Are the weak $L^p$ spaces, namely $L^{p,\infty}$, weak or weak-$\star$ compact?
Specifically I would like to know if we have a sequence $(u_\varepsilon)_{\varepsilon>0}$ which is bounded in $L^{p,\infty}(\mathbb{R}^d)$ independently on $\varepsilon > 0$, do we know if there exists an element $u\in L^{p,\infty}(\mathbb{R}^d)$ such that $u_\varepsilon \rightharpoonup u$ weak or weak-$\star$ in $L^{p,\infty}(\mathbb{R}^d)$ maybe up to a subsequence?