## Schatten properties of commutators

par - publié le

Jeudi 12 mai 2022
Kai Zeng
(Université de Franche-Comté)

Given a quantum tori $\mathbb{T}_\theta^d$, we can define the Riesz transforms $\mathfrak{R}_j$ on the quantum tori and the commutator $dx_i := [\mathfrak{R}_i, M_x]$, where $M_x$ is the operator on $L_2(\mathbb{T}_\theta^d)$ of pointwise multiplication by $x \in L_\infty(\mathbb{T}_\theta^d)$. In this talk, we will characterize the Schatten properties of the commutator $[\mathfrak{R}_i, M_x]$ by showing that $x \in B^\alpha_{p,q}(\mathbb{T}_\theta^d)$, where $B^\alpha_{p,q}(\mathbb{T}_\theta^d)$ is the Besov space on quantum tori. Futhermore, we will extend this characterisation to the more general case where $\mathfrak{R}_j$ replaced by an arbitrary Calderon-Zygmund operator. To date, these new results treat the quantum differentiability in the strictly noncommutative setting