# Generalized bilinear estimates

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