Наредни састанак Семинара биће одржан онлајн у четвртак,29. априла 2021. са почетком у 17:15.
Предавач: Стеван Пилиповић, академик САНУ
Наслов предавања: HORMANDER WEYL CALCULUS: ELLIPTICITY AND FREDHOLMNESS
Апстракт: One of the most important property all of the frequently used global pseudodifferential calculi have, is the spectral invariance. This amounts to the following. If A is a ΨDO with 0 order symbol (thus continuous on L^2) which is invertible on L^2 then the inverse is again a ΨDO with a 0 order symbol. This property has been proved by several authors for various global calculi including the Shubin calculus, the SG (scattering) calculus, the Beals-Fefferman calculus, e.t.c. In their seminal paper, Bony and Chemin generalised these results by proving the spectral invariance for the Weyl-Hormander calculus when the H¨ormander metric satisfies the so-called geodesic temperance. After an appropriate introduction, In the second part of the talk I will show that this process of taking inverses preserves continuity and smoothness in the following sense. If λ → a_λ is a continuous (resp., smooth) mapping with values in S(1,g) such that awλ is invertible in L,^2, then the mapping λ → b_λ, where b^w_λ is the inverse of a^w_λ, remains to be continuous (resp., smooth) (in fact, we have this result for matrix valued symbols). In the third part of the talk, I will present the Fredholm properties of ΨDOs with symbols in the Weyl-H¨ormander classes when acting between the Sobolev spaces naturally associated to these classes. The main result is that the Fredholm property of a ΨDO can be characterised by ellipticity of the symbol, that is a ΨDO is Fredholm operator between appropriate Sobolev spaces if and only if its symbol is elliptic. This result heavily relies on the main result in the first part (as well as on the spectral invariance of the Weyl-Hormander calculus) and consequently, on the geodesic temperance of the metric.
S. Pilipovic, B. Prangoski, Equvalence of Elipticity and the Fredholm Property in the Weyl-HormandserCalculus, J Inst. Math. Jussieu (2020), pp. 1–27.
Meeting number: 183 958 4971
Meeting password: 9NJb2B3VMPM