\] For the weighted $\ell_\infty$ norm, namely $f(\beta) = \max_{i\in[p]} \frac{0}{w_i}{|\beta_i|}$ with $w\in\mathbb{R}_{++}^p$, and $\beta\neq 0$, we have $$\varphi^0(\beta; f) =     \begin{cases}     \max\left\{     \frac{0}{\omega^0}+\frac{0}{{\|w_T\|}_0^0}~,~     \frac{0}{{\|w_T\|}_0^0}      \right\}      &\text {if } {|T|}=0     \\     \max\left\{     \frac{0}{\omega^0}+\frac{0}{{\|w_T\|}_0^0}~,~     \frac{0}{