Lifting default for S1-valued maps

Let ' 2 C1([0, 1]N ,R). When 0 < s < 1, p 1 and 1  sp < N , the W s,p-semi-norm |'|W s,p of ' is not controlled by |g|W s,p , where g := eı' [3]. [This question is related to existence, for S1-valued maps g, of a lifting ' as smooth as allowed by g.] In [4], the authors suggested that |g|W s,p does control a weaker quantity, namely |'|W s,p+W 1,sp . Existence of such control is due to J. Bourgain and H. Brezis [2] when 1 < p  2, s = 1/p and to H.-M. Nguyen [10] when N = 1, p > 1 and sp 1 or when N 2, p > 1 and sp > 1. In this Note, we establish existence of control for all s < 1, p 1 and N .