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On the index of elliptic translators. (English. Russian original) Zbl 1234.35333
Dokl. Math. 83, No. 1, 76-79 (2011); translation from Dokl. Akad. Nauk 436, No. 4, 443-447 (2011).
The paper stresses some properties of pseudodifferential operators (\(\psi\)DO) acting on a closed manifold \(M\) when the boundary submanifolds \(Y^{p}\) and \(Y^{q}\) have singularities and in an elliptic case. The authors first study the case of isolated singularities. They define the notion of pseudo-differential translator \(T_{pq}\) from a function space \(H^{s}(Y^{q})\) on the submanifold \(Y^{q}\) to a function space \( H^{s-\sum_{j}d_{j}-\frac{\nu _{p}+\nu _{q}}{2}}(Y^{p})\) on the submanifold \(Y^{p}\) , through \(T_{pq}=D_{1}i*_{p}D_{2}i_{q^{\ast }}D_{3}\), where the \( D_{i}\) are \(\psi\)DOs. The main result here proves that operators of the type \( I+T=\left( \begin{matrix} 1 & T_{12} \\ T_{21} & 1 \end{matrix} \right) \) are Fredholm and the authors compute their index. In the case of multidimensional singularities, the authors consider operators of the type \( I+T\) and they obtain a quite similar but for the so-called elliptic rigging of these operators. The paper ends with an application to some Sobolev problem.

MSC:
35S05 Pseudodifferential operators as generalizations of partial differential operators
58J05 Elliptic equations on manifolds, general theory
58J40 Pseudodifferential and Fourier integral operators on manifolds
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