Refine
Has Fulltext
- no (101)
Year of publication
Document Type
- Article (101) (remove)
Language
- English (101)
Is part of the Bibliography
- yes (101)
Keywords
- Pseudo-differential operators (4)
- Asymptotics of solutions (2)
- Edge calculus (2)
- Mellin symbols with values in the edge calculus (2)
- Meromorphic operator-valued symbols (2)
- operator-valued symbols (2)
- 35J70 (1)
- 47G30 (1)
- 58J40 (1)
- Anisotropic pseudo-differential operators (1)
- Atiyah-Bott obstruction (1)
- Boutet de Monvel's calculus (1)
- Cone and edge pseudo-differential operators (1)
- Corner pseudo-differential operators (1)
- Edge degenerate operators (1)
- Edge symbols (1)
- Elliptic complexes (1)
- Ellipticity and parametrices (1)
- Ellipticity of corner-degenerate operators (1)
- Ellipticity of edge-degenerate operators (1)
- Fourier and Mellin transform (1)
- Fourier and Mellin transforms (1)
- Fourier transform (1)
- Fredholm property (1)
- Kegel space (1)
- Manifolds with boundary (1)
- Mellin and Green operators edge symbols (1)
- Mellin operators (1)
- Mellin oscillatory integrals (1)
- Mellin quantization (1)
- Mellin quantizations (1)
- Mellin transform (1)
- Operator-valued symbols (1)
- Operator-valued symbols of Mellin type (1)
- Operators on singular cones (1)
- Operators on singular manifolds (1)
- Parametrices of elliptic operators (1)
- Quantizations (1)
- Schrodinger equation (1)
- Singular analysis (1)
- Singular cones (1)
- Stratified spaces (1)
- Toeplitz-type pseudodifferential operators (1)
- Twisted symbolic estimates (1)
- Volterra operator (1)
- Weighted edge spaces (1)
- Zaremba problem (1)
- algebra (1)
- asymptotic properties of eigenfunctions (1)
- boundary value problems (1)
- corner Sobolev spaces with double weights (1)
- corner parametrices (1)
- distribution with asymptotics (1)
- ellipticity (1)
- exit calculus (1)
- iterated asymptotics (1)
- manifolds with corners (1)
- manifolds with edge and boundary (1)
- many-electron systems (1)
- operator valued symbols (1)
- operators with corner symbols (1)
- parametrices of elliptic operators (1)
- pseudo-differential boundary value problems (1)
- pseudo-differential operators (1)
- weighted Sobolev spaces (1)
- weighted edge and corner spaces (1)
Necessary and sufficient conditions for the representation of the index of elliptic operators on manifolds with edges in the form of the sum of homotopy invariants of symbols on the smooth stratum and on the edge are found. An index formula is obtained for elliptic operators on manifolds with edges under symmetry conditions with respect to the edge covariables
The regularity of solutions to elliptic equations on a manifold with singularities, say, an edge, can be formulated in terms of asymptotics in the distance variable r > 0 to the singularity. In simplest form such asymptotics turn to a meromorphic behaviour under applying the Mellin transform on the half-axis. Poles, multiplicity, and Laurent coefficients form a system of asymptotic data which depend on the specific operator. Moreover, these data may depend on the variable y along the edge. We then have y-dependent families of meromorphic functions with variable poles, jumping multiplicities and a discontinuous dependence of Laurent coefficients on y. We study here basic phenomena connected with such variable branching asymptotics, formulated in terms of variable continuous asymptotics with a y-wise discrete behaviour.
A manifold M with smooth edge Y is locally near Y modelled on X-Delta x Omega for a cone X-Delta := ( (R) over bar (+) x X)/({0} x X) where Xis a smooth manifold and Omega subset of R-q an open set corresponding to a chart on Y. Compared with pseudo-differential algebras, based on other quantizations of edge-degenerate symbols, we extend the approach with Mellin representations on the r half-axis up to r = infinity, the conical exit of X-boolean AND = R+ x X (sic) (r, x) at infinity. The alternative description of the edge calculus is useful for pseudo-differential structures on manifolds with higher singularities.
By edge algebra we understand a pseudo-differential calculus on a manifold with edge. The operators have a two-component principal symbolic hierarchy which determines operators up to lower order terms. Those belong to a filtration of the corresponding operator spaces. We give a new characterisation of this structure, based on an alternative representation of edge amplitude functions only containing holomorphic edge-degenerate Mellin symbols.