Elliptic G-Operators on Manifolds with Isolated Singularities


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Abstract

We study elliptic operators on manifolds with singularities such that a discrete group G acts on the manifold. Following the standard elliptic theory approach, we de ne the Fredholm property of an operator by its principal symbol. For this problem, we prove that the symbol is a pair consisting of the symbol on the principal stratum (the inner symbol) and the symbol at the conical point (the conormal symbol). We establish the Fredholm property of elliptic elements.

About the authors

A. Yu. Savin

Peoples’ Friendship University of Russia

Email: antonsavin@mail.ru
Moscow, Russia

B. Yu. Sternin

Peoples’ Friendship University of Russia

Email: sternin@mail.ru
Moscow, Russia

References

  1. Агранович М. С., Вишик М. И. Эллиптические задачи с параметром и параболические задачи общего вида// Усп. мат. наук. - 1964. - 19, № 3. - С. 53-161.
  2. Кондратьев В. А. Краевые задачи для эллиптических уравнений в областях с коническими и угловыми точками// Тр. Моск. Мат. об-ва. - 1967. - 16. - С. 209-292.
  3. Назайкинский В. Е., Савин А. Ю., Стернин Б. Ю. Псевдодифференциальные операторы на стратифицированных многообразиях. II// Дифф. уравн. - 2007. - 43, № 5. - С. 685-696.
  4. Стернин Б. Ю. Квазиэллиптические операторы на бесконечном цилиндре. - М.: МИЭМ, 1972.
  5. Стернин Б. Ю. Эллиптическая теория на компактных многообразиях с особенностями. - М.: МИЭМ, 1974.
  6. Савин А. Ю. О символе нелокальных операторов в пространствах Соболева// Дифф. уравн. - 2011. - 47, № 6. - С. 890-893.
  7. Antonevich A., Belousov M., Lebedev A. Functional di erential equations. II. C∗-applications. Ч. 1, 2. - Harlow: Longman, 1998.
  8. Antonevich A., Lebedev A. Functional di erential equations. I. C∗-theory. - Harlow: Longman, 1994.
  9. Nazaikinskii V. E., Savin A. Yu., Schulze B.-W., Sternin B. Yu. Elliptic theory on singular manifolds. - Boca Raton: CRC-Press, 2005.
  10. Nazaikinskii V. E., Savin A. Yu., Sternin B. Yu. Elliptic theory and noncommutative geometry. - Basel: Birkha¨user, 2008.
  11. Nazaikinskii V., Savin A., Sternin B. Elliptic theory on manifolds with corners. I. Dual manifolds and pseudodi erential operators// В сб.: C∗-algebras and elliptic theory. II. - Basel: Birkha¨user, 2008. - С. 183-206.
  12. Pedersen G. K. C∗-Algebras and their automorphism groups. - London-New York: Academic Press, 1979.
  13. Savin A. Yu., Sternin B. Yu. Uniformization of nonlocal elliptic operators and KK-theory// Russ. J. Math. Phys. - 2013. - 20, № 3. - С. 345-359.
  14. Savin A. Yu., Sternin B. Yu. Elliptic theory for operators associated with di eomorphisms of smooth manifolds// В сб.: Pseudo-di erential operators, generalized functions and asymptotics. Selected papers of the 8th ISAAC congress, Moscow, Russia, August 22-27, 2011. - Basel: Birkha¨user, 2013. - С. 1-26.
  15. Savin A. Yu., Sternin B. Yu. Index of elliptic operators for di eomorphisms of manifolds//j. Noncommut. Geom. - 2014. - 8, № 3. - С. 695-734.

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