Pseudo-Differential Operators, Singularities, Applications [Paperback]

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 Sobolevs 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 G?rding inequality.- 2.5.1 G?rding inequality for elliptic differential operators.- 2.5.2 Sharp G?rding 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 opl“=