Complexes of Differential Operators

Transcription

1 Complexes of Differential Operators by Nikolai N. Tarkhanov Institute of Physics, Siberian Academy of Sciences, Krasnoyarsk, Russia KLUWER ACADEMIC PUBLISHERS DORDRECHT / BOSTON / LONDON

2 Contents Preface to the English Translation Preface to the Russian Edition xv xvii Introduction Timeliness Directions Purpose Methods Approach Results Authorship 7 List of Main Notations 9 1 Resolution of Differential Operators Differential Complexes and Their Cohomology 11, Manifolds countable at infinity Partitions of unity i Sections of vector bundles Spaces of smooth sections Holder spaces Sobolev spaces Besov spaces Spaces of generalized sections Differential operators Information from homological algebra Differential complexes Elliptic complexes Transposed complex The Hilbert Resolution of a Differential Operator with Constant Coefficients Compatibility problem for overdetermined systems of differential equations Formal theory in the case of constant coefficients (Hilbert complexes) 30

3 Vlll Elliptic Hilbert complexes Existence theory in the case of constant coefficients Recovering a solution of an elliptic system by means of a "scalar" component The de Rham complex The Dolbeault complex The Koszul complexes The Asada complexes The Spencer Resolution of a Formally Integrable Differential Operator Geometrical viewpoint for differential operators Regularity conditions Formally exact differential complexes Formal integrability Involutive differential operators Normalized differential operators Commutativity relations Compatibility complexes for normalized operators Compatibility complexes for arbitrary regular operators Existence theory in the real analytic case The first Spencer sequence The second Spencer sequence Existence theory in the differentiate case Tensor products of differential complexes and Kiinneth's formula Topological tensor product of locally convex spaces Grothendieck's theorem on the topological tensor product of complexes Tensor product of differential complexes. A Kunneth formula, Solvability of systems of differential equations with a parameter Examples Cochain mappings of differential complexes A brief survey of operator kernels. Schwartz's kernel theorem Kernel of the identity operator Pseudo-differential operators Kernels of pseudo-differential operators. Seeley's theorem The complex of homomorphisms A differential version of the complex of homomorphisms A Kunneth formula Examples 88 Parametrices and Fundamental Solutions of Differential Complexes Parametrices of Differential Complexes Parametrices and the homotopy formula Parametrices and cohomologies of differential complexes Formal adjoint operator 95

4 Table of Contents ix Parametrices of elliptic complexes. A theorem of Atiyah and Bolt Hypoellipticity of elliptic complexes Analyticity of cohomologies of elliptic complexes Hodge Theory for Elliptic Complexes on Compact Manifolds Harmonic spaces Hodge theory Orthogonal decompositions Fundamental Solutions of Differential Complexes Fundamental solutions Existence of local fundamental solutions Fundamental solution at degree q. A generalization of a theorem of Malgrange Kernels of convolution type Fundamental solutions of convolution type of complexes of differential operators with constant coefficients in R Green Operators for Differential Operators and the Homotopy Formula on Manifolds with Boundary Poly differential operators with values in the space of differential forms Green operator of a differential operator The homotopy formula on manifolds with boundary Conservation laws for solutions of overdetermined systems The Most Immediate Corollaries and Examples The homotopy formula for elliptic complexes on a manifold with boundary Fundamental solution of convolution type for the de Rham ; complex inr n Fundamental solution of convolution type for the Dolbeault complex inc Green formulas Fundamental solution of convolution type for the Koszul complexl Homotopy formula on manifolds with boundary for Asada complexes Sokhotskii-Plemelj Formulas for Elliptic Complexes Formally Non-characteristic Hypersurfaces for Differential Complexes. The Tangential Complex Preliminaries Quotient complexes Complexes of jet spaces on S Cauchy data complexes on S Tangential complexes Formally non-characteristic hypersurfaces Decomposition of a complex 152

5 3.1.8 Existence and uniqueness of formal solutions to a non-characteristic local Cauchy problem A condition for a hypersurface to be formally non-characteristic locally Completion of the proof of Theorem Sokhotskii-Plemelj Formulas for Elliptic Complexes of First Order Differential Operators Historical reference Tangential and normal components of sections The Sokhotskii-Plemelj formulas Structure of the Cauchy data for the differential operator A; Formal version of the Cauchy-Kovalevskaya Theorem for the differential operator A Green's identity for the differential operator A Proof of Theorem Further results A description of the tangential complex Generalization of the Sokhotskii-Plemelj Formulas to the Case of Arbitrary Elliptic Complexes Special parametrix of an elliptic complex Behaviour of potentials near the surface of integration Jump formulas Formal version of the Cauchy-Kovalevskaya Theorem for the differential operator A (general case) Decomposition in jet spaces Dual decompositions Formal version of the Cartan-Kahler Theorem for elliptic complexes A description of the tangential complex for an arbitrary elliptic): complex Jump formulas for the volume potential Jump formulas for potentials with smooth densities Integral Formulas for Elliptic Complexes. Morera's Theorem Historical remarks Preliminary results A complete description of cocycles of the complex (E\y) The spaces W( F ) A complete description of harmonic sections of E'\y Morera's theorem Multiplication of Currents via Their Harmonic Representations A brief survey of representations of distributions Weak boundary values of harmonic forms of finite order of growth Harmonic representations of currents with compact support Harmonic representation of arbitrary currents 202

6 Table of Contents xi On multiplication of distributions Exterior product of currents Index of intersection of currents Boundary Problems for Differential Complexes The Neumann-Spencer Problem Representation of cohomologies of differential complexes on manifolds with boundary I?-version of the Neumann problem Examples Reducing the question of solvability to a subelliptic estimate.. Ill A geometrical condition for solvability of the Neumann problem The Z 2 -Cohomologies of Differential Complexes and the Bergman Projector L 2 -cohomologies Adjoint complex Weak version of the Neumann problem Weak orthogonal decomposition The Neumann operator The Dirichlet norm A representation of I?-cohomologies of differential complexes The Bergman projector and a generalization of a theorem of Bungart The Mayer-Vietoris sequence Basic example Long cohomological sequence The Mayer-Vietoris sequence for open sets (the case of arbi- i trary supports) The Mayer- Vietoris sequence for open sets (the case of compact supports) The Mayer-Vietoris sequence for closed sets The Mayer-Vietoris sequence for elliptic complexes Examples The Cauchy problem for cohomology classes of differential complexes The Cauchy problem for sections Reducing to boundary differential complexes f The Cauchy problem for cohomology classes Representation of cohomologies of a differential complex by means of cohomologies of the tangential complex Removable singularities of solutions of overdetermined systems and a generalization of Bochner 's theorem The Dirichlet problem and representation of cohomologies of the complex of spaces of sections with zero Cauchy data A generalization of the theorem of Kohn and Rossi The Kernel Approach to Solving the Equation Pu f 262

7 Xll General remark Decomposition of the fundamental solution Special homotopy formulas on manifolds with boundary Kernels for solutions of the equation Pu = f 266 Duality Theory for Cohomologies of Differential Complexes The Poincare Duality and the Alexander-Pontryagin Duality Duality Lemma Poincare duality Alexander-Pontryagin duality The Weil Homomorphism Preliminaries The Weil homomorphism Properties of the Weil homomorphism A localized version of the Weil homomorphism The adjoint Weil homomorphism Properties of the adjoint Weil homomorphism A localized version of the adjoint Weil homomorphism Pairing at sequences of Weil representatives A remark on hypoelliptic complexes Integral Formulas Connected by the Weil Homomorphism U-resolution of a singular cycle Homologies generated by cycles and conservation laws Abstract integral representations for solutions of overdetermined systems Generalization of the Cauchy- Weil formula C polyhedra in general position with respect to a covering Overdetermined homogeneous systems of differential equations with constant coefficients '" A Cauchy-Fantappie formula for Koszul complexes Cech parametrices Grothendieck's Theorem on Cohomology Classes Regular at Infinity History of the question Sections regular at infinity Cohomology classes regular at infinity Example Grothendieck Duality for Elliptic Complexes Survey of results The space H q ( {E\ Y )) The space HimSiE'lxw)) The approximation condition Grothendieck duality A preparatory lemma Proof of the-grothendieck duality theorem Some consequences 326

8 TaWe of Contents xiii 6 The Atiyah-Bott-Lefschetz Theorem on Fixed Points for Elliptic Complexes The Argument Principle for Elliptic Complexes A brief history Chains defined by smooth mappings The argument principle Proof of Theorem The logarithmic residue formula for holomorphic functions A generalization of the Kronecker formula Mappings preserving the dimension An analog of the Cauchy-Fantappie formula for closed differential forms An Integral Formula for the Lefschetz Number A brief survey of the Lefschetz theory for elliptic complexes The Lefschetz number of an endomorphism of an elliptic complex Main integral formula for the Lefschetz number Geometric endomorphisms A modification of the main formula for geometric endomorphisms Localization of the Lefschetz number Local index of an isolated component of the set of fixed points The Atiyah-Bott Formula for Simple Fixed Points Preliminaries The Atiyah-Bott formula Isolated Components of the Set of Fixed Points General remarks Uniqueness of the local index A cohomological formula for the local index Some Examples for the Classical Complexes The classical Lefschetz formula The holomorphic Lefschetz formula Other examples and applications 367 Bibliography 369 Name Index 387 Subject Index 390 Index of Notation 394