May one conclude from $u \in W^{-1,2}(\Omega)$ and $\phi \in L^\infty(\Omega)$ that also $u\phi \in W^{-1,2}(\Omega) $?
Intuitively this seems clear and I tried to do this in the same way as the usual $L^p$-estimates by pulling out the $L^\infty$-norm but my problem is that I can't apply this in the definition of the $W^{-1,2}$-norm. I also tried to estimate
$\| u\phi \|_{W^{-1,2}} \leq \| u\phi \|_{L^2} \leq \| \phi \|_{L^\infty}\| u \|_{L^2}$
but as far as I know we can't estimate $\|\cdot\|_{L^2} \leq \| \cdot \|_{W^{-1,2}}$. Is there maybe some other estimate like $\|\cdot\|_{L^p} \leq \| \cdot \|_{W^{-1,2}}$ for some $p<2$ I could use or another embedding theorem I don't know? Thank you for any hints.