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On a class of nonlocal elliptic operators for compact Lie groups. Uniformization and finiteness theorem

100%
Sternin B.
Open Mathematics
|
2011
|
tom 9
|
nr 4
814-832
EN
We consider a class of nonlocal operators associated with an action of a compact Lie group G on a smooth closed manifold. Ellipticity condition and Fredholm property for elliptic operators are obtained. This class of operators is studied using pseudodifferential uniformization, which reduces the problem to a pseudodifferential operator acting in sections of infinite-dimensional bundles.
2
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Exponential coordinates and regularity of groupoid heat kernels

76%
So B.
Open Mathematics
|
2014
|
tom 12
|
nr 2
284-297
EN
We prove that on an asymptotically Euclidean boundary groupoid, the heat kernel of the Laplacian is a smooth groupoid pseudo-differential operator.
3
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Elliptic operators on refined Sobolev scales on vector bundles

64%
Zinchenko T.
Open Mathematics
|
2017
|
tom 15
|
nr 1
907-925
EN
We introduce a refined Sobolev scale on a vector bundle over a closed infinitely smooth manifold. This scale consists of inner product Hörmander spaces parametrized with a real number and a function varying slowly at infinity in the sense of Karamata. We prove that these spaces are obtained by the interpolation with a function parameter between inner product Sobolev spaces. An arbitrary classical elliptic pseudodifferential operator acting between vector bundles of the same rank is investigated on this scale. We prove that this operator is bounded and Fredholm on pairs of appropriate Hörmander spaces. We also prove that the solutions to the corresponding elliptic equation satisfy a certain a priori estimate on these spaces. The local regularity of these solutions is investigated on the refined Sobolev scale. We find new sufficient conditions for the solutions to have continuous derivatives of a given order.
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