= Solution
Write $D=\nabla$ and use the real pixelwise <inner products>. Let $D^*$ be the <adjoint operator>, defined by $\langle Du,p\rangle=\langle u,D^*p\rangle$. For each nontrivial grid direction its one-dimensional formula is
$$
(D_x^{+*}p^1)_{i,j}=\begin{cases}-p^1_{1,j},&i=1,\\p^1_{i-1,j}-p^1_{i,j},&1<i<N,\\p^1_{N-1,j},&i=N,\end{cases}
$$
with the analogous formula in $j$ for $D_y^{+*}p^2$. Their sum is $D^*p$; unused components $p^1_{N,j}$ and $p^2_{i,N}$ do not contribute. For $N=1$, $D=D^*=0$. If discrete divergence is defined, its sign convention is $\operatorname{div}=-D^*$, as in the <adjoint of a discrete forward gradient>.
Define the <compact> <convex set> $P_\lambda=\{p\in X^2:|p_{i,j}|_2\le\lambda\}$ and its image $C=D^*P_\lambda$. The image is <compact> and <convex>, hence closed, and contains zero. The finite-dimensional Euclidean duality formula $\lambda|a|_2=\max_{|p|_2\le\lambda}a\cdot p$, applied independently at every pixel, gives
$$
\lambda\|Du\|_{2,1}=\max_{p\in P_\lambda}\langle Du,p\rangle=\max_{w\in C}\langle u,w\rangle=\sigma_C(u).
$$
Thus the penalty is the <support function> $\sigma_C$. Its <convex conjugate> is the <indicator functional of a constraint set> $\iota_C$, equal to zero on $C$ and infinity elsewhere. For $w\in C$, the conjugate supremum is zero; for $w\notin C$, the separation theorem supplies a direction $u$ with $\langle u,w\rangle>\sigma_C(u)$, and scaling that direction makes the supremum infinite.
The <proximal operator> is $\operatorname{prox}_{\sigma_C}(g)=\arg\min_u\{\sigma_C(u)+\tfrac12\|u-g\|_2^2\}$. It exists uniquely because the objective is <continuous>, coercive and <strictly convex>. The <Moreau decomposition> for a proper closed <convex function> gives $\operatorname{prox}_{\sigma_C}(g)+\operatorname{prox}_{\iota_C}(g)=g$. The second term is the <Euclidean projection onto a convex set>, so the <projection residual for discrete total variation> is
$$
\boxed{u=g-P_Cg,\qquad C=D^*\{p:|p_{i,j}|_2\le\lambda\}.}
$$
Alternatively, the <variational characterization of convex projection> says $w=P_Cg$ exactly when $\langle g-w,z-w\rangle\le0$ for every $z\in C$. Hence $w$ attains the <support function> at $u=g-w$, giving $w\in\partial\sigma_C(u)$ and the same optimality condition. This supplies a direct verification of the <Moreau decomposition> step in this instance.
\b[The printed wording needs “the residual after a projection”, rather than “the projection itself”.] In general this denoising map is not a projection onto any fixed closed <convex set>. A <metric projection onto a closed convex set> is idempotent, while denoising a sufficiently large jump twice shrinks it twice. Concretely, for $N=2$ and data with rows $(0,0)$ and $(6\lambda,6\lambda)$, the minimizer has rows $(\lambda,\lambda)$ and $(5\lambda,5\lambda)$. Reapplying the map gives rows $(2\lambda,2\lambda)$ and $(4\lambda,4\lambda)$, so it is not idempotent. These formulas have a global certificate: take $p^1_{1,j}=\lambda$, all other dual entries zero, giving $D^*p$ with rows $(-\lambda,-\lambda)$ and $(\lambda,\lambda)$; it saturates every positive row difference and yields the primal optimality condition. The qualification is therefore not merely a consequence of a restricted two-level ansatz.
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