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Multi-parameter singular integrals /
This book develops a new theory of multi-parameter singular integrals associated with Carnot-Carathéodory balls. Brian Street first details the classical theory of Calderón-Zygmund singular integrals and applications to linear partial differential equations. He then outlines the theory of multi-pa...
| Clasificación: | Libro Electrónico |
|---|---|
| Autor principal: | |
| Formato: | Electrónico eBook |
| Idioma: | Inglés |
| Publicado: |
Princeton :
Princeton University Press,
[2014]
|
| Colección: | Annals of mathematics studies ;
no. 189. |
| Temas: | |
| Acceso en línea: | Texto completo |
Tabla de Contenidos:
- Cover; Contents; Preface; 1 The Calderón-Zygmund Theory I: Ellipticity; 1.1 Non-homogeneous kernels; 1.2 Non-translation invariant operators; 1.3 Pseudodifferential operators; 1.4 Elliptic equations; 1.5 Further reading and references; 2 The Calderón-Zygmund Theory II: Maximal Hypoellipticity; 2.1 Vector fields with formal degrees; 2.2 The Frobenius theorem; 2.2.1 Scaling techniques; 2.2.2 Ideas in the proof; 2.3 Vector fields with formal degrees revisited; 2.4 Maximal hypoellipticity; 2.4.1 Subellipticity; 2.4.2 Scale invariance; 2.5 Smooth metrics and bump functions; 2.6 The sub-Laplacian.
- 2.7 The algebra of singular integrals2.7.1 More on the cancellation condition; 2.8 The topology; 2.9 The maximal function; 2.10 Non-isotropic Sobolev spaces; 2.11 Maximal hypoellipticity revisited; 2.11.1 The Kohn Laplacian; 2.12 Exponential maps; 2.13 Nilpotent Lie groups; 2.14 Pseudodifferential operators; 2.15 Beyond Hörmander's condition; 2.15.1 More on the assumptions; 2.15.2 When the vector fields span; 2.15.3 When the vector fields do not span; 2.15.4 A Littlewood-Paley theory; 2.15.5 The role of real analyticity; 2.16 Further reading and references.
- 3 Multi-parameter Carnot-Carathéodory Geometry3.1 Assumptions on the vector fields; 3.2 Some preliminary estimates; 3.3 The maximal function; 3.4 A Littlewood-Paley theory; 3.5 Further reading and references; 4 Multi-parameter Singular Integrals I: Examples; 4.1 The product theory of singular integrals; 4.1.1 Non-isotropic Sobolev spaces; 4.1.2 Further reading and references; 4.2 Flag kernels on graded groups and beyond; 4.2.1 Non-isotropic Sobolev spaces; 4.2.2 Further reading and references; 4.3 Left and right invariant operators; 4.3.1 An example of Kohn.
- 4.3.2 Further reading and references4.4 Carnot-Carathéodory and Euclidean geometries; 4.4.1 The [omitted]-Neumann problem; 4.4.2 Further reading and references; 5 Multi-parameter Singular Integrals II: General Theory; 5.1 The main results; 5.1.1 Non-isotropic Sobolev spaces; 5.1.2 Multi-parameter pseudodifferential operators; 5.1.3 Adding parameters; 5.1.4 Pseudolocality; 5.2 Schwartz space and product kernels; 5.3 Pseudodifferential operators and A[sub(3)] ⁶"A[sub(4)]; 5.4 Elementary operators and A[sub(4)] ⁶"A[sub(3)]; 5.5 A[sub(4)] ⁶"A[sub(2)] ⁶"A[sub(1)]; 5.6 A[sub(1)] ⁶"A[sub(4)].
- 5.7 The topology5.8 Non-isotropic Sobolev spaces; 5.9 Adding parameters; 5.10 Pseudolocality; 5.10.1 Operators on a compact manifold; 5.11 Examples; 5.11.1 Euclidean vector fields; 5.11.2 Hörmander vector fields and other geometries; 5.11.3 Carnot-Carathéodory and Euclidean geometries; 5.11.4 An Example of Kohn; 5.11.5 The product theory of singular integrals; 5.12 Some generalizations; 5.13 Closing remarks; A Functional Analysis; A.1 Locally convex topological vector spaces; A.1.1 Duals and distributions; A.2 Tensor Products; B Three Results from Calculus; B.1 Exponential of vector fields.