= Jump-amplitude inequality for a bounded ROF minimizer
{title2=$[w]([g]-[w])\ge0\quad\mathcal H^{n-1}\text{-a.e. on }J_w$}
Assume $w\in BV\cap L^\infty$ minimizes scalar <total variation denoising> for $g\in BV\cap L^2$. The data $g$ may be unbounded. For the <local flow> of $\varphi e_j$, use $(1-\theta)w+\theta w\circ\Phi_{\pm t}$ as two competitors. The <total variation under opposite smooth flows> and convexity of the <total variation seminorm> bound the sum of regularizer changes by $O(t^2)$. Minimality and the <opposite-flow fidelity identity for quadratic data>, evaluated using the <BV jump-product limit with one bounded factor>, imply
$$
\int_{J_w}([g][w]-(1-\theta)[w]^2)\varphi|\nu_w\cdot e_j|\,d\mathcal H^{n-1}\ge0.
$$
Let $\theta\downarrow0$. The integrands define finite signed <Radon measures>, since $w$ is bounded and the jump variation of $g$ is finite. Arbitrary nonnegative smooth $\varphi$ make their densities nonnegative. Testing every coordinate direction removes the factor $|\nu_w\cdot e_j|$ and proves the stated inequality.
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