We provide answers to some questions raised in a recent work by H. Brezis, J. 3] concerning the Gagliardo semi-norm |u| W s,q computed at s = 1, when the strong L q is replaced by weak L q . In particular, we address generalization of the results in [2, 3] for a general domain and non-smooth functions.
Proof of Theorem 1.5This section is devoted to the proof of Theorem 1.5. Its special case r = q is essential for the proof of the main results Theorem 1.3 and Corollary 1.1. Theorem 1.5 is a particular case of the following more general statment.