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- K-theory (3)
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- Atiyah-Bott condition (1)
- Atiyah-Patodi-Singer theory (1)
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The topological significance of the spectral Atiyah-Patodi-Singer η-invariant is investigated. We show that twice the fractional part of the invariant is computed by the linking pairing in K-theory with the orientation bundle of the manifold. The Pontrjagin duality implies the nondegeneracy of the linking form. An example of a nontrivial fractional part for an even-order operator is presented.
We prove a general theorem on the behavior of the relative index under surgery for a wide class of Fredholm operators, including relative index theorems for elliptic operators due to Gromov-Lawson, Anghel, Teleman, Booß-Bavnbek-Wojciechowski, et al. as special cases. In conjunction with additional conditions (like symmetry conditions), this theorem permits one to compute the analytical index of a given operator. In particular, we obtain new index formulas for elliptic pseudodifferential operators and quantized canonical transformations on manifolds with conical singularities.
We continue the investigation of the calculus of Fourier Integral Operators (FIOs) in the class of symbols with exit behaviour (SG symbols). Here we analyse what happens when one restricts the choice of amplitude and phase functions to the subclass of the classical SG symbols. It turns out that the main composition theorem, obtained in the environment of general SG classes, has a "classical" counterpart. As an application, we study the Cauchy problem for classical hyperbolic operators of order (1, 1); for such operators we refine the known results about the analogous problem for general SG hyperbolic operators. The material contained here will be used in a forthcoming paper to obtain a Weyl formula for a class of operators defined on manifolds with cylindrical ends, improving the results obtained in [9].
The problem of analytic representation of integrable CR functions on hypersurfaces with singularities is treated. The nature o singularities does not matter while the set of singularities has surface measure zero. For simple singularities like cuspidal points, edges, corners, etc., also the behaviour of representing analytic functions near singular points is studied.
We prove the existence of Hp(D)-limit of iterations of double layer potentials constructed with the use of Hodge parametrix on a smooth compact manifold X, D being an open connected subset of X. This limit gives us an orthogonal projection from Sobolev space Hp(D) to a closed subspace of Hp(D)-solutions of an elliptic operator P of order p ≥ 1. Using this result we obtain formulae for Sobolev solutions to the equation Pu = f in D whenever these solutions exist. This representation involves the sum of a series whose terms are iterations of double layer potentials. Similar regularization is constructed also for a P-Neumann problem in D.
We construct a new calculus of boundary value problems with the transmission property on a non-compact smooth manifold with boundary and conical exits to infinity. The symbols are classical both in covariables and variables. The operators are determined by principal symbol tuples modulo operators of lower orders and weights (such remainders are compact in weighted Sobolev spaces). We develop the concept of ellipticity, construct parametrices within the algebra and obtain the Fredholm property. For the existence of Shapiro-Lopatinskij elliptic boundary conditions to a given elliptic operator we prove an analogue of the Atiyah-Bott condition.
Crack problems are regarded as elements in a pseudo-differential algbra, where the two sdes int S± of the crack S are treated as interior boundaries and the boundary Y of the crack as an edge singularity. We employ the pseudo-differential calculus of boundary value problems with the transmission property near int S± and the edge pseudo-differential calculus (in a variant with Douglis-Nirenberg orders) to construct parametrices od elliptic crack problems (with extra trace and potential conditions along Y) and to characterise asymptotics of solutions near Y (expressed in the framework of continuous asymptotics). Our operator algebra with boundary and edge symbols contains new weight and order conventions that are necessary also for the more general calculus on manifolds with boundary and edges.
We prove the existence of a limit in Hm(D) of iterations of a double layer potential constructed from the Hodge parametrix on a smooth compact manifold with boundary, X, and a crack S ⊂ ∂D, D being a domain in X. Using this result we obtain formulas for Sobolev solutions to the Cauchy problem in D with data on S, for an elliptic operator A of order m ≥ 1, whenever these solutions exist. This representation involves the sum of a series whose terms are iterations of the double layer potential. A similar regularisation is constructed also for a mixed problem in D.
Content: Introduction 1 Anisotropic operators in a cylinder with a conical base 1.1 Manifolds with conical singularities and opertors of Fuchs type 1.2 Typical operators and symbol structures 2 Weighted wedge Sobolev spaces and edge asymptotics 2.1 Discrete edge asymptotics 2.2 Continuos edge asymptotics with discrete limit at infinity 2.3 Calculus with operator valued symbols 3 Corner asymptotics at infinity 3.1 The structure of singular functions 3.2 Operators with trace and potential conditions 3.3 Asymptotics and (anisotropic) elliptic regularity
For a domain D subset of IRn with singular points on the boundary and a weight function ω infinitely differentiable away from the singularpoints in D, we consider a C*-algebra G (D; ω) of operators acting in the weighted space L² (D, ω). It is generated by the operators XD F-¹ σ F XD where σ is a homogeneous function. We show that the techniques of limit operators apply to define a symbol algebra for G (D; ω). When combined with the local principle, this leads to describing the Fredholm operators in G (D; ω).
Content: 0.1 Preliminary Remarks Chapter 1: Noncommutative Analysis: Main Ideas, Definitions, and Theorems 1.1 Functions of One Operator (Functional Calculi) 1.2 Functions of Several Operators 1.3 Main Formulas of Operator Calculus 1.4 Main Tools of Noncommutative Analysis 1.5 Composition Laws and Ordered Representations