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Let $\Omega$ be a bounded domain in $\mathbb{R}^N, N\geq 2$.

Let $v\in H^{-1}(\Omega)$-dual space of $H_0^1(\Omega)$ and I want to find the assumptions on $u\in X, (X=?)$ such that the following inequality holds:

$$\|uv\|_{H^{-1}(\Omega)}\leq \|u\|_{X}\|v\|_{H^{-1}(\Omega)}$$

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It is sufficient to require $u\in W^{1,\infty}(\Omega)\cap H^1_0(\Omega)$. Suppose this. Take $v\in H^{-1}(\Omega)$ and $w\in H^1(\Omega)$.

The notation $uv$ is kind of sloppy, I guess you meant something like $$ (uv)(w) := v(uw) \quad w\in H^1_0(\Omega). $$ Then $$ |uv(w)| = |v(uw)| \le \|v\|_{H^{-1}} \|uw\|_{H^1(\Omega)} \le \|v\|_{H^{-1}} ( \|u\|_{L^\infty(\Omega)}\|\nabla w\|_{L^2(\Omega)} + \|\nabla u\|_{L^\infty(\Omega)}\| w\|_{L^2(\Omega)} ) \le 2\|v\|_{H^{-1}} \|u\|_{W^{1,\infty}} \|w\|_{H^1}. $$ This proves, that the mapping $w\mapsto v(uw)$ is bounded.

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  • $\begingroup$ Thank you for your answer! $\endgroup$
    – lamtran
    Jan 19, 2015 at 8:06
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    $\begingroup$ @lamtran if the answer is correct and help you to solve your problem, then you should consider to up-vote and accept it. :) $\endgroup$
    – spatially
    Jan 19, 2015 at 23:10

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