Trilinear Airy estimates

Algebraic identity

Much of the trilinear estimate theory for Airy equation rests on (various permutations of) the following "four-wave resonance identity":

$\xi_1^3 + \xi_2^3 + \xi_3^3 + \xi_4^3 = 3 (\xi_1+\xi_4) (\xi_2+\xi_4) (\xi_3+\xi_4)$ whenever

ξ 1 + ξ 2 + ξ 3 + ξ 4 = 0

The following trilinear estimates are known:

$|| (uvw)_x ||_{1/4, -1/2+} <~ || u ||_{1/4, 1/2+} || v ||_{1/4, 1/2+} || w ||_{1/4, 1/2+}$

The 1/4 is sharp KnPoVe1996.We also have

$|| uvw ||_{-1/4, -5/12+} <~ || u ||_{-1/4, 7/12+} || v ||_{-1/4, 7/12+} || w ||_{-1/4, 7/12+};$

$|| (uvw)_x ||_{1/2, -1/2} <~ || u ||_{1/2, 1/2*} || v ||_{1/2, 1/2*} || w ||_{1/2, 1/2*}$

The 1/2 is sharp KnPoVe1996.

Remark: the trilinear estimate always needs one more derivative of regularity than the bilinear estimate; this is consistent with the heuristics from the Miura transform from mKdV to KdV.