3
$\begingroup$

How to derive this inequality?

$$\|F(x,·)\|_{L^{4,1}(E)}^2 \leq C\|F(x,·)\|_{L^{\infty}(E)}\|F(x,·)\|_{L^{2,2}(E)},$$ where $C$ is constant and $$\|F(x,·)\|_{L^{p,n}(E)}=\left(\int_E\displaystyle\sum_{m\leq n}\left|\partial_y^m F(x,y)\right|^pdy\right)^{1/p},$$ $E$ is a bounded domain of $\mathbb{R}^2$.

  • 0
    Sobolev + interpolation gives $$\|g\|^2_{L^{4,1}} \lesssim \left(\|g\|_\infty + \|g\|_{L^{2,2}}\right)\|g\|_{L^{2,2}}$$ but I don't yet see how to get rid of the additional factor of $\|g\|_{L^{2,2}}$.2012-07-24

1 Answers 1