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 Ufimsk. Mat. Zh., 2014, Volume 6, Issue 3, Pages 35–71 (Mi ufa252)

Singular integral operators on a manifold with a distinguished submanifold

Yu. A. Kordyukova, V. A. Pavlenkob

a Institute of Mathematics, Russian Academy of Sciences, 112, Chernyshevsky str., 450008 Ufa, Russia
b Bashkir State Agrarian University, 34 50-letiya Oktyabrya Str., 450001 Ufa, Russia

Abstract: Let $X$ be a compact manifold without boundary and $X^0$ its smooth submanifold of codimension one. In this work we introduce classes of integral operators on $X$ with kernels $K_A(x,y)$, being smooth functions for $x\notin X^0$ and $y\notin X^0$, and admitting an asymptotic expansion of certain type, if $x$ or $y$ approaches $X^0$. For operators of these classes we prove theorems about action in spaces of conormal functions and composition. We show that the trace functional can be extended to a regularized trace functional $\operatorname{r-Tr}$ defined on some algebra $\mathcal K(X,X^0)$ of singular integral operators described above. We prove a formula for the regularized trace of the commutator of operators from this class in terms of associated operators on $X^0$. The proofs are based on theorems about pull-back and push-forward of conormal functions under maps of manifolds with distinguished codimension one submanifolds.

Keywords: manifolds, singular integral operators, conormal functions, regularized trace, pull-back, push-forward.

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English version:
Ufa Mathematical Journal, 2014, 6:3, 35–68 (PDF, 701 kB); https://doi.org/10.13108/2014-6-3-35

UDC: 515.168+517.983
MSC: 47G10, 58J40,47C05

Citation: Yu. A. Kordyukov, V. A. Pavlenko, “Singular integral operators on a manifold with a distinguished submanifold”, Ufimsk. Mat. Zh., 6:3 (2014), 35–71; Ufa Math. J., 6:3 (2014), 35–68

Citation in format AMSBIB
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\by Yu.~A.~Kordyukov, V.~A.~Pavlenko
\paper Singular integral operators on a~manifold with a~distinguished submanifold
\jour Ufimsk. Mat. Zh.
\yr 2014
\vol 6
\issue 3
\pages 35--71
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\elib{http://elibrary.ru/item.asp?id=22370777}
\transl
\jour Ufa Math. J.
\yr 2014
\vol 6
\issue 3
\pages 35--68
\crossref{https://doi.org/10.13108/2014-6-3-35}
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This publication is cited in the following articles:
1. Y. A. Kordyukov, V. A. Pavlenko, “On Lefschetz formulas for flows on foliated manifolds”, Ufa Math. J., 7:2 (2015), 71–101
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