Let $(\cdot,\cdot)$ be the $L^2(\Omega)$ scalar product, and let $V=L^1(\Omega)$, $W=H^{-1}(\Omega)$ (the dual space of $H_0^1(\Omega)$). My question is if there exists a constant $C$, such that for all $a,b>0$ it holds
$$ \inf_{f=v+w} \{a^{-1}\|v\|_{V}+b^{-1}\|w\|_{W} \}\leq C \sup_{a\|g\|_{V^*}+b\|g\|_{W^*}=1}(f,g)\quad\forall f \in C_c^\infty(\Omega). $$
EDIT: Note that I switched the roles of spaces and duals. Also, the reason I am asking is that I read that
$$(\phi_n,\psi)\leq \|\psi\|_{L^\infty}+o(n)\|\psi\|_{H^1_0}\quad\forall \psi\in C^\infty_c(\Omega) $$ implies that $\phi_n=\phi^{(1)}_n+\phi^{(2)}_n$ with $\phi^{(1)}$ bounded in $L^1$ and $\phi_n^{(2)}$ compact in $H^{-1}$. Maybe this is true for another reason?