Lifting, degree, and distributional Jacobian revisited

Let g : I = (0, 1) → S1. If g ∈ VMO, we may write g = eiφ for some φ ∈ VMO; this φ is unique modulo 2π (see [13] and the earlier work [14]). There is no control of |φ|BMO in terms of |g|BMO, since we always have |g|BMO ≤ 2 and |φ|BMO can be arbitrarily large; recall, however, that, when |g|BMO is sufficiently small, there is a linear estimate |φ|BMO ≤ C |g|BMO (see [13, theorem 4], [14], and Remark 0.2 below). We are going to establish that a norm slightly stronger than |g|BMO does control |φ|BMO. Consider, for 1 < p < ∞, 0 < s < 1, the fractional Sobolev space Ws,p(I ), equipped with its standard seminorm