Bilinear Airy estimates

Algebraic identity

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

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

Estimates

The following bilinear estimates are known:

• The $-3/4+$ estimate KnPoVe1996 on R:

$\|u\partial _{x}v\|_{X^{-3/4+,-1/2+}}\lesssim \|u\|_{X^{-3/4+,1/2+}}\|v\|_{X^{-3/4+,1/2+}}$

• • The above estimate fails at the endpoint $-3/4$. 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

$\|u\partial _{x}v\|_{X^{0,-1/2+}}\lesssim \|u\|_{X^{-3/8+,1/2+}}\|v\|_{X^{-3/8+,1/2+}}$

• The -1/2 estimate KnPoVe1996 on T: if u,v have mean zero, then for all $s\geq -1/2$

$\|u\partial _{x}v\|_{X^{s,-1/2}}\lesssim \|u\|_{X^{s,1/2}}\|v\|_{X^{s,1/2}}$

• • The above estimate fails for $s<-1/2$. Also, one cannot replace $1/2,-1/2$ by $1/2+,-1/2+$. KnPoVe1996
  • This estimate also holds in the large period case if one is willing to lose a power of \lambda^{0+} in the constant. CoKeStTkTa2003
• Remark: In principle, a complete list of bilinear estimates could be obtained from Ta2001.