Multilinear Airy estimates

The following multilinear estimates for the Airy equation are known:

• We have the quintilinear estimate on R: CoKeStTkTa2003

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

• The analogue for this on T is: CoKeStTkTa2003, CoKeStTkTa-p3

$intu^{3}v^{2}dxdt<~||u||_{1/2,1/2*}^{3}||v||_{-1/2,1/2*}^{2}$

In fact, this estimate also holds for large period, but a loss of lambda^{0+}.