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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 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.

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I cleaned up the LaTeX for you –  Yemon Choi Jun 4 '13 at 17:42
    
The JDE paper being referenced can be found at labma.ufrj.br/~rrosa/publications.html or directly at labma.ufrj.br/~rrosa/dvifiles/kdvl2.pdf –  Barry Cipra Jun 4 '13 at 18:16
    
Why do you expect the product uv to be as smooth as u? –  Piero D'Ancona Jun 4 '13 at 23:04
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