Let $(X,Sigma,mu)$ be a finite measure space and let $L^2(mu,mathbb{R}^d)$ denote the Bochner space of strongly measurable functions taking values in $mathbb{R}^d$. Let

$$

begin{aligned}

T &:L^2(mu,mathbb{R}^d)rightarrow L^2(mu,mathbb{R})\

& fmapsto |f|_2,

end{aligned}

$$

where $|cdot|_2$ is the Euclidean norm on $mathbb{R}^d$. Fix a $phiinGamma_0(L^2(mu,mathbb{R}))$.

Is there a known expression for

$

operatorname{Prox}_{Tcirc phi}

$

in terms of $operatorname{Prox}_{phi}$, $phi$, and $T$?