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Type: Journal article
Title: Quantum differentiability of essentially bounded functions on Euclidean space
Author: Lord, S.
McDonald, E.
Sukochev, F.
Zanin, D.
Citation: Journal of Functional Analysis, 2017; 273(7):2353-2387
Publisher: Elsevier
Issue Date: 2017
ISSN: 0022-1236
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Steven Lord, Edward McDonald, Fedor Sukochev, Dmitry Zanin
Abstract: We investigate the properties of the singular values of the quantised derivatives of essentially bounded functions on R d with d > 1. The commutator i [sgn( D ) , 1 ⊗ M f ]of an essentially bounded function f on R d acting by pointwise multiplication on L 2 ( R d )and the sign of the Dirac operator D acting on C 2 d/ 2 ⊗ L 2 ( R d )is called the quantised derivative of f . We prove the condition that the function x →‖ ( ∇ f )( x ) ‖ d 2 := (( ∂ 1 f )( x ) 2 + ... +( ∂ d f )( x ) 2 ) d/ 2 , x ∈ R d , being integrable is necessary and sufficient for the quantised derivative of f to belong to the weak Schatten d -class. This problem has been previously studied by Rochberg and Semmes, and is also explored in a paper of Connes, Sullivan and Telemann. Here we give new and complete proofs using the methods of double operator integrals. Furthermore, we prove a formula for the Dixmier trace of the d -th power of the absolute value of the quantised derivative. Fo r real valued f , when x →‖ ( ∇ f )( x ) ‖ d 2 is integrable, there exists a constant c d > 0such that for every continuous normalised trace φ on the weak trace class L 1 , ∞ we have φ ( | [sgn( D ) , 1 ⊗ M f ] | d ) = c d ∫ R d ‖ ( ∇ f )( x ) ‖ d 2 dx .
Keywords: Quantum derivative; quantised calculus; trace formula; noncommutative geometry
Rights: © 2017 Elsevier Inc. All rights reserved.
DOI: 10.1016/j.jfa.2017.06.020
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