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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.

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