Inhalt

1 Sobolev spaces.- 1.1 Fourier transform.- 1.1.1 Definition.- 1.1.2 The Fourier transform in the Schwartz spaces.- 1.2 The first definition of the Sobolev space.- 1.2.1 The classical definition.- 1.2.2 The completeness of the classical Sobolev space.- 1.3 General definition of Sobolev spaces in ?n.- 1.3.1 General definition.- 1.3.2 Some properties of Sobolev spaces in ?n.- 1.4 Representation of a linear functional over Hs.- 1.5 Embedding theorems.- 1.5.1 Sobolev's theorem.- 1.5.2 Distributions with compact supports.- 1.5.3 Traces on the boundary.- 1.6 Sobolev spaces in a domain.- 1.6.1 Definition.- 1.6.2 The invariance under diffeomorphisms.- 1.6.3 The compactness of embeddings.- 2 Pseudo-differential Operators.- 2.1 The algebra of differential operators.- 2.1.1 Differential operators in ?n.- 2.1.2 Differential operators on a manifold.- 2.1.3 The cotangent space and the characteristic form.- 2.1.4 Fundamental solutions of differential operators with constant coefficients.- 2.1.5 Examples of fundamental solutions.- 2.1.6 Hypoelliptic operators.- 2.2 Basic properties of pseudo-differential operators.- 2.2.1 Definition and basic properties.- 2.2.2 Pseudo-differential operators as integral operators.- 2.2.3 Continuity in the Sobolev spaces.- 2.3 Calculus of pseudo-differential operators.- 2.3.1 A technical Lemma.- 2.3.2 The composition of pseudo-differential operators.- 2.3.3 A more general definition.- 2.3.4 Formally adjoint operators.- 2.4 Pseudo-differential operators on closed manifolds.- 2.4.1 Transformation of operators under a change of variables.- 2.4.2 Pseudo-differential operators on a manifold.- 2.5 Gårding inequality.- 2.5.1 Gårding inequality for elliptic differential operators.- 2.5.2 Sharp Gårding inequality for pseudo-differential operators.- 2.5.3 Some generalizations.- 3 Elliptic pseudo-differential operators.- 3.1 Parametrices of the elliptic operators.- 3.1.1 Definitions and a technical lemma.- 3.1.2 The construction of a parametrix.- 3.2 Elliptic operators on a manifold.- 3.2.1 Definitions.- 3.2.2 The parametrix construction.- 3.2.3 A priori estimates and regularity of solutions.- 3.2.4 The Fredholm property.- 3.2.5 Vanishing of the index.- 4 Elliptic boundary value problems.- 4.1 Model elliptic boundary value problems.- 4.1.1 Statement of the problem and the condition of ellipticity.- 4.1.2 Construction of a parametrix.- 4.2 Elliptic boundary value problems in a domain.- 4.2.1 Ellipticity condition.- 4.2.2 Examples.- 4.2.3 Construction of a parametrix.- 4.2.4 Continuity of the parametrices.- 4.2.5 Fredholm property.- 4.2.6 Necessity of the ellipticity condition.- 5 Kondratiev's theory.- 5.1 A model problem.- 5.2 The general problem.- 5.2.1 The conditions on a domain and differential operators.- 5.2.2 Functional spaces.- 5.2.3 The statement of the general boundary value problem and main results.- 5.3 The boundary value problem in an infinite cone for operators with constant coefficients.- 5.3.1 The statement of the problem and its transformations.- 5.3.2 The resolution of the model boundary value problem.- 5.3.3 The asymptotics of the solution.- 5.3.4 Lemmas.- 5.3.5 The proof of Theorem 3.- 5.4 Equations with variable coefficients in an infinite cone.- 5.4.1 Conditions on the coefficients.- 5.4.2 Lemmas.- 5.4.3 The existence of the solution.- 5.4.4 The smoothness of the solution.- 5.5 The boundary value problem in a bounded domain.- 5.5.1 Lemmas.- 5.5.2 The construction of the parametrix.- 5.5.3 Proof of Theorem 1.- 5.5.4 Smoothness of solutions.- 5.5.5 The solution of the boundary value problem in usual Sobolev spaces.- 6 Non-elliptic operators; propagation of singularities.- 6.1 Canonical transformations and Fourier integral operators.- 6.1.1 Definitions.- 6.1.2 Examples.- 6.1.3 Fourier integral operators and canonical transformations.- 6.1.4 Canonical transformations and quadratic forms.- 6.1.5 Reduction of operators of principal type to canonical forms.- 6.2 Wave fronts of distributions.- 6.2.1 Definitions.- 6.2.2 Examples.- 6.2.3 Properties of wave fronts.- 6.2.4 Wave fronts under push-forwards and pull-backs.- 6.2.5 Wave fronts and traces of distributions on a manifold of a lower dimension.- 6.2.6 Products of distributions.- 6.3 Wave fronts and Fourier integral operators.- 6.3.1 Wave fronts and integral operators.- 6.3.2 Wave fronts and pseudo-differential operators.- 6.4 Propagation of singularities.- 6.4.1 Propagation of singularities for operators of real principal type.- 6.4.2 Propagation of singularities in the Sobolev spaces.- 6.4.3 Solvability of real principal type.- 6.5 The Cauchy problem for a strongly hyperbolic equation.- 6.5.1 The Cauchy problem for the wave equation.- 6.5.2 The Cauchy problem for a hyperbolic equation.- 6.5.3 The construction of the phase function and the symbol.- 6.5.4 The Cauchy problem for a hyperbolic system of first order.- 7 Pseudo-differential operators on manifolds with conical and edge singularities; motivation and technical preparations.- 7.1 The general background.- 7.1.1 The program of the analysis of manifolds with singularities.- 7.1.2 Typical differential operators on manifolds with conical singularities.- 7.1.3 The typical differential operators on manifolds with edges and corners.- 7.2 Parameter-dependent pseudo-differential operators and operator-valued Mellin symbols.- 7.2.1 Additional material on pseudo-differential operators on closed compact C? manifolds.- 7.2.2 The parameter-dependent calculus; reductions of orders.- 7.2.3 Mellin pseudo-differential operators with operator-valued symbols.- 7.2.4 Kernel cut-off and operator-valued holomorphic Mellin symbols.- 7.2.5 Meromorphic Fredholm families.- 8 Pseudo-differential operators on manifolds with conical singularities.- 8.1 The cone algebra with asymptotics.- 8.1.1 Weighted Sobolev spaces with asymptotics and Green operators.- 8.1.2 Smoothing Mellin operators.- 8.1.3 A Mellin operator convention.- 8.1.4 The cone algebra.- 8.1.5 Ellipticity and regularity with asymptotics.- 8.2 The algebra on the infinite cone.- 8.2.1 Symbols in ?n with exit behaviour.- 8.2.2 Classical symbols.- 8.2.3 Pseudo-differential operators in ?n with exit behaviour.- 8.2.4 The calculus on the infinite cylinder.- 8.2.5 The cone algebra on X^.- 9 Pseudo-differential operators on manifolds with edges.- 9.1 Pseudo-differential operators with operator-valued symbols.- 9.1.1 The operator-valued symbol spaces.- 9.1.2 Pseudo-differential operators.- 9.1.3 Abstract wedge Sobolev spaces.- 9.2 The edge symbolic calculus.- 9.2.1 Green symbols.- 9.2.2 Smoothing Mellin symbols.- 9.2.3 Complete edge symbols.- 9.3 Edge pseudo-differential operators.- 9.3.1 Edge Sobolev spaces with discrete asymptotics.- 9.3.2 The algebra of edge pseudo-differential operators.- 9.3.3 Ellipicity and regularity with discrete edge asymptotics.- 9.3.4 Global constructions and Fredholm property.- 9.4 Applications, examples and remarks.- 9.4.1 Boundary value problems as particular edge problems.- 9.4.2 The nature of asymptotics in singular configurations.- 9.4.3 Remarks on the role of the edge trace and potential conditions.