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. [[CoKeStTkTa-p2]]
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** 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>[[Ta-p2]].
 
* ''Remark''<nowiki>: In principle, a complete list of bilinear estimates could be obtained from </nowiki>[[Ta-p2]].
  
 
[[Category:Estimates]]
 
[[Category:Estimates]]

Revision as of 22:47, 14 August 2006

Algebraic identity

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

whenever

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 Ta-p2.