Continuous imbedding theorems in Musielak spaces

Abstract

We provide continuous Sobolev-type imbedding results for Musielak spaces. In the first result the target space is a Musielak space bigger than the considered one, while the second is that of bounded and continuous functions. Continuous imbeddings into a particular Musielak space built upon Sobolev conjugate functions generate serious difficulties and does not holds in general for arbitrary Musielak functions, as it is well known in variable exponent Sobolev spaces. We overcome this problem by imposing only a regularity assumption on the Musielak functions without resorting to other restrictions such as the Δ2/∇2 conditions. Nevertheless, the continuous embedding in the space of bounded continuous functions is obtained by using the convex envelope, which then allows us to use a result already obtained in Orlicz spaces. The results we obtain extend and unify those obtained in classical Sobolev spaces, variable exponent Sobolev spaces as well as those obtained by Donaldson-Trudinger in the setting of Orlicz spaces.