Total variation calibration

For an absolutely one-homogeneous functional $J=\mathrm{TV}$ on a real Hilbert space, a subgradient $q$ at $u$ is characterized by $⟨q,v⟩\le J(v)$ for every $v$ and $⟨q,u⟩=J(u)$. A bounded vector field with $|z|\le 1$ and $q=\mathrm{div}z$ proves the inequality through the dual definition, provided cutoff approximation is valid in the ambient space. Equality is called a calibration. It certifies a global minimizer of total variation denoising, rather than only a minimizer within a chosen ansatz.