Some approximation results in Musielak-Orlicz spaces

Poincaré-type inequalities in Musielak spaces

“…However, proving the Poincaré integral inequality for functions in C ∞ 0 (Ω) and then extending it by a density argument (as is often done for a constant exponent) is not an easy task since the passage to the limits is not allowed because of the lack in general of density of smooth functions in W m 0 L M (Ω) at least in the modular sense (see Definition 2.1). This is mainly due to the fact that the shift operator is not acting in general on Musielak spaces unless some regularity conditions on the Musielak function M are satisfied see [2,21].…”

New characterizations of Musielak-Orlicz-Sobolev spaces via sharp ball averaging functions

“…To prove Theorem 1.10, we need the following reflexivity of the Musielak-Orlicz space L Φ (R n ), which was obtained in [39,Theorem 1.4].…”

“…It is worth pointing out that Musielak-Orlicz spaces or Musielak-Orlicz-Sobolev spaces naturally appear in the study of the regularity for solutions of some nonlinear elliptic equations or minimizers of functionals with non-standard growth (see, for example, [1,2,9,10]). We also refer the reader to [31,32,38,39] for some recent progresses about the real-variable theory of both Musielak-Orlicz-Sobolev spaces and function spaces of Musielak-Orlicz type.…”

“…(iv) More examples of Musielak-Orlicz functions satisfying Assumption 1.7 were given in [4] (see also [39,Example 2.1]). Moreover, some necessary and sufficient conditions for the Musielak-Orlicz function Φ satisfying Assumption 1.8 were established in [13,24].…”