Brezis–Van Schaftingen–Yung formula in Orlicz spaces

Abstract

In [13], Brezis–Van Schaftingen–Yung discovered new expression of the Sobolev semi-norm by replacing the Lp(R2N) norm in the Gagliardo Ws,p(RN) semi-norm with weak Lp(R2N) quasi-norm and setting s=1. In this paper we prove the same formula for functions in Orlicz–Sobolev spaces, in which the role of the power function t↦tp is performed by more general finite valued Young functions.