Sobolev embeddings in Musielak-Orlicz spaces

Abstract

An embedding theorem for Sobolev spaces built upon general Musielak-Orlicz norms is offered. These norms are defined in terms of generalized Young functions which also depend on the x variable. Under minimal conditions on the latter dependence, a Sobolev conjugate is associated with any function of this type. Such a conjugate is sharp, in the sense that, for each fixed x, it agrees with the sharp Sobolev conjugate in classical Orlicz spaces. Both Sobolev inequalities in the whole Rn and Sobolev-Poincaré inequalities in domains are established. Compact Sobolev embeddings are also presented. In particular, optimal embeddings for standard Orlicz-Sobolev spaces, variable exponent Sobolev spaces, and double-phase Sobolev spaces are recovered and complemented in borderline cases. A key tool, of independent interest, in our approach is a new weak type inequality for Riesz potentials in Musielak-Orlicz spaces involving a sharp fractional-order Sobolev conjugate.