Asymptotics of singular values for quantum derivatives

We obtain Weyl type asymptotics for the quantised derivative \dj ƒ of a function ƒ from the homgeneous Sobolev space Ẇ^(1)_(d) on (ℝᵈ). The asymptotic coefficient ||∇ƒ||_[L_(d)(ℝᵈ)] is equivalent to the norm of \dj ƒ in the principal ideal L_(d,∞), thus, providing a non-asymptotic, uniform bound on the spectrum of \dj ƒ. Our methods are based on the C*-algebraic notion of the principal symbol mapping on ℝᵈ, as developed recently by the last two authors and collaborators.