= Mean control for positive imaging operators
{title2=$u_\Omega\|T1\|_1\le\|Tu\|_1+C_\Omega\|T\|\operatorname{TV}(u)$}
On a bounded connected <Lipschitz domain>, let $u\ge0$ belong to the <BV space>, and let $T$ be a <positivity-preserving operator> with $T1\ne0$. Writing $u=c1+(u-c1)$, $c=u_\Omega\ge0$, the <triangle inequality> and <Poincaré inequality for total variation> give $c\|T1\|_1\le\|Tu\|_1+\|T\|\|u-c\|_1\le\|Tu\|_1+C_\Omega\|T\|\operatorname{TV}(u)$. Thus forward-image and variation bounds control the missing constant mode, and hence the full $BV$ <norm>.
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