Bilinear Airy estimates

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Algebraic identity

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



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
    • The above estimate fails for . Also, one cannot replace by . 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.