BV mollification error estimate ($\Vert w\ast {\rho }_{ϵ}-w{\Vert }_1\le ϵ|Dw|({\mathbb{R}}^n)$)

For a nonnegative normalized mollifier supported in the radius-$ϵ$ ball, integrate the BV translation estimate with weight ${\rho }_{ϵ}(h)$. The full-space derivative variation controls the full-space error. Replacing it by variation on a smaller domain is invalid without a support condition: a nonconstant bump outside that domain has zero derivative measure inside but a nonzero mollification error.