BV translation estimate ($\Vert w(\cdot -h)-w{\Vert }_1\le |h||Dw|({\mathbb{R}}^n)$)

For a smooth function, integrate its directional derivative along each segment of length $|h|$ and then integrate in space. This gives the displayed bound with $\int |\nabla w|$. Smooth BV approximation and sequential lower semicontinuity give the same inequality for general functions in the BV space. Averaging it over a mollifier proves the BV mollification error estimate.

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.