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:

|| (uv)_x ||_{-3/4+, -1/2+} <~ || u ||_{-3/4+, 1/2+} || v ||_{-3/4+, 1/2+}

$\|uv\|_{-3/4+,-1/2+}<~\|u\|_{-3/4+,1/2+}\|v\|_{-3/4+,1/2+}$

• • 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

|| (uv)_x ||_{0, -1/2+} <~ || u ||_{-3/8+, 1/2+} || v ||_{-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

|| (uv)_x ||_{s, -1/2} <~ || u ||_{s, 1/2} || v ||_{s, 1/2}

• • The above estimate fails for s < -1/2. Also, one cannot replace 1/2, -1/2 by 1/2+, -1/2+. 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]].