Generalized Orlicz spaces and related PDE

We prove the boundedness of the maximal operator in generalized Orlicz spaces defined on subsets of RnRn. The proof is based on an extension result for ΦΦ-functions. We study generalized Sobolev–Orlicz spaces and establish density of smooth functions and the Poincaré inequality. As applications we establish the existence of solutions of the φφ-Laplace equation with zero and non-zero right-hand side. Further, we systematize assumptions for ΦΦ-functions and prove several basic tools needed for the study of differential equations of generalized Orlicz growth.