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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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##### References:
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