A useful criterion in dispersive PDE.

Question

I would like to prove the following theorem: Consider two Banach spaces $X\hookrightarrow Y$ and $1<p,q\leq\infty$. Let $(f_n)_{n\geq 0}$ be a bounded sequence in $L^q(I,Y)$ and let $f:I\mapsto Y$ be such that $f_{n}(t)\rightharpoonup f(t)$ in $Y$ for a.a. $t\in I$. If $f_n$ is bounded in $L^p(I,X)$ and if $X$ is reflexive, then $f\in L^p(I,X)$ and $\|f\|_{L^p(I,X)}\leq\liminf_{n\rightarrow}\|f_n\|_{L^p(I,X)}$.

I don't have any references.