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arXiv · 2610.26463

Limits of Besov Norms on Quantum Euclidean Spaces

Qingquan Deng·Guixiang Hong·Ping Li·Yanzhen Long
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Abstract

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 \[ K\bigl(\x,\varepsilon^k; L_p(\R_θ^d),\dot W_{p,0}^k(\R_θ^d)\bigr) \approx ω_p^k(\x,\varepsilon). \] 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 \[ \lim_{|h|\to\infty}\|T_h\x-\x\|_p = 2^{1/p}\|\x\|_p, \qquad \x\in L_p(\R_θ^d),\quad 1\leq p<\infty, \] whose proof combines oscillatory decay of the non-pure terms in the tracial expansion with weighted spectral-measure arguments adapted to the semifinite trace.

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BibTeXRIS

Qingquan Deng, Guixiang Hong, Ping Li, Yanzhen Long. 2026-10-19. Limits of Besov Norms on Quantum Euclidean Spaces. https://arxiv.org/abs/2610.26463

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