Limits of Besov Norms on Quantum Euclidean Spaces

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.