Let be a bounded open subset of with Lipschitz boundary and let be Lipschitz-continuous. The authors consider the generalised Lebesgue space and the corresponding Sobolev space , consisting of all with first-order distributional derivatives in .
It is shown that if for all , then there is a constant such that for all ,
Here is the norm on an appropriate space of Orlicz-Musielak type and is the norm on . The inequality reduces to the usual Sobolev inequality if . Corresponding results are proved for the case in which for all .