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Recently a series of interesting theorems have been proved by Bourgain, Brezis, Mironescu, and Van Schaftingen,  , that involve the divergence and curl of vector fields. Among the many results obtained is the following surprising inequality: Theorem A. (Bourgain and Brezis ). Suppose Z is a smooth vector field, Z(x) = (Z 1 (x), . . . , Z n (x)), of compact support in R n , with n ≥ 3. If curl Z = f and div Z = 0, thendoi:10.4310/mrl.2005.v12.n1.a6 fatcat:6l3bv4nw6fcq3ptreyeum5txiy