Failure of a mixed Sobolev estimate for scalar potentials

We construct smooth real mean-zero functions on the two-dimensional unit torus whose gradients stay bounded in $H^{-1/2}+L^1$, while their $H^{1/2}+W^{1,1}$ norms tend to infinity. This disproves the mixed potential estimate proposed by Bourgain and Brezis. The construction uses a finite stationary reflected walk to produce a bounded vector field with a uniformly bounded $H^{-1/2}$ divergence. Pairing this divergence with a smoothed sheet potential gives the divergent lower bound.