Asymptotics of singular values for quantum derivatives

Abstract

We obtain Weyl type asymptotics for the quantised derivative <inline-formula content-type="math/tex"> <tex-math>\dj \mkern 1muf</tex-math> </inline-formula> of a function <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f"> <mml:semantics> <mml:mi>f</mml:mi> <mml:annotation encoding="application/x-tex">f</mml:annotation> </mml:semantics> </mml:math> </inline-formula> from the homgeneous Sobolev space <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="ModifyingAbove upper W With dot Subscript d Superscript 1 Baseline left-parenthesis double-struck upper R Superscript d Baseline right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msubsup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mover> <mml:mi>W</mml:mi> <mml:mo> ˙ </mml:mo> </mml:mover> </mml:mrow> <mml:mi>d</mml:mi> <mml:mn>1</mml:mn> </mml:msubsup> <mml:mo stretchy="false">(</mml:mo> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mi>d</mml:mi> </mml:msup> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\dot {W}^1_d(\mathbb {R}^d)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper R Superscript d Baseline period"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mi>d</mml:mi> </mml:msup> <mml:mo>.</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb {R}^d.</mml:annotation> </mml:semantics> </mml:math> </inline-formula> The asymptotic coefficient <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-vertical-bar nabla f double-vertical-bar Subscript upper L Sub Subscript d Subscript left-parenthesis double-struck upper R Sub Superscript d Subscript right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo fence="false" stretchy="false"> ‖ </mml:mo> <mml:mi mathvariant="normal"> ∇ </mml:mi> <mml:mi>f</mml:mi> <mml:msub> <mml:mo fence="false" stretchy="false"> ‖ </mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>L</mml:mi> <mml:mi>d</mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mi>d</mml:mi> </mml:msup> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">\|\nabla f\|_{L_d(\mathbb R^d)}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is equivalent to the norm of <inline-formula content-type="math/tex"> <tex-math>\dj \mkern 1muf</tex-math> </inline-formula> in the principal ideal <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper L Subscript d comma normal infinity Baseline comma"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">L</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>d</mml:mi> <mml:mo>,</mml:mo> <mml:mi mathvariant="normal"> ∞ </mml:mi> </mml:mrow> </mml:msub> <mml:mo>,</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {L}_{d,\infty },</mml:annotation> </mml:semantics> </mml:math> </inline-formula> thus, providing a non-asymptotic, uniform bound on the spectrum of <inline-formula content-type="math/tex"> <tex-math>\dj \mkern 1muf.</tex-math> </inline-formula> Our methods are based on the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C Superscript asterisk"> <mml:semantics> <mml:msup> <mml:mi>C</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo> ∗ </mml:mo> </mml:mrow> </mml:msup> <mml:annotation encoding="application/x-tex">C^{\ast }</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -algebraic notion of the principal symbol mapping on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper R Superscript d"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mi>d</mml:mi> </mml:msup> <mml:annotation encoding="application/x-tex">\mathbb {R}^d</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , as developed recently by the last two authors and collaborators.

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