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.