Quantum differentiability of essentially bounded functions on Euclidean space

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 .