I have the following question, please help me.
Let $1 < p \leq \infty$ and $u \in L^p(\Omega)$, $\Omega \subset \mathbb{R}^n$ an open set. Show that if there exists a constant $C >0$ such that \begin{equation} \left|\,\int\limits_{\Omega} u \frac{\partial \phi}{\partial x_j}\mathrm{d}x\right| \leq C|\phi|_{0,q,\Omega}, \label{1} \tag{1} \end{equation} for every $\phi \in \mathcal{D}(\Omega)$ and $1 \leq j \leq n$, $q$ the conjugate exponent of $p$, then $u \in W^{1,p}(\Omega)$.
My work: to prove that $u \in W^{1,p}(\Omega)$ we need to show that $u \in L^p(\Omega)$ which is already given in the question, and we also need to show that $\frac{\partial u}{\partial x_j} \in L^p(\Omega)$. I tried to apply Holder's inequality with the aid of equation \eqref{1} but I am not sure how to work with $\phi$ and eliminate it from the calculations.
Any help is appreciated!
Note that $\int u \phi_j = - \langle u_j, \phi\rangle_{D' \times D}$ where $u_j \in D'$ is the distributional derivative of $u$. You know $|\langle u_j, \phi \rangle_{D' \times D}| \leq C|\phi|_{L^q}$ for all $\phi \in D \subset L^q$, which means $\sup_{\phi \in L^q} \frac{|\langle u_j, \phi \rangle_{D' \times D}|}{|\phi|_{L^q}} \leq C$. Now, note that this is just the norm of $(L^q)^*=L^p$. Hence $u_j \in L^p$.