Trilinear Airy estimates

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Algebraic identity

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

  • The key algebraic fact is (various permutations of)


The following trilinear estimates are known:

The 1/4 is sharp KnPoVe1996.We also have

see Cv2004.

  • The 1/2 estimate CoKeStTkTa-p3 on T: if have mean zero, then

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.