Introduction .................................................... 1
1 Description of the main results .............................. 1
2 Properties of the Dirichlet-Neumann operator ................. 5
3 Paradifferential normal forms method ........................ 11
4 Iterated vector fields Z .................................... 15
1 Statement of the main results ............................... 19
1.1 Definitions and properties of the Dirichlet-Neumann
operator ............................................... 19
1.2 Main Sobolev estimate .................................. 34
2 Estimates for the Dirichlet-Neumann operator ................ 39
2.1 Main results ........................................... 40
2.2 Sharp estimates ........................................ 43
2.3 Tame estimates ......................................... 56
2.4 Paralinearization of the Dirichlet-Neumann operator .... 58
2.5 Linearization of the Dirichlet-Neumann operator ........ 60
2.6 Taylor expansions ...................................... 62
2.7 Smooth domains ......................................... 69
3 Normal form for the water waves equation .................... 77
3.1 Quadratic approximations without losses ................ 77
3.2 Quadratic and cubic terms in the equations ............. 84
3.3 Quadratic normal form: strategy of the proof ........... 87
3.4 Paradifferential operators ............................. 88
3.5 The main equations ..................................... 95
3.6 System for the new unknown ............................ 109
3.7 Energy estimate ....................................... 115
4 Commutation of the Z-field with the equations .............. 119
4.1 Action of the Z-field on the Dirichlet-Neumann
operator .............................................. 119
4.2 Other identities ...................................... 123
4.3 Estimates for the action of iterated vector fields .... 126
4.4 Nonlinear estimates ................................... 142
4.5 Estimate of the remainder terms ....................... 150
5 Energy estimates for the Z-field system .................... 167
5.1 Notations ............................................. 167
5.2 Normal form for the Z-systems ......................... 171
6 Appendices ................................................. 209
A.l. Paradifferential calculus ............................. 209
A.2. Estimates in Holder spaces ............................ 215
A.3. Identities ............................................ 217
A.4. Local existence results ............................... 219
Index ......................................................... 233
Bibliography .................................................. 235
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