Quantum differentiability of essentially bounded functions on Euclidean space
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...
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| Format: | Journal article |
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Elsevier
2017
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| _version_ | 1797104314969227264 |
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| author | Lord, S McDonald, E Sukochev, F Zanin, D |
| author_facet | Lord, S McDonald, E Sukochev, F Zanin, D |
| author_sort | Lord, S |
| collection | OXFORD |
| description | 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. |
| first_indexed | 2024-03-07T06:32:03Z |
| format | Journal article |
| id | oxford-uuid:f6510c00-e0e8-467d-af13-7062ba45b5e2 |
| institution | University of Oxford |
| last_indexed | 2024-03-07T06:32:03Z |
| publishDate | 2017 |
| publisher | Elsevier |
| record_format | dspace |
| spelling | oxford-uuid:f6510c00-e0e8-467d-af13-7062ba45b5e22022-03-27T12:34:18ZQuantum differentiability of essentially bounded functions on Euclidean spaceJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:f6510c00-e0e8-467d-af13-7062ba45b5e2Symplectic Elements at OxfordElsevier2017Lord, SMcDonald, ESukochev, FZanin, DWe 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. |
| spellingShingle | Lord, S McDonald, E Sukochev, F Zanin, D Quantum differentiability of essentially bounded functions on Euclidean space |
| title | Quantum differentiability of essentially bounded functions on Euclidean space |
| title_full | Quantum differentiability of essentially bounded functions on Euclidean space |
| title_fullStr | Quantum differentiability of essentially bounded functions on Euclidean space |
| title_full_unstemmed | Quantum differentiability of essentially bounded functions on Euclidean space |
| title_short | Quantum differentiability of essentially bounded functions on Euclidean space |
| title_sort | quantum differentiability of essentially bounded functions on euclidean space |
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