勷勤数学•专家报告

题      目：Schatten properties of Riesz transform commutator in the two weight setting

报  告  人：李冀 教授  (邀请人：韩彦昌)

                                             Macquarie University, Australia

时      间：6月29日  09：00-10：00

地     点：数科院东楼507

报告人简介:

       李冀，澳大利亚Macquarie大学数学系Associate Professor。2009年在中山大学获得博士学位，毕业后在中山大学数学系任讲师并于2012年任副教授。2014年1月入职Macquarie大学任Lecturer，2016年11月任Senior Lecturer，2022年11月任Associate Professor。主要在调和分析方向从事研究工作，在多参数调和分析，奇异积分交换子，与微分算子相关的函数空间等方面取得一定的进展，部分成果发表在Memoirs of the American Mathematical Society, Journal de Mathématiques Pures et Appliquées, Mathematische Annalen, Analysis and PDE, Advances in Mathematics等国际知名刊物上。现任澳大利亚数学会期刊Journal of the Australian Mathematical Society及Bulletin of the Australian Mathematical Society编委。

摘      要：

       Schatten class estimates of the commutator of Riesz transform in R^n link to the quantised derivative of A. Connes. A general setting for quantised calculus is a spectral triple (A, H, D), which consists of a Hilbert space H, a pre-C∗-algebra A, represented faithfully on H and a self-adjoint operator D acting on H such that every a ∈ A maps the domain of D into itself and the commutator

[D, a] = Da−aD extends from the domain of D to a bounded linear endomorphism of H. Here, the quantised differential d¯a of a ∈ A is deffned to be the bounded

operator i[sgn(D), a], i^2 = −1. We provide full characterisation of the Schatten properties of [Mb, R_j ], the commutator  of the j-th Riesz transform on R

n with symbol b (Mbf(x) := b(x)f(x)), in the two weight setting. The approach is not depending on the Euclidean structure or Fourier, and hence it can be applied to other settings. This talk is based on my recent work joint with Michael Lacey, Brett Wick and Liangchuan Wu.