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<d. The question whether quantised derivative can fall into a more complicated ideal (non-comparable with \mathcal{L}_{p,\infty}-scale) was left open.
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.
