Limits of Besov Norms on Quantum Euclidean Spaces
Abstract
We extend the Bourgain--Brézis--Mironescu (BBM) and Maz'ya--Shaposhnikova (MS) limit formulae to higher-order Besov norms on quantum Euclidean spaces Rdθ, a fundamental class of noncompact noncommutative manifolds obtained by deformation quantization of Rd. A central ingredient is a Banach realization of the homogeneous Sobolev endpoint that forms a compatible couple with Lp(Rθd), together with the equivalence K(x,εk; Lp(Rθd),Ẇp,0k(Rθd)) ≈ ωpk(x,ε). This framework yields a unified proof of both endpoint formulae through the endpoint continuity of real interpolation scales. The principal new ingredient at the MS endpoint is the sharp large-translation asymptotic lim|h|→∞|Thx−x|p = 21/p|x|p, x∈ Lp(Rθd), 1≤ p<∞, whose proof combines oscillatory decay of the non-pure terms in the tracial expansion with weighted spectral-measure arguments adapted to the semifinite trace.
Related
- Coefficient duality and interpolation of Euclidean flag Besov spaces
- Quantum integration in Besov spaces with mixed smoothness
- Endpoint Schatten class properties of commutators on quantum Euclidean spaces
- Calderón Commutators on Quantum Euclidean Spaces and Quantum Tori
- Besov and Triebel--Lizorkin Spaces on Filtered Lie Groups, III: the Spaces F∞,qα
BibTeX
@misc{deng2026limits,
title = {{Limits of Besov Norms on Quantum Euclidean Spaces}},
author = {Qingquan Deng and Guixiang Hong and Ping Li and Yanzhen Long},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.26463},
eprint = {2610.26463},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.26463},
}
RIS
TY - UNPB
TI - Limits of Besov Norms on Quantum Euclidean Spaces
AU - Qingquan Deng
AU - Guixiang Hong
AU - Ping Li
AU - Yanzhen Long
PY - 2026
DA - 2026/10/19
PB - arXiv
DO - 10.48550/arxiv.2610.26463
UR - https://arxiv.org/abs/2610.26463
AB - We extend the Bourgain--Brézis--Mironescu (BBM) and Maz'ya--Shaposhnikova (MS) limit formulae to higher-order Besov norms on quantum Euclidean spaces R^d_θ, a fundamental class of noncompact noncommutative manifolds obtained by deformation quantization of R^d. A central ingredient is a Banach realization of the homogeneous Sobolev endpoint that forms a compatible couple with L_p(R_θ^d), together with the equivalence K(x,ε^k; L_p(R_θ^d),Ẇ_p,0^k(R_θ^d)) ≈ ω_p^k(x,ε). This framework yields a unified proof of both endpoint formulae through the endpoint continuity of real interpolation scales. The principal new ingredient at the MS endpoint is the sharp large-translation asymptotic lim_|h|→∞|T_hx−x|_p = 2^1/p|x|_p, x∈ L_p(R_θ^d), 1≤ p<∞, whose proof combines oscillatory decay of the non-pure terms in the tracial expansion with weighted spectral-measure arguments adapted to the semifinite trace.
ER -