Let $u\in W^{1,2}(\Omega)$.
I need to prove that:
$\|(\nabla u)u\|_{W^{0,2}(\Omega)}\leq \|\nabla u\|_{W^{0,4}(\Omega)}\;\|u\|_{W^{0,4}(\Omega)}$
I'm using the usual Sobolev notation (see for example wikipedia: http://en.wikipedia.org/wiki/Sobolev_space)
$$ \lVert f \rVert_{W^{0,2}(\Omega)} = \left(\int_{\Omega} \lvert f \rvert^2\right)^{1/2} = \lVert f \rVert_{L^2(\Omega)}. $$ Hölder's inequality is $$ \left\lvert \int_{\Omega} f g \right\rvert \leqslant \left( \int_{\Omega} \lvert f \rvert^2 \right)^{1/2} \left( \int_{\Omega} \lvert g \rvert^2 \right)^{1/2} \, $$ or $$ \lVert f g \rVert_{L^1(\Omega)} \leqslant \lVert f \rVert_{L^2(\Omega)} \lVert g \rVert_{L^2(\Omega)}. $$ It should be fairly clear what to do from here by mucking about with the indices.