Quantum differentiability of essentially bounded functions on Euclidean space

Journal article

We investigate the properties of the singular values of the quantised derivatives of essentially bounded functions on R d with d & #x003E;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)‖ 2 d :=((∂ 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. For real valued f, when x↦‖(∇f)(x)‖ 2 d is integrable, there exists a constant c d & #x003E;0 such 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)‖ 2 d dx.

Authors: Lord, S; McDonald, E; Sukochev, F; Zanin, D

• Publication status: Published
• Peer review status: Peer reviewed
• Publisher: Elsevier
• Journal: Journal of Functional Analysis
• Volume: 273
• Issue: 7
• Pages: 2353-2387
• Publication date: 2017-07-04
• Acceptance date: 2017-06-22
• DOI: 10.1016/j.jfa.2017.06.020
• EISSN: 1096-0783
• ISSN: 0022-1236
• Keywords: trace formula; quantised calculus; noncommutative geometry; quantum derivative