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Hormander Operators [Kõva köide]

(Politecnico Di Milano, Italy), (Univ Of Bergamo, Italy)
  • Formaat: Hardback, 724 pages
  • Ilmumisaeg: 27-Dec-2022
  • Kirjastus: World Scientific Publishing Co Pte Ltd
  • ISBN-10: 9811261687
  • ISBN-13: 9789811261688
Teised raamatud teemal:
Hormander OperatorsSuurem pilt
  • Formaat: Hardback, 724 pages
  • Ilmumisaeg: 27-Dec-2022
  • Kirjastus: World Scientific Publishing Co Pte Ltd
  • ISBN-10: 9811261687
  • ISBN-13: 9789811261688
Teised raamatud teemal:
Püsilink: https://www.kriso.ee/db/9789811261688.html
Märksõnad:
Hörmander operators are a class of linear second order partial differential operators with nonnegative characteristic form and smooth coefficients, which are usually degenerate elliptic-parabolic, but nevertheless hypoelliptic, that is highly regularizing. The study of these operators began with the 1967 fundamental paper by Lars Hörmander and is intimately connected to the geometry of vector fields.Motivations for the study of Hörmander operators come for instance from Kolmogorov-Fokker-Planck equations arising from modeling physical systems governed by stochastic equations and the geometric theory of several complex variables. The aim of this book is to give a systematic exposition of a relevant part of the theory of Hörmander operators and vector fields, together with the necessary background and prerequisites.The book is intended for self-study, or as a reference book, and can be useful to both younger and senior researchers, already working in this area or aiming to approach it.
Foreword xi
Introduction xiii
0.1 Prologue xiii
0.2 Scope and structure of the book xviii
0.3 Why study Hormander operators? xxii
1 Basic geometry of vector fields
1(66)
1.1 Introduction
1(1)
1.2 Exponentials and commutators of vector fields
2(6)
1.3 Lie algebras, Hormander's condition, Hormander operators
8(9)
1.4 The control distance
17(4)
1.5 The weighted control distance
21(3)
1.6 Connectivity
24(11)
1.7 Other properties related to connectivity
35(2)
1.8 Maximum principles for degenerate elliptic operators
37(4)
1.9 Propagation of maxima and strong maximum principle for sum of squares operators
41(7)
1.10 Propagation of maxima for operators with drift
48(8)
1.11 Some examples of explicit computations with the control distance
56(9)
1.12 Notes
65(2)
2 Function spaces defined by systems of vector fields
67(26)
2.1 Sobolev spaces induced by vector fields
67(13)
2.2 Holder spaces induced by Hormander vector fields
80(10)
2.3 Notes
90(3)
3 Homogeneous groups in RN
93(60)
3.1 Homogeneous groups
94(12)
3.2 Homogeneous Lie algebras of invariant vector fields on a homogeneous group
106(10)
3.3 Exponential maps on a homogeneous group
116(2)
3.4 Convolution and mollifiers on a homogeneous group
118(5)
3.5 Homogeneous stratified Lie groups and Lie algebras, and their control distance
123(5)
3.6 Connectivity matters and Poincare inequality on stratified groups
128(4)
3.7 Weak solutions to Dirichlet problems for divergence form equations structured on vector fields
132(3)
3.8 Homogeneous stratified Lie algebras and Lie groups of type II
135(6)
3.9 Distributions on homogeneous groups
141(2)
3.10 Examples of homogeneous groups and homogeneous Hormander operators
143(7)
3.11 Notes
150(3)
4 Hypoellipticity of sublaplacians on Carnot groups
153(38)
4.1 Introduction, statement of the main results and strategy of the proofs
153(3)
4.2 Notation and preliminary facts about Sobolev spaces and finite differences
156(7)
4.3 Regularity estimates for the canonical sublaplacian
163(11)
4.4 Hypoellipticity of the canonical sublaplacian
174(10)
4.5 General sublaplacians and uniform estimates
184(6)
4.6 Notes
190(1)
5 Hypoellipticity of general Hormander operators
191(56)
5.1 Introduction
191(1)
5.2 The Fourier transform on the Schwartz space S(Rn) and on tempered distributions
192(3)
5.3 Fractional order Sobolev spaces
195(4)
5.4 Some classes of operators on S(Rn)
199(18)
5.5 Subelliptic estimates
217(12)
5.6 Localized subelliptic estimate
229(3)
5.7 Hypoellipticity of Hormander operators
232(6)
5.8 Uniform subelliptic estimates
238(4)
5.9 Some applications of the subelliptic estimates
242(3)
5.10 Notes
245(2)
6 Fundamental solutions of Hormander operators
247(44)
6.1 Fundamental solutions and solvability of general Hormander operators
248(6)
6.2 Homogeneous Hormander operators
254(9)
6.3 Existence of a global homogeneous fundamental solution and uniform estimates
263(5)
6.4 Properties of the global fundamental solution
268(17)
6.5 Some explicit examples of fundamental solutions on homogeneous groups
285(3)
6.6 Notes
288(3)
7 Real analysis and singular integrals in locally doubling metric spaces
291(46)
7.1 Introduction
291(4)
7.2 Locally doubling metric spaces
295(3)
7.3 Localized kernels of singular and fractional integrals
298(3)
7.4 Singular and fractional integrals on Holder spaces
301(8)
7.5 L2 continuity of singular integrals via continuity on Cα
309(3)
7.6 Local maximal function and fractional integrals on Lp spaces
312(6)
7.7 Calderon-Zygmund theory in locally doubling metric spaces
318(9)
7.8 Integral characterization of Holder continuity
327(4)
7.9 Some geometric results
331(4)
7.10 Notes
335(2)
8 Sobolev and Holder estimates for Hormander operators on groups
337(62)
8.1 Introduction
337(8)
8.2 Homogeneous kernels on G, fractional integrals and Sobolev embeddings
345(9)
8.3 Singular integrals associated to homogeneous kernels of type 0
354(6)
8.4 Global Sobolev estimates
360(10)
8.5 Local Sobolev estimates
370(8)
8.6 Holder estimates for solutions of Lu = ƒ
378(19)
8.7 Notes
397(2)
9 More geometry of vector fields: metric balls and equivalent distances
399(80)
9.1 Introduction and statement of the main results
399(7)
9.2 Dependence of the constants
406(2)
9.3 The Baker-Campbell-Hausdorff formula
408(12)
9.4 Suboptimal bases and their properties
420(17)
9.5 Structure of metric balls
437(21)
9.6 Local equivalence of the distances d, d*
458(4)
9.7 Segment properties and the global doubling condition
462(4)
9.8 Proof of the BCH formula for formal series and other consequences
466(11)
9.9 Notes
477(2)
10 Lifting and approximation
479(56)
10.1 Motivation and statement of the main results
479(7)
10.2 Lifting of Hormander vector fields
486(12)
10.3 Approximation of free vector fields with left invariant homogeneous vector fields
498(15)
10.4 Some geometry of free lifted vector fields
513(10)
10.5 Abstract free Lie algebras and Lie groups
523(10)
10.6 Notes
533(2)
11 Sobolev and Holder estimates for general Hormander operators
535(78)
11.1 Introduction and general overview
535(7)
11.2 Operators of type λ
542(18)
11.3 Parametrix and representation formulas
560(6)
11.4 Continuity of operators of type λ
566(11)
11.5 A priori estimates in Sobolev or Holder spaces for solutions to Lu = f
577(28)
11.6 Smoothing of distributional solutions and solvability in Holder or Sobolev spaces
605(6)
11.7 Notes
611(2)
12 Nonvariational operators constructed with Hormander vector fields
613(54)
12.1 Introduction
613(6)
12.2 Operators of type A and representation formulas
619(8)
12.3 Holder estimates
627(4)
12.4 LP estimates
631(4)
12.5 IP continuity of variable operators of type 0 and their commutators
635(11)
12.6 Regularization of solutions
646(8)
12.7 Proof of the estimates on spherical harmonics
654(10)
12.8 Notes
664(3)
Appendix A: Short summary of distribution theory 667(14)
Bibliography 681(10)
Index 691