Asymptotics of singular values for quantum derivatives

We obtain Weyl type asymptotics for the quantised derivative $\text{dbar}f$ of a function $f$ from the homgeneous Sobolev space ${\dot{W}}_d^1({\mathbb{R}}^d)$ on ${\mathbb{R}}^d\mathrm{.}$ The asymptotic coefficient $\mathrm{\parallel }\mathrm{\nabla }f{\mathrm{\parallel }}_{L_d({\mathbb{R}}^d)}$ is equivalent to the norm of $\text{dbar}f$ in the principal ideal ${\mathcal{L}}_{d,\mathrm{\infty }},$ thus, providing a non-asymptotic, uniform bound on the spectrum of $\text{dbar}f\mathrm{.}$ Our methods are based on the $C^{\ast }$-algebraic notion of the principal symbol mapping on ${\mathbb{R}}^d$, as developed recently by the last two authors and collaborators.