Given a function, $$f = \ell^{(\mathcal{\alpha})}_{ij}(a) = \max \left\{ \alpha a_{i}, (\alpha-1) a_{i} \right\}$$ I assume its conjugate has the form, $$f^*(y) = \sup_{x \in dom f} (y^Tx-f(x)) = \sup_{x \in dom f} (y^Ta - \max \left\{ \alpha a_{i}, (\alpha-1) a_{i} \right\}) \\ = \begin{cases} ? \quad \\ \infty \quad \text{Otherwise} \end{cases}$$ However I stopped here and don't know how to derive further.
In fact, I want to derive the proximal operator of $\ell^{(\mathcal{\alpha})}_{ij}(a)$, which I think maybe this is the intermediate step?