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 references.html#KnPoVe1996 KnPoVe1996 on R:

Failed to parse (syntax error): {\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 -3/4. references.html#NaTkTs-p NaTkTs2001
  • As a corollary of this estimate we have the -3/8+ estimate references.html#CoStTk1999 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 references.html#KnPoVe1996 KnPoVe1996 on T: if u,v have mean zero, then for all s >= -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 <math>1/2, -1/2 by 1/2+, -1/2+<math>. references.html#KnPoVe1996 KnPoVe1996
  • This estimate also holds in the large period case if one is willing to lose a power of \lambda^{0+} in the constant. references.html#CoKeStTaTk-p2 CoKeStTkTa-p2
• Remark: In principle, a complete list of bilinear estimates could be obtained from [[references.html#Ta-p2 Ta-p2]].