Let $v(x) = |x|^{2-N}$, let $a = (1, 0, \ldots, 0)$ be a unit vector, let $$\Delta_a f(x) = f(x + a) - f(x)$$ be the difference operator, and finally let $u$ be the third-order finite difference of $v$: $$u(x) = \Delta_a^3 v(x) .$$ Then:

* $\Delta u$ is a finite measure concentrated at $\{0, a, 2a, 3a\}$;

* $u$ is locally integrable, $u \approx |x|^{-1-N}$ at infinity, and hence $u$ is integrable;

* $|\nabla u| \approx |x|^{-N}$ at infinity, and hence $\nabla u$ is not integrable.

Thus, $u$ is a counter-example.