We obtain Weyl type asymptotics for the quantised derivative $\dbar f$ of a function $f$ from the homgeneous Sobolev space $\dot{ W}^1_d(\mathbb{ R}^d)$ on $\mathbb{ R}^d.$ The asymptotic coefficient $\|\nabla f\|_{ L_d(\mathbb R^d)}$ is equivalent to the norm of $\dbar f$ in the principal ideal $\mathcal{ L}_{ d,\infty},$ thus, providing a non-asymptotic, uniform bound on the spectrum of $\dbar f.$ 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.
Categories: math.SP math-ph math.FA math.MP math.OA