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Elliptic translators on manifolds with point singularities. (English. Russian original) Zbl 1267.35269
Differ. Equ. 48, No. 12, 1577-1585 (2012); translation from Differ. Uravn 48, No. 12, 1612-1620 (2012).
Consider a smooth closed manifold $$M$$ and two closed submanifolds $$Y^p$$, $$p= 1,2$$, transversally intersecting along a smooth submanifold $$X$$, $$\dim X= 0$$, $$X$$ being the set of singularities of $$Y= Y^1\cup Y^2$$. Following the second author, the authors define a pseudodifferential translator $$T_{12}$$ acting from $$Y^2$$ to $$Y^1$$ and develop an elliptic theory for the operators of the form $1+ T: H^s(Y)\to H^s(Y),\quad H^2(Y)= H^{s_1}(Y^1)\oplus H^{s_2}(Y^2).$ The elliptic operators are of Fredholm type (Th. 1) and an index formula for them is found in Th. 2. The index of $$(1+T)$$ is expressed by the winding number of the invertible on the weight line $$\text{Re\,}z= \alpha$$ function $$1+\sigma(T)(z)$$, $$\sigma(T)(z)$$ being the symbol of corresponding operator family.

##### MSC:
 35S15 Boundary value problems for PDEs with pseudodifferential operators 35S35 Topological aspects for pseudodifferential operators in context of PDEs: intersection cohomology, stratified sets, etc. 58J32 Boundary value problems on manifolds 58J20 Index theory and related fixed-point theorems on manifolds 47A53 (Semi-) Fredholm operators; index theories 58B15 Fredholm structures on infinite-dimensional manifolds
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