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# Pseudo-Differential Operators, Singularities, Applications

## byIouri Egorov, Bert-Wolfgang Schulze

### Paperback | October 16, 2012

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Pseudo-differential operators belong to the most powerful tools in the analysis of partial differential equations. Basic achievements in the early sixties have initiated a completely new understanding of many old and important problems in analy sis and mathematical physics. The standard calculus of pseudo-differential and Fourier integral operators may today be considered as classical. The development has been continuous since the early days of the first essential applications to ellip ticity, index theory, parametrices and propagation of singularities for non-elliptic operators, boundary-value problems, and spectral theory. The basic ideas of the calculus go back to Giraud, Calderon, Zygmund, Mikhlin, Agranovich, Dynin, Vishik, Eskin, and Maslov. Subsequent progress was greatly stimulated by the classical works of Kohn, Nirenberg and Hormander. In recent years there developed a new vital interest in the ideas of micro local analysis in connection with analogous fields of applications over spaces with singularities, e.g. conical points, edges, corners, and higher singularities. The index theory for manifolds with singularities became an enormous challenge for analysists to invent an adequate concept of ellipticity, based on corresponding symbolic structures. Note that index theory was another source of ideas for the later development of the theory of pseudo-differential operators. Let us mention, in particular, the fundamental contributions by Gelfand, Atiyah, Singer, and Bott.

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Title:Pseudo-Differential Operators, Singularities, ApplicationsFormat:PaperbackDimensions:353 pages, 0.01 × 0.01 × 0.02 inPublished:October 16, 2012Publisher:Springer-Verlag/Sci-Tech/TradeLanguage:English

The following ISBNs are associated with this title:

ISBN - 10:3034898207

ISBN - 13:9783034898201

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Table of Contents

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.