广州数学大讲坛第二期

第二十讲——新南威尔士州大学Dmitriy Zanin教授学术报告


题目:Asymptotics of singular values for quantised derivatives

时间:2024年4月2日(星期二)下午 16:00-18:00

地点:理科南楼107

报告人:Dmitriy Zanin教授

摘要:In 2017 paper by Lord, McDonald, Sukochev and Zanin the following criterion was proved. Bounded measurable function f on \mathbb{R}^d, d\geq 2, belongs to the Sobolev space \dot{W}^1_d(\mathbb{R}^d) iff its quantised derivative

[{\rm sgn}(D),1\otimes M_f]

belongs to the Schatten class \mathcal{L}_{d,\infty}. Furthermore, it was established that

\varphi(|[{\rm sgn}(D),1\otimes M_f]|^d)=c_d\| |\nabla f| \|_d^d

for every continuous singular trace on \mathcal{L}_{1,\infty}. In particular, quantised derivative cannot belong to \mathcal{L}_{p,\infty} for p

In recent paper by Frank, Sukochev and Zanin, the following asymptotics was proved:

t^{\frac1d}\mu(t,[{\rm sgn}(D),1\otimes M_f])\to c_d\| |\nabla f| \|_d, t\to\infty.

In particular, if a good (say, smooth compactly supported) function belongs to some ideal, then this ideal must contain \mathcal{L}_{d,\infty}.

The proof relies on Birman-Solomyak methods for spectral asymptotics combined with the principal symbol calculus designed by McDonald, Sukochev and Zanin.