Assume we have $u_{k} \rightharpoonup u$ in $L^{q}(\Omega)$, does it then follow that $u_{k} \rightharpoonup u$ in $L^{p}(\Omega)$, given that $q > p$ and $\Omega \subset \mathbb{R}^{n}$ is bounded?
I know strong convergence $u_{k} \rightarrow u$ in $L^{q}(\Omega)$ implies strong convergence $u_{k} \rightarrow u$ in $L^{p}(\Omega)$ given that $\Omega \subset \mathbb{R}^{n}$ is bounded. (using Jensen's Inequality to prove)
Thanks.
Since $\Omega$ is bounded, we have a continuous inclusion
$$j \colon L^q(\Omega) \hookrightarrow L^p(\Omega).$$
Now we have the general fact that a continuous linear map $T\colon X \to Y$ between two normed spaces (it holds more generally for Hausdorff locally convex spaces) is also continuous if we endow both spaces with their respective weak topology. Since a continuous map maps convergent sequences to convergent sequences, we have indeed the conclusion that weak convergence in $L^q(\Omega)$ implies weak convergence in $L^p(\Omega)$ for $\Omega$ of finite measure and $p \leqslant q$.