Hello. Let $ X^{s,b} $ be the Bourgain space generated by $ \tau - \xi^3 $. It is proved that, for $ s\in (-\frac{1}{2}, 0] $, we have

$$

| (u^2)_x | _{ X^{s,b'-1} } \leq c |u| { X^{s,b} } |u|{ X^{-\frac{1}{2},b} }

$$

for some $ b, b' \in (\frac{1}{2},1) $. I want to know wether it holds that

$$

| (uv)x |{ X^{s,b'-1} } \leq c |u|{ X^{s,b} } |v|{ X^{-\epsilon,b} }

$$

for some $ \epsilon>0 $. Some references include similar results are welcome.