1 Hardy—Littlewood inequalities ................................ 1
1.1 Differential forms ...................................... 1
1.1.1 Basic elements ................................... 1
1.1.2 Definitions and notations ........................ 5
1.1.3 Poincare lemma ................................... 6
1.2 A-harmonic equations .................................... 8
1.2.1 Quasiconformal mappings .......................... 9
1.2.2 A-harmonic equations ............................ 10
1.3 p-Harmonic equations ................................... 14
1.3.1 Two equivalent forms ............................ 14
1.3.2 Three-dimensional cases ......................... 15
1.3.3 The equivalent system ........................... 17
1.3.4 An example ...................................... 19
1.4 Some weight classes .................................... 21
1.4.1 Ar(Ω)-weights ................................... 21
1.4.2 Ar(λ,E)-weights ................................. 23
1.4.3 Aλr(E)-weights .................................. 25
1.4.4 Some classes of two-weights ..................... 27
1.5 Inequalities in John domains ........................... 29
1.5.1 Local inequalities .............................. 29
1.5.2 Weighted inequalities ........................... 32
1.5.3 Global inequalities ............................. 34
1.6 Inequalities in averaging domains ...................... 37
1.6.1 Averaging domains ............................... 37
1.6.2 Ls(μ)-averaging domains ......................... 38
1.6.3 Other weighted inequalities ..................... 41
1.7 Two-weight cases ....................................... 43
1.7.1 Local inequalities .............................. 43
1.7.2 Global inequalities ............................. 45
1.8 The best integrable condition .......................... 45
1.8.1 An example ...................................... 45
1.8.2 Remark .......................................... 47
1.9 Inequalities with Orlicz norms ......................... 47
1.9.1 Norm comparison theorem ......................... 48
1.9.2 Lp(log L)α-norm inequality ...................... 49
1.9.3 Ar(Ω)-weighted case ............................. 52
1.9.4 Global Ls(log L)α-norm inequality ............... 55
2 Norm comparison theorems .................................... 57
2.1 Introduction ........................................... 57
2.2 The local unweighted estimates ......................... 58
2.2.1 Basic Lp-inequalities ........................... 58
2.2.2 Special cases ................................... 60
2.3 The local weighted estimates ........................... 61
2.3.1 Ls-estimates for d*υ ............................ 61
2.3.2 Ls-estimates for du ............................. 64
2.3.3 The norm comparison between d* and d ............ 65
2.4 The global estimates ................................... 67
2.4.1 Global estimates for d*υ ........................ 67
2.4.2 Global estimates for du ......................... 68
2.4.3 Global Lp-estimates ............................. 69
2.4.4 Global Ls-estimates ............................. 70
2.5 Applications ........................................... 72
2.5.1 Imbedding theorems for differential forms ....... 72
3 Poincare-type inequalities .................................. 75
3.1 Introduction ........................................... 75
3.2 Inequalities for differential forms .................... 75
3.2.1 Basic inequalities .............................. 75
3.2.2 Weighted inequalities ........................... 76
3.2.3 Inequalities for harmonic forms ................. 78
3.2.4 Global inequalities in averaging domains ........ 83
3.2.5 Aλr-weighted inequalities ....................... 84
3.3 Inequalities for Green's operator ...................... 86
3.3.1 Basic estimates for operators ................... 88
3.3.2 Weighted inequality for Green's operator ........ 90
3.3.3 Global inequality for Green's operator .......... 92
3.4 Inequalities with Orlicz norms ......................... 92
3.4.1 Local inequality ................................ 93
3.4.2 Weighted inequalities ........................... 96
3.4.3 The proof of the global inequality .............. 98
3.5 Two-weight inequalities ............................... 100
3.5.1 Statements of two-weight inequalities .......... 100
3.5.2 Proofs of the main theorems .................... 101
3.5.3 Aλr(Ω)-weighted inequalities ................... 104
3.6 Inequalities for Jacobians ............................ 107
3.6.1 Some notations ................................. 108
3.6.2 Two-weight estimates ........................... 109
3.7 Inequalities for the projection operator .............. 1ll
3.7.1 Statement of the main theorem .................. 111
3.7.2 Inequality for Δ and G ......................... 112
3.7.3 Proof of the main theorem ...................... 116
3.8 Other Poincare-type inequalities ...................... 116
4 Caccioppoli inequalities ................................... 119
4.1 Preliminary results ................................... 119
4.2 Local and global weighted cases ....................... 120
4.2.1 Ar(Ω)-weighted inequality ...................... 120
4.2.2 Ar(λ,Ω)-weighted inequality .................... 123
4.2.3 Aλr(Ω)-weighted inequality ..................... 125
4.2.4 Parametric version ............................. 126
4.2.5 Inequalities with two parameters ............... 127
4.3 Local and global two-weight cases ..................... 127
4.3.1 An unweighted inequality ....................... 127
4.3.2 Two-weight inequalities ........................ 129
4.4 Inequalities with Orlicz norms ........................ 133
4.4.1 Basic || • ||Lp(log L)α(E) estimates .............. 133
4.4.2 Weak reverse Holder inequalities ............... 136
4.4.3 Ar(M)-weighted cases ........................... 137
4.5 Inequalities with the codifferential operator ......... 140
4.5.1 Lq-estimate for d*υ ............................ 140
4.5.2 Two-weight estimate for d*υ .................... 141
5 Imbedding theorems ......................................... 145
5.1 Introduction .......................................... 145
5.2 Quasiconformal mappings ............................... 145
5.3 Solutions to the nonhomogeneous equation .............. 146
5.4 Imbedding inequalities for operators .................. 147
5.4.1 The gradient and homotopy operators ............ 148
5.4.2 Some special cases ............................. 152
5.4.3 Global imbedding theorems ...................... 153
5.5 Other weighted cases .................................. 154
5.5.1 Ls-estimates for T ............................. 154
5.5.2 Ls-estimates for ∇ º Т ......................... 156
5.5.3 Ar(λ,Ω)-weighted imbedding theorems ............ 158
5.5.4 Some corollaries ............................... 159
5.5.5 Global imbedding theorems ...................... 162
5.5.6 Aλr(Ω)-weighted estimates ...................... 164
5.6 Compositions of operators ............................. 165
5.6.1 Ar(Ω)-weighted estimates for T º d º G ......... 165
5.6.2 Ar(Ω)-weighted estimates for ∇ º T º G .......... 169
5.6.3 Imbedding for T º d º G ........................ 171
5.7 Two-weight cases ...................................... 172
5.7.1 Two-weight imbedding for the operator T ........ 173
5.7.2 Ar,λ(E)-weighted imbedding ..................... 178
5.7.3 Ar(λ,E)-weighted imbedding ..................... 181
5.7.4 Global imbedding theorem ....................... 184
6 Reverse Holder inequalities ................................ 187
6.1 Preliminaries ......................................... 187
6.1.1 Gehring's lemma ................................ 187
6.1.2 Inequalities for supersolutions ................ 188
6.2 The first weighted case ............................... 189
6.2.1 Ar(Ω)-weighted inequalities .................... 190
6.2.2 Inequalities in Ls(μ)-averaging domains ........ 193
6.2.3 Inequalities in John domains ................... 195
6.2.4 Parametric inequalities ........................ 196
6.2.5 Estimates for du ............................... 200
6.3 The second weighted case .............................. 201
6.3.1 Inequalities with Aαr(Ω)-weights ............... 201
6.3.2 Inequalities with two parameters ............... 205
6.3.3 Aλr(Ω)-weighted inequalities for du ............ 209
6.4 The third weighted case ............................... 209
6.4.1 Local inequalities ............................. 210
6.4.2 Global inequality .............................. 212
6.4.3 Analogies for du ............................... 213
6.5 Two-weight inequalities ............................... 214
6.5.1 Ar,λ(Ω)-weighted cases ......................... 214
6.5.2 Ar(λ,Ω)-weighted cases ......................... 216
6.5.3 Two-weight inequalities for du ................. 218
6.6 Inequalities with Orlicz norms ........................ 219
6.6.1 Elementary inequalities ........................ 219
6.6.2 Lp(log L)α-norm inequalities ................... 220
7 Inequalities for operators ................................. 225
7.1 Introduction .......................................... 225
7.2 Some basic estimates .................................. 226
7.2.1 Estimates related to Green's operator .......... 226
7.2.2 Estimates for ∇ º T ............................ 229
7.2.3 Estimates for d º Т ............................ 234
7.3 Compositions of operators ............................. 236
7.3.1 Estimates for ∇ º Т º G and T º Δ º G .......... 236
7.3.2 Global estimates on manifolds .................. 242
7.3.3 Ls-estimates for T º G ......................... 244
7.3.4 Local imbedding theorems for T º G ............. 248
7.3.5 Global imbedding theorems for T º G ............ 250
7.3.6 Some special cases ............................. 251
7.3.7 Ls-estimates for Δ º G º d ..................... 252
7.4 Poincaré-type inequalities for operators .............. 256
7.4.1 Poincaré-type inequalities for T º G ........... 256
7.4.2 Poincaré-type inequalities for G º Т ........... 275
7.5 The homotopy operator ................................. 281
7.5.1 Basic estimates for T .......................... 281
7.5.2 Ar(Ω)-weighted estimates for T ................. 283
7.5.3 Poincaré-type imbedding for T .................. 284
7.5.4 Two-weight Poincaré-type imbedding for T ....... 285
7.6 Homotopy and projection operators ..................... 288
7.6.1 Basic estimates for T º Н ...................... 288
7.6.2 Ar(Ω)-weighted inequalities for T º Н .......... 290
7.6.3 Other single weighted cases .................... 291
7.6.4 Inequalities with two-weights in Ar,(λ,Ω) ...... 292
7.6.5 Inequalities with two-weights in Aλr(Ω) ........ 293
7.6.6 Basic estimates for H º Т ...................... 295
7.6.7 Weighted inequalities for H º Т ................ 297
7.6.8 Two-weight inequalities for H º Т .............. 299
7.6.9 Some global inequalities ....................... 302
7.7 Compositions of three operators ....................... 304
7.7.1 Basic estimates for T º d º Н .................. 304
7.7.2 Ar(Ω)-weighted inequalities for T º d º Н ...... 306
7.7.3 Cases of other weights ......................... 307
7.7.4 Cases of two-weights ........................... 308
7.7.5 Estimates for T º H º d ........................ 311
7.7.6 Estimates for H º Т º d ........................ 312
7.8 The maximal operators ................................. 313
7.8.1 Global Ls-estimates ............................ 313
7.8.2 The norm comparison theorem .................... 315
7.8.3 The fractional maximal operator ................ 316
7.9 Singular integrals .................................... 318
8 Estimates for Jacobians .................................... 323
8.1 Introduction .......................................... 323
8.2 Global integrability .................................. 324
8.2.1 Preliminary lemmas ............................. 325
8.2.2 LP(log L)α(Ω)-integrability .................... 327
8.2.3 Applications ................................... 329
8.2.4 Examples ....................................... 331
8.2.5 The norm comparison ............................ 333
9 Lipschitz and BMO norms .................................... 339
9.1 Introduction .......................................... 339
9.2 BMO spaces and Lipschitz classes ...................... 340
9.2.1 Some recent results ............................ 340
9.2.2 Sharpness of integrability exponents ........... 342
9.3 Global integrability .................................. 344
9.3.1 Estimates for du ............................... 344
9.3.2 Estimates for du+ .............................. 345
9.4 Lipschitz and BMO norms ............................... 346
9.4.1 Estimates for Lipschitz norms .................. 346
9.4.2 Lipschitz norms of T º G ....................... 350
9.4.3 Lipschitz norms of G º T ....................... 353
9.4.4 Lipschitz norms of T º H and H º T ............. 354
9.4.5 Estimates for BMO norms ........................ 355
9.4.6 Weighted norm inequalities ..................... 357
9.4.7 Estimates in averaging domains ................. 360
9.4.8 Applications ................................... 366
References .................................................... 369
Index ......................................................... 385
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