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 Rd with d>1. The commutator i[sgn(D),1⊗Mf] of an essentially bounded function f on Rd acting by pointwise multiplication on L2(Rd) and the sign of the Dirac operator D acting on C2⌊d/2⌋⊗L2(Rd) is called the quantised derivative of f. We prove the condition that the function x↦‖(∇f)(x)‖2d:=((∂1f)(x)2+…+(∂df)(x)2)d/2, x∈Rd, 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)‖2d is integrable, there exists a constant cd>0 such that for every continuous normalised trace φ on the weak trace class L1,∞ we have φ(|[sgn(D),1⊗Mf]|d)=cd∫Rd‖(∇f)(x)‖2ddx.