# Bilinear Airy estimates

## Algebraic identity

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

${\displaystyle \xi _{1}^{3}+\xi _{2}^{3}+\xi _{3}^{3}=3\xi _{1}\xi _{2}\xi _{3}}$ whenever ${\displaystyle \xi _{1}+\xi _{2}+\xi _{3}=0}$

## Estimates

The following bilinear estimates are known:

• The ${\displaystyle -3/4+}$ estimate KnPoVe1996 on R:
${\displaystyle \|u\partial _{x}v\|_{X^{-3/4+,-1/2+}}\lesssim \|u\|_{X^{-3/4+,1/2+}}\|v\|_{X^{-3/4+,1/2+}}}$

• The above estimate fails at the endpoint ${\displaystyle -3/4}$. 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
${\displaystyle \|u\partial _{x}v\|_{X^{0,-1/2+}}\lesssim \|u\|_{X^{-3/8+,1/2+}}\|v\|_{X^{-3/8+,1/2+}}}$
• The -1/2 estimate KnPoVe1996 on T: if u,v have mean zero, then for all s >= -1/2
${\displaystyle \|u\partial _{x}v\|_{X^{s,-1/2}}\lesssim \|u\|_{X^{s,1/2}}\|v\|_{X^{s,1/2}}}$
• The above estimate fails for ${\displaystyle s<-1/2}$. Also, one cannot replace ${\displaystyle 1/2,-1/2by1/2+,-1/2+}$. 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
• Remark: In principle, a complete list of bilinear estimates could be obtained from Ta-p2.