Hypoelliptic Laplacian and orbital integrals / Jean-Michel Bismut.

Format:

Book

Author/Creator:

Bismut, Jean-Michel.

Series:

Annals of mathematics studies ; no. 177.

Annals of mathematics studies ; no. 177

Language:

English

Subjects (All):

Differential equations, Hypoelliptic.

Laplacian operator.

Definite integrals.

Orbit method.

Physical Description:

330 pages ; 24 cm.

Place of Publication:

Princeton ; Oxford : Princeton University Press, 2011.

Summary:

Princeton University Press is proud to have published the Annals of Mathematics Studies since 1940. One of the oldest and most respected series publishing, it has included many of the most important and influential mathematical works of the twentieth century. The series continues this tradition into the twenty-first century as Princeton looks forward to publishing the major works of the new millennium.

To mark the continued success of the series, all books are again available in paperback. For a complete list of titles, please visit the Princeton University Press Web site: press.princeton.edu. The most recently published volumes include: Book jacket.

Contents:

1 Clifford and Heisenberg algebras 12

1.1 The Clifford algebra of a real vector space 12

1.2 The Clifford algebra of V + V* 14

1.3 The Heisenberg algebra 15

1.4 The Heisenberg algebra of V + V* 17

1.5 The Clifford-Heisenberg algebra of V + V* 18

1.6 The Clifford-Heisenberg algebra of V + V* when V is Euclidean 19

2 The hypoelliptic Laplacian on X = G/K 22

2.1 A pair (G, K) 23

2.2 The flat connection on TX + N 25

2.3 The Clifford algebras of g 25

2.4 The flat connections on Λ (T* X + N*) 25

2.5 The Casimir operator 27

2.6 The form κ^g 28

2.7 The Dirac operator of Kostant 30

2.8 The Clifford-Heisenberg algebra of g + g* 32

2.9 The operator D_b 33

2.10 The compression of the operator D_b 34

2.11 A formula for D²_b 34

2.12 The action of D_b on quotients by K 35

2.13 The operators L^X and L^X_b 39

2.14 The scaling of the form B 41

2.15 The Bianchi identity 41

2.16 A fundamental identity 41

2.17 The canonical vector fields on X 45

2.18 Lie derivatives and the operator L^X_b 46

3 The displacement function and the return map 48

3.1 Convexity, the displacement function, and its critical set 49

3.2 The norm of the canonical vector fields 50

3.3 The subset X (γ) as a symmetric space 54

3.4 The normal coordinate system on X based at X (γ) 57

3.5 The return map along the minimizing geodesies in X (γ) 62

3.6 The return map on X 64

3.7 The connection form in the parallel transport trivialization 65

3.8 Distances and pseudodistances on X and X 67

3.9 The pseudodistance and Toponogov's theorem 68

3.10 The flat bundle (TX + N) (γ) 75

4 Elliptic and hypoelliptic orbital integrals 76

4.1 An algebra of invariant kernels on X 77

4.2 Orbital integrals 78

4.3 Infinite dimensional orbital integrals 81

4.4 The orbital integrals for the elliptic heat kernel of X 84

4.5 The orbital supertraces for the hypoelliptic heat kernel 84

4.6 A fundamental equality 85

4.7 Another approach to the orbital integrals 86

4.8 The locally symmetric space Z 87

5 Evaluation of supertraces for a model operator 92

5.1 The operator and the function 92

5.2 A conjugate operator 94

5.3 An evaluation of certain infinite dimensional traces 95

5.4 Some formulas of linear algebra 103

5.5 A formula for 110

6 A formula for semisimple orbital integrals 113

6.1 Orbital integrals for the heat kernel 113

6.2 A formula for general orbital integrals 114

6.3 The orbital integrals for the wave operator 116

7 An application to local index theory 120

7.1 Characteristic forms on X 120

7.2 The vector bundle of spinors on X and the Dirac operator 122

7.3 The McKean-Singer formula on Z 124

7.4 Orbital integrals and the index theorem 125

7.5 A proof of (7.4.4) 126

7.6 The case of complex symmetric spaces 130

7.7 The case of an elliptic element 131

7.8 The de Rham-Hodge operator 134

7.9 The integrand of de Rham torsion 136

8 The case where 138

8.1 The case where G = K 138

8.1 Thecase 139

8.3 The case where G = SL₂ (R) 140

9 A proof of the main identity 142

9.1 Estimates on the heat kernel away from 142

9.2 A rescaling on the coordinates (f, Y) 145

9.3 A conjugation of the Clifford variables 147

9.4 The norm of α 150

9.5 A conjugation of the hypoelliptic Laplacian 150

9.6 The limit of the rescaled heat kernel 152

9.7 A proof of Theorem 6.1.1 153

9.8 A translation on the variable Y^TX 153

9.9 A coordinate system and a trivialization of the vector bundles 156

9.10 The asymptotics of the operator 158

9.11 A proof of Theorem 9.6.1 159

10 The action functional and the harmonic oscillator 161

10.1 A variational problem 162

10.2 The Pontryagin maximum principle 164

10.3 The variational problem on an Euclidean vector space 166

10.4 Mehler's formula 173

10.5 The hypoelliptic heat kernel on an Euclidean vector space 175

10.6 Orbital integrals on an Euclidean vector space 177

10.7 Some computations involving Mehler's formula 182

10.8 The probabilistic interpretation of the harmonic oscillator 183

11 The analysis of the hypoelliptic Laplacian 187

11.1 The scalar operators on X 188

11.2 The Littlewood-Paley decomposition along the fibres TX 189

11.3 The Littlewood-Paley decomposition on X 192

11.4 The Littlewood Paley decomposition on X 193

11.5 The heat kernels for 201

11.6 The scalar hypoelliptic operators on X 205

11.7 The scalar hypoelliptic operator on X with a quartic term 206

11.8 The heat kernel associated with the operator 210

12 Rough estimates on the scalar heat kernel 212

12.1 The Malliavin calculus for the Brownian motion on X 214

12.2 The probabilistic construction of exp ( ) over X 217

12.3 The operator and the wave equation 219

12.4 The Malliavin calculus for the operator 222

12.5 The tangent variational problem and integration by parts 223

12.6 A uniform control of the integration by parts formula as 6 → 0 226

12.7 Uniform rough estimates on for bounded b 228

12.8 The limit as b → 4 230

12.9 The rough estimates as b → +∞ 237

12.10 The heat kernel 241

12.11 The heat kernel 244

13 Refined estimates on the scalar heat kernel for bounded b 248

13.1 The Hessian of the distance function 248

13.2 Bounds on the scalar heat kernel on X for bounded b 251

13.3 Bounds on the scalar heat kernel on X for bounded b 260

14 The heat kernel for bounded b 262

14.1 A probabilistic construction of exp ( ) 263

14.2 The operator and the wave equation 263

14.3 Changing Y into - Y 264

14.4 A probabilistic construction of exp ( ) 265

14.5 Estimating V 266

14.6 Estimating W 267

14 7 A proof of (4.5.3) when E is trivial 268

14.8 A proof of the estimate (4.5.3) in the general case 270

14.9 Rough estimates on the derivatives of for bounded b 274

14.10 The behavior of V. as b → 0 280

14.11 The limit of as b → 0 287

15 The heat kernel for b large 290

15.1 Uniform estimates on the kernel over X 291

15 2 The deviation from the geodesic flow for large b 292

15.3 The scalar heat kernel on X away from 294

15.4 Gaussian estimates for near 299

15 5 The scalar heat kernel on X away from 299

15.6 Estimates on the scalar heat kernel on X near 306

15.7 A proof of Theorem 9.1.1 310

15.8 A proof of Theorem 9.1.3 311

15.9 A proof of Theorem 9.5.6 312

15.10 A proof of Theorem 9.11.1 313.

Notes:

Includes bibliographical references (pages [317]-321) and index.

ISBN:

9780691151298

0691151296

OCLC:

724663133