报告地点：行健楼学术活动室526
邀请人：黄益副教授
摘要：Cwikel-type estimates appear to be very useful in Mathematical Physics. However, they are also useful in Harmonic Analysis --- see Lord-McDonald-Sukochev-Zanin-JFA-2017 (Euclidean setting) and Fan-Li-McDonald-Sukochev-Zanin-JFA-2024 (Heisenberg group setting). To accommodate the needs of the latter paper, one had to develop Cwikel-type estimates on stratified Lie groups. Just like in Euclidean setting, those estimates for ideals \mathcal{L}_{p,\infty} differ dramatically for p>2, p=2 and p<2. The case p>2 is well served by the Abstract Cwikel Estimates as proved in Levitina-Sukochev-Zanin-PLMS-2020. In contrast, for p<2 and p=2 our approach is much more specific and requires the analogues of the Birman-Solomyak spaces l_p(l_q(\mathbb{R}^d)) for a given group G. The most problematic bit is the definition of such spaces --- the most obvious scenario of using the fundamental domains of co-compact lattice fails because existence of such a lattice requires structural constants of the Lie algebra to be rational. In this talk, we discuss in details the covering lemmas needed to define Birman-Solomyak spaces for a given stratified group.
The second part of the talk concerns the spectral asymptotics. The key device here is the Tauberian theorem due to Wiener and Ikehara. However, in the setting of a locally compact group, the sub-Laplacian is no longer compact (in fact, its spectrum is absolutely continuous). This creates a severe problem in using Wiener-Ikehara theorem for proving the asymptotics. The problem was resolved only recently in Sukochev-Zanin-Asterisque-2023. Interestingly, the condition in Sukochev-Zanin-Asterisque-2023 requires the analyticity of a different \zeta-function, not the one typically considered in Non-commutative Geometry. In this talk, we combine the technique from Sukochev-Zanin-Asterisque-2023 and Wiener-Ikehara theorem and derive the spectral asymptotics.
