Is this particular subset of $L^{1}([0,1])$ a weakly compact set?

Question

I was wondering if this set $$A_{3}=\bigl\{ u \in L^{1}([0,1]): \lvert\lvert u \rvert\rvert_{L^{1}([0,1])} \leq 1 \quad \int_{0}^{1}\frac{u(x)}{sin(x-\theta)} dx \leq1 \quad \forall \theta \in (0,\pi)\bigr\},$$ is weakly compact.