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":

$\xi_1^3 + \xi_2^3 + \xi_3^3 = 3 \xi_1 \xi_2 \xi_3$ whenever

ξ 1 + ξ 2 + ξ 3 = 0

The following bilinear estimates are known:

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

- 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.