Given a lattice in R2 and a function f in L2 (R), the associated Gabor system is a collection of functions obtained by taking modulations and translations of f associated to points in the lattice. The classical Balian-Low theorem is a

manifestation of the uncertainty principle in the setting of Gabor systems; it states that if both f and its Fourier transform are in the Sobolev space H1 (R), then the Gabor system associated to f and the integer lattice is not an orthonormal basis (or, more generally, a frame) for L2 (R). We generalize this result by showing that if f is in Hp/2 and its Fourier transform is in Hq/2 with p and q conjugate exponents, then the associated Gabor system is not a frame. We accomplish this by proving a variant of the endpoint Sobolev embedding into VMO.