Preface ........................................................ xi
Chapter 1. Laplace operator and the heat equation in  n ........ 1
1.1. Historical background ..................................... 1
1.2. The Green formula ......................................... 2
1.3. The heat equation ......................................... 4
Notes .......................................................... 13
Chapter 2. Function spaces in n .............................. 15
2.1. Spaces Ck and Lp ......................................... 15
2.2. Convolution and partition of unity ....................... 17
2.3. Approximation of integrable functions by smooth ones ..... 20
2.4. Distributions ............................................ 23
2.5. Approximation of distributions by smooth functions ....... 28
2.6. Weak derivatives and Sobolev spaces ...................... 34
2.7. Heat semigroup in n ..................................... 40
Notes .......................................................... 47
Chapter 3. Laplace operator on a Riemannian manifold .......... 49
3.1. Smooth manifolds ......................................... 49
3.2. Tangent vectors .......................................... 53
3.3. Riemannian metric ........................................ 56
3.4. Riemannian measure ....................................... 59
3.5. Divergence theorem ....................................... 64
3.6. Laplace operator and weighted manifolds .................. 67
3.7. Submanifolds ............................................. 70
3.8. Product manifolds ........................................ 72
3.9. Polar coordinates in n,  n,  n .......................... 74
3.10. Model manifolds .......................................... 80
3.11. Length of paths and the geodesic distance ................ 85
3.12. Smooth mappings and isometries ........................... 91
Notes .......................................................... 95
Chapter 4. Laplace operator and heat equation in L2(M) ........ 97
4.1. Distributions and Sobolev spaces ......................... 97
4.2. Dirichlet Laplace operator and resolvent ................ 103
4.3. Heat semigroup and L2-Cauchy problem .................... 112
Notes ......................................................... 122
Chapter 5. Weak maximum principle and related topics ......... 123
5.1. Chain rule in W01 ........................................ 123
5.2. Chain rule in W1 ........................................ 127
5.3. Markovian properties of resolvent and the heat
semigroup ............................................... 130
5.4. Weak maximum principle .................................. 135
5.5. Resolvent and the heat semigroup in subsets ............. 143
Notes ......................................................... 149
Chapter 6. Regularity theory in n ........................... 151
6.1. Embedding theorems ...................................... 151
6.2. Two technical lemmas .................................... 159
6.3. Local elliptic regularity ............................... 162
6.4. Local parabolic regularity .............................. 170
Notes ......................................................... 181
Chapter 7. The heat kernel on a manifold ..................... 183
7.1. Local regularity issues ................................. 183
7.2. Smoothness of the semigroup solutions ................... 190
7.3. The heat kernel ......................................... 198
7.4. Extension of the heat semigroup ......................... 201
7.5. Smoothness of the heat kernel in t, x, у ................ 208
7.6. Notes ................................................... 215
Chapter 8. Positive solutions ................................ 217
8.1. The minimality of the heat semigroup .................... 217
8.2. Extension of resolvent .................................. 219
8.3. Strong maximum/minimum principle ........................ 222
8.4. Stochastic completeness ................................. 231
Notes ......................................................... 241
Chapter 9. Heat kernel as a fundamental solution ............. 243
9.1. Fundamental solutions ................................... 243
9.2. Some examples ........................................... 248
9.3. Eternal solutions ....................................... 259
Notes ......................................................... 263
Chapter 10. Spectral properties ............................... 265
10.1. Spectra of operators in Hilbert spaces ................. 265
10.2. Bottom of the spectrum ................................. 271
10.3. The bottom eigenfunction ............................... 275
10.4. The heat kernel in relatively compact regions .......... 277
10.5. Minimax principle ...................................... 284
10.6. Discrete spectrum and compact embedding theorem ........ 287
10.7. Positivity of λ1 ....................................... 291
10.8. Long time asymptotic of log pt ......................... 292
Notes ......................................................... 293
Chapter 11. Distance function and completeness ............... 295
11.1. The notion of completeness ............................. 295
11.2. Lipschitz functions .................................... 296
11.3. Essential self-adjointness ............................. 301
11.4. Stochastic completeness and the volume growth .......... 303
11.5. Parabolic manifolds .................................... 313
11.6. Spectrum and the distance function ..................... 317
Notes ......................................................... 319
Chapter 12. Gaussian estimates in the integrated form ........ 321
12.1. The integrated maximum principle ....................... 321
12.2. The Davies-Gaffney inequality .......................... 324
12.3. Upper bounds of higher eigenvalues ..................... 327
12.4. Semigroup solutions with a harmonic initial function ... 331
12.5. Takeda's inequality .................................... 333
Notes ......................................................... 339
Chapter 13. Green function and Green operator ................ 341
13.1. The Green operator ..................................... 341
13.2. Superaveraging functions ............................... 348
13.3. Local Harnack inequality ............................... 351
13.4. Convergence of sequences of α-harmonic functions ....... 355
13.5. The positive spectrum .................................. 357
13.6. Green function as a fundamental solution ............... 359
Notes ......................................................... 362
Chapter 14. Ultracontractive estimates and eigenvalues ....... 365
14.1. Ultracontractivity and heat kernel bounds .............. 365
14.2. Faber-Krahn inequalities ............................... 367
14.3. The Nash inequality .................................... 368
14.4. The function classes L and Г ........................... 371
14.5. Faber-Krahn implies ultracontractivity ................. 380
14.6. Ultracontractivity implies a Faber-Krahn inequality .... 381
14.7. Lower bounds of higher eigenvalues ..................... 384
14.8. Faber-Krahn inequality on direct products .............. 386
Notes ......................................................... 388
Chapter 15. Pointwise Gaussian estimates I ................... 391
15.1. L2-mean value inequality ............................... 391
15.2. Faber-Krahn inequality in balls ........................ 397
15.3. The weighted L2-norm of heat kernel .................... 399
15.4. Faber-Krahn inequality in unions of balls .............. 402
15.5. Off-diagonal upper bounds .............................. 404
15.6. Relative Faber-Krahn inequality and Li-Yau upper
bounds ................................................. 409
Notes ......................................................... 414
Chapter 16. Pointwise Gaussian estimates II .................. 417
16.1. The weighted L2-norm of Ptƒ ............................ 417
16.2. Gaussian upper bounds of the heat kernel ............... 422
16.3. On-diagonal lower bounds ............................... 424
16.4. Epilogue: alternative ways of constructing the heat
kernel ................................................. 428
Notes and further references .................................. 429
Appendix A. Reference material ............................... 431
A.l. Hilbert spaces .......................................... 431
A.2. Weak topology ........................................... 432
A.3. Compact operators ....................................... 434
A.4. Measure theory and integration .......................... 434
A.5. Self-adjoint operators .................................. 444
A.6. Gamma function .......................................... 455
Bibliography .................................................. 457
Some notation ................................................. 475
Index ......................................................... 477
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