If region $\Omega$ is bounded and $u_n$ has weak star convergence in $L^\infty ( \Omega)$ to some $u\in L^\infty(\Omega)$ , does it imply that $u_n$ converges weakly in any $L^p(\Omega) $ ?
I think i got it : If $sup$ of a function is finite then integral over a bounded region is finite with any $p$ norm . is it right ?