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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) $, see JDE 2002 (185) 25-53. 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. I have try my best to edit the tex, but it looks far from nice.

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) $, see JDE 2002 (185) 25-53. 

I want to know whether 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 which include similar results are welcome.

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) $, see JDE 2002 (185) 25-53. 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. I have try my best to edit the tex, but it looks far from nice.

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) $, see JDE 2002 (185) 25-53. 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. I have try my best to edit the tex, but it looks far from nice.

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.

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) $, see JDE 2002 (185) 25-53. 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. I have try my best to edit the tex, but it looks far from nice.