounded convex domain of finite type. We characterize absolute summability on its Bergman spaces and estimate approximation errors in the summing norm. For 1<p<∞ and 1≤ q,r<∞, we characterize absolutely r-summing Carleson embeddings A^p(Ω)→ L^q(μ) by diagonal operators formed from normalized masses on McNeal balls. For 1<p,q<∞ and 1≤ r<∞, we obtain a corresponding big Hankel criterion using local L^q distances to holomorphic functions, with the appropriate volume weights. Both criteria give upper and lower norm estimates. The proofs combine uniform local nuclear estimates, kernel synthesis, and an analytic decomposition at two fixed scales. Local Taylor polynomials give explicit rank and error bounds for the embeddings. These bounds transfer to general Hankel operators when p=2 or q=2. Suitable atomic embeddings have the same best approximation errors as their diagonal models, up to constants. A family of Hankel operators contains a complemented copy of the diagonal ideal and has matching approximation rates for power sequences in the Hilbert case. For positive Toeplitz operators on finite-type ellipsoids, we compute exact spectral asymptotics in three regimes, including the critical logarithmic factor and the leading constants for geometric symbols. These asymptotics give optimal approximation rates in the summing norm. Applications to composition and Volterra operators include the target endpoint q=1.
ER -