Asymptotics of singular values for quantum derivatives

We obtain Weyl type asymptotics for the quantised derivative \dj \mkern 1muf of a function content-type="math/mathml"> f f from homgeneous Sobolev space alttext="ModifyingAbove upper W With dot Subscript d Superscript 1 Baseline left-parenthesis double-struck R right-parenthesis"> W ˙ d 1 ( R stretchy="false">) encoding="application/x-tex">\dot {W}^1_d(\mathbb {R}^d) on alttext="double-struck period"> . encoding="application/x-tex">\mathbb {R}^d. The asymptotic coefficient alttext="double-vertical-bar nabla f double-vertical-bar L Sub fence="false" stretchy="false">‖ L encoding="application/x-tex">\|\nabla f\|_{L_d(\mathbb R^d)} is equivalent to norm in principal ideal alttext="script comma normal infinity comma"> class="MJX-tex-caligraphic" mathvariant="script">L , mathvariant="normal">∞<!-- ∞ encoding="application/x-tex">\mathcal {L}_{d,\infty }, thus, providing non-asymptotic, uniform bound spectrum 1muf. Our methods are based alttext="upper C asterisk"> C ∗<!-- ∗ encoding="application/x-tex">C^{\ast } -algebraic notion symbol mapping d"> {R}^d , as developed recently by last two authors and collaborators.