Quadratic estimates and functional calculi of perturbed Dirac operators

We prove quadratic estimates for complex perturbations of Dirac-type operators, and thereby show that such operators have a bounded functional calculus. As an application we show that spectral projections of the Hodge–Dirac operator on compact manifolds depend analytically on L_∞ changes in the metric. We also recover a unified proof of many results in the Calderón program, including the Kato square root problem and the boundedness of the Cauchy operator on Lipschitz curves and surfaces.     

The boundedness and the fredholm property of integral operators with anisotropically homogeneous kernels of compact type and variable coefficients

In the space L _p (? ⁿ ), 1 < p < ??, we study a new wide class of integral operators with anisotropically homogeneous kernels. We obtain sufficient conditions for the boundedness of operators from this class. We consider the Banach algebra generated by operators with anisotropically homogeneous kernels of compact type and multiplicatively slowly oscillating coefficients. We establish a relationship between this algebra and multidimensional convolution operators, and construct a symbolic calculus for it. We also obtain necessary and sufficient conditions for the Fredholm property of operators from this algebra.     

Index in K-Theory for Families of Fibred Cusp Operators

A families index theorem in K-theory is given for the setting of Atiyah, Patodi, and Singer of a family of Dirac operators with spectral boundary condition. This result is deduced from such a K-theory index theorem for the calculus of cusp, or more generally fibred-cusp, pseudodifferential operators on the fibres (with boundary) of a fibration; a version of Poincaré duality is also shown in this setting, identifying the stable Fredholm families with elements of a bivariant K-group. (Received: February 2006)     

Quadratic centers defining elliptic surfaces

Let X be a quadratic vector field with a center whose generic orbits are algebraic curves of genus one. To each X we associate an elliptic surface (a smooth complex compact surface which is a genus one fibration). We give the list of all such vector fields and determine the corresponding elliptic surfaces.     

Inverse problems and index formulae for Dirac operators

We consider a Dirac-type operator DP on a vector bundle V over a compact Riemannian manifold (M,g) with a non-empty boundary. The operator DP is specified by a boundary condition P(u_|∂M)=0 where P is a projector which may be a non-local, i.e., a pseudodifferential operator. We assume the existence of a chirality operator which decomposes L²(M,V) into two orthogonal subspaces X₊⊕X₋. Under certain conditions, the operator DP restricted to X₊ and X₋ defines a pair of Fredholm operators which maps X₊→X₋ and X₋→X₊ correspondingly, giving rise to a superstructure on V. In this paper we consider the questions of determining the index of DP and the reconstruction of and DP from the boundary data on ∂M. The data used is either the Cauchy data, i.e., the restrictions to ∂M×R₊ of the solutions to the hyperbolic Dirac equation, or the boundary spectral data, i.e., the set of the eigenvalues and the boundary values of the eigenfunctions of DP. We obtain formulae for the index and prove uniqueness results for the inverse boundary value problems. We apply the obtained results to the classical Dirac-type operator in M×C⁴, M⊂R³.     

On the algebra of multidimensional integral operators with homogeneous SO(n)-invariant kernels and weakly radially oscillating coefficients

In this paper, we study the Banach algebra B generated by multidimensional integral operators whose kernels are homogeneous functions of degree (?n) invariant with respect to the rotation group SO(n) and by the operators of multiplication by radial weakly oscillating functions. A symbolic calculus is developed for the algebra 25. The Fredholm property and the formula for calculating the index are described in terms of this calculus.     

Products of longitudinal pseudodifferential operators on flag varieties

Associated to each set S of simple roots of SL(n,C) is an equivariant fibration X→XS of the complete flag variety X of Cn. To each such fibration we associate an algebra JS of operators on L²(X), or more generally on L²-sections of vector bundles over X. This ideal contains, in particular, the longitudinal pseudodifferential operators of negative order tangent to the fibres. Together, they form a lattice of operator ideals whose common intersection is the compact operators. Thus, for instance, the product of negative order pseudodifferential operators along the fibres of two such fibrations, X→XS and X→XT, is a compact operator if S∪T is the full set of simple roots. The construction of the ideals uses noncommutative harmonic analysis, and hinges upon a representation theoretic property of subgroups of SU(n), which may be described as ‘essential orthogonality of subrepresentations’.     

Polynomially Compact-Like Strongly Continuous Semigroups

J. R. Cuthbert gave some results about the class of semigroups of operators (T(t)) _t0 on a Banach space X which have the property that for some t>0, T(t)–I is compact. Cuthbert's results were extended to various classes of operators generalizing the set of compact operators such as the ideal of Fredholm perturbations or the set of Riesz operators. The purpose of the present paper is to give further results in this direction. Thus we consider semigroups for which there exists a non-trivial polynomial p()C[z] such that, for some t>0, p(T(t))J(X) where J(X) is an arbitrary proper two-sided ideal of the algebra (X) contained in the set of Fredholm perturbations.     

Cyclic cohomology and the family Lefschetz theorem

Any closed current on the base of a compact fibration gives rise to a cyclic cocycle on the smooth convolution algebra. We prove that such cocycle furnishes additive maps from the vertically equivariant K-theory to the scalars. This enables to associate to any closed current on the base of the fibration, a Lefschetz formula for fiber-preserving isometries. Using geometric operators on the base, we deduce the integrality of some characteristic numbers. Received: 28 June 2001 / Published online: 1 February 2002     

Patterson–Sullivan distributions in higher rank

For a compact locally symmetric space X _Γ of non-positive curvature, we consider sequences of normalized joint eigenfunctions which belong to the principal spectrum of the algebra of invariant differential operators. Using an h-pseudo-differential calculus on X _Γ , we define and study lifted quantum limits as weak*-limit points of Wigner distributions. The Helgason boundary values of the eigenfunctions allow us to construct Patterson–Sullivan distributions on the space of Weyl chambers. These distributions are asymptotic to lifted quantum limits and satisfy additional invariance properties, which makes them useful in the context of quantum ergodicity. Our results generalize results for compact hyperbolic surfaces obtained by Anantharaman and Zelditch.     

Shape fibrations and strong shape theory

The notion of shape fibration was introduced by Marde?i? and Rushing. In this paper we use ‘fibrant space’ techniques in strong shape theory to prove that every shape fibration p:E → B of compact metric spaces is contained in a map of fibrant spaces p′:E′→B′ which enjoys a certain lifting property and whose homotopy properties reflect the strong shape properties of the map p. Standard methods for studying Hurewicz fibrations are readily applied to the map p' and in this way we obtain a number of strong shape generalizations of results of Marde?i? and Rushing. We also prove the following theorem which answers a question of Rushing: A shape fibration of compact metric spaces which is a strong shape equivalence is an hereditary shape equivalence. Since the converse was known, this gives a characterization of hereditary shape equivalences.     

Geometric BVPs, Hardy spaces, and the Cauchy integral and transform on regions with corners

In this paper we give a new perspective on the Cauchy integral and transform and Hardy spaces for Dirac-type operators on manifolds with corners of codimension two. Instead of considering Banach or Hilbert spaces, we use polyhomogeneous functions on a geometrically “blown-up” version of the manifold called the total boundary blow-up introduced by Mazzeo and Melrose [R.R. Mazzeo, R.B. Melrose, Analytic surgery and the eta invariant, Geom. Funct. Anal. 5 (1) (1995) 14-75]. These polyhomogeneous functions are smooth everywhere on the original manifold except at the corners where they have a “Taylor series” (with possible log terms) in polar coordinates. The main application of our analysis is a complete Fredholm theory for boundary value problems of Dirac operators on manifolds with corners of codimension two.     

Reduction of the calculus of pseudodifferential operators on a noncompact manifold to the calculus on a compact manifold of doubled dimension

The paper is devoted to the exposition of results announced in [1]. We construct a reduction (following an idea of S. P. Novikov) of the calculus of pseudodifferential operators on Euclidean space ? ⁿ to a similar calculus in the space of sections of a one-dimensional fiber bundle ξ on the 2n-dimensional torus $\mathbb{T}^{2n} $ . This reduction enables us to identify the Schwartz space on ? ⁿ with the space of smooth sections Γ^∞(T ²ⁿ , ξ), compare the Sobolev norms on the corresponding spaces and pseudodifferential operators in them, and describe the class of elliptic operators that reduce to Fredholm operators in Sobolev norms. Thus, for a natural class of elliptic pseudodifferential operators on a noncompact manifold of ? ⁿ , we construct an index formula in accordance with the classical Atya-Singer formula.     

Computation of the Maslov index and the spectral flow via partial signatures

Given a smooth Lagrangian path, both in the finite and in the infinite dimensional (Fredholm) case, we introduce the notion of partial signatures at each isolated intersection of the path with the Maslov cycle. For real-analytic paths, we give a formula for the computation of the Maslov index using the partial signatures; a similar formula holds for the spectral flow of real-analytic paths of Fredholm self-adjoint operators on real separable Hilbert spaces. As applications of the theory, we obtain a semi-Riemannian version of the Morse index theorem for geodesics with possibly conjugate endpoints, and we prove a bifurcation result at conjugate points along semi-Riemannian geodesics. To cite this article: R. Giambò et al., C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

On stability of index of Fredholm complexes on the C* -algebra

This article deals with the index of Fredholm complexes of Λ-operators of the Hilbert Λ-modulus on C*-algebra. For this class of operators necessary and sufficient conditions in order to be a Fredholm, are obtained. Based on these results, a notion of Fredholm complex and its index is introduced. For this index, a stability theorem related to various perturbations is proved. In the second part of the article, a completation of a semigroup Fredholm complexes is analysed. It is proved that the group K _G (X, Λ) is the completation of G ? Λ-fibration of the above group on the compact space X.     

Logarithmic Terms in Trace Expansions of Atiyah–Patodi–Singer Problems

For Dirac-type operator D on a manifold X with a spectral boundarycondition (defined by a pseudodifferential projection), the associated heatoperator trace has an expansion in integer and half-integer powers and log-powersof t; the interest in the expansion coefficients goes back to the work of Atiyah,Patodi and Singer. In the product case considered by APS, it is known that allthe log-coefficients vanish when dim X is odd, whereas the log-coefficients atinteger powers vanish when dim X is even. We investigate here whether this partialvanishing of logarithms holds more generally. One type of result, shown forgeneral D with well-posed boundary conditions, is that a perturbation of Dby a tangential differential operator vanishing to order k on the boundaryleaves the first k log-power terms invariant (and the nonlocal power termsof the same degree are only locally perturbed). Another type of result is thatfor perturbations of the APS product case by tangential operators commuting withthe tangential part of D, all the logarithmic terms vanish when dim X is odd(whereas they can all be expected to be nonzero when dim X is even). The treatmentis based on earlier joint work with R. Seeley and a recent systematic parameter-dependentpseudodifferential boundary operator calculus, applied to the resolvent.     

On the constant and step in Jackson’s inequality for best approximations by trigonometric polynomials and by Haar polynomials

We consider the C*-algebra generated by multidimensional integral operators with (?n)th-order homogeneous kernels and by the operators of multiplication by oscillating coefficients of the form |x|^iα. For this algebra, we construct an operator symbolic calculus and obtain necessary and sufficient conditions for the Fredholm property of an operator in terms of this calculus.     

On a calculus for a generalized scalar operator

Let $\text{X}$ be a Banach space; S and T bounded scalar-type operators in $\text{X}$. Define Δ on the space of bounded operators on $\text{X}$ by ΔX = TX ? XS if X is a bounded operator. We set up a calculus for Δ which allows us to consider f(Δ), for f a complex-valued bounded Borel measurable function on the spectrum of Δ, as an operator in the space of bounded operators whose domain is a subspace of operators which we call measure generating. This calculus is used to obtain some results on when the kernel of Δ is a complemented subspace of the space of bounded operators on $\text{X}$.     

Boundary value problems with global projection conditions

Parametrices of elliptic boundary value problems for differential operators belong to an algebra of pseudodifferential operators with the transmission property at the boundary. However, generically, smooth symbols on a manifold with boundary do not have this property, and several interesting applications require a corresponding more general calculus. We introduce here a new algebra of boundary value problems that contains Shapiro-Lopatinskij elliptic as well as global projection conditions; the latter ones are necessary, if an analogue of the Atiyah-Bott obstruction does not vanish. We show that every elliptic operator admits (up to a stabilisation) elliptic conditions of that kind. Corresponding boundary value problems are then Fredholm in adequate scales of spaces. Moreover, we construct parametrices in the calculus.