Arbre de Noël du GDR "Géométrie non-commutative"

Schatten Properties of Commutators

2 déc. 2022, 14:45

45m

Centre de conférences Marilyn et James Simons (Le Bois-Marie)

Orateur

Prof. Kai ZENG (Université de Bourgogne-Franche-Comté)

Description

Given a quantum tori ${\mathbb{T}}_{\theta }^d$, we can define the Riesz transforms ${\mathfrak{R}}_j$ on the quantum tori and the commutator $đx_i$ := [${\mathfrak{R}}_i,M_x$], where $M_x$ is the operator on $L^2({\mathbb{T}}_{\theta }^d)$ of pointwise multiplication by $x\in L^{\mathrm{\infty }}({\mathbb{T}}_{\theta }^d)$. In this talk, we will characterize the Schatten properties of the commutator [${\mathfrak{R}}_i,M_x$] by showing that $x\in B_{p,q}^{\alpha }({\mathbb{T}}_{\theta }^d)$, where $B_{p,q}^{\alpha }({\mathbb{T}}_{\theta }^d)$ 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.

Documents de présentation

Vidéo