# Difference between revisions of "Bilinear Airy estimates"

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** The above estimate fails for <math>s < -1/2</math>. Also, one cannot replace <math>1/2, -1/2 by 1/2+, -1/2+</math>. [[KnPoVe1996]] | ** The above estimate fails for <math>s < -1/2</math>. Also, one cannot replace <math>1/2, -1/2 by 1/2+, -1/2+</math>. [[KnPoVe1996]] | ||

** This estimate also holds in the large period case if one is willing to lose a power of \lambda^{0+} in the constant. [[CoKeStTkTa2003]] | ** This estimate also holds in the large period case if one is willing to lose a power of \lambda^{0+} in the constant. [[CoKeStTkTa2003]] | ||

− | * ''Remark''<nowiki>: In principle, a complete list of bilinear estimates could be obtained from </nowiki>[[ | + | * ''Remark''<nowiki>: In principle, a complete list of bilinear estimates could be obtained from </nowiki>[[Ta2001]]. |

[[Category:Estimates]] | [[Category:Estimates]] |

## Revision as of 19:35, 4 March 2007

## Algebraic identity

Much of the bilinear estimate theory for Airy equation rests on the following "three-wave resonance identity":

## Estimates

The following bilinear estimates are known:

- The estimate KnPoVe1996 on
**R**:

- The above estimate fails at the endpoint . NaTkTs2001
- As a corollary of this estimate we have the -3/8+ estimate CoStTk1999 on
**R**: If u and v have no low frequencies ( |\xi| <~ 1 ) then

- The -1/2 estimate KnPoVe1996 on
**T**: if u,v have mean zero, then for all s >= -1/2

- The above estimate fails for . Also, one cannot replace . KnPoVe1996
- This estimate also holds in the large period case if one is willing to lose a power of \lambda^{0+} in the constant. CoKeStTkTa2003

*Remark*: In principle, a complete list of bilinear estimates could be obtained from Ta2001.