Inequality in Sobolev Space

Question

Given $\Omega \subset \mathbb{R}^3$, prove $\forall u, v, w \in H^{1,2} (\Omega)$ it holds that $ | \int_{\Omega} u \frac{\partial v}{ \partial x} w dx | \leq \| u \|_{1,2,\Omega}\|v \|_{1,2,\Omega}\| w \|_{1,2,\Omega}, $ where $\| \cdot \|_{1,2,\Omega}$ denotes the norm on $ H^{1,2} (\Omega)$.

Can somebody help me with this doubt?

Regards.

Answer 1

The Sobolev embedding theorem yields the existence of $C$, such that $$\lVert u \rVert_{L^4(\Omega)} \le C \, \lVert u \rVert_{H^{1,2}(\Omega)}.$$ Together with Hölder's inequality, this is enough to prove your estimate.