# Bilinear Airy estimates

From DispersiveWiki

## Algebraic identity

Much of the bilinear estimate theory for Airy equation rests on the following "three-wave resonance identity":

## Estimates

The following bilinear estimates are known:

- The estimate KnPoVe1996 on
**R**:

- The above estimate fails at the endpoint . 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

- The -1/2 estimate KnPoVe1996 on
**T**: if u,v have mean zero, then for all

- The above estimate fails for . Also, one cannot replace by . 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.