Some approximation results in Musielak-Orlicz spaces

Abstract

We give sufficient conditions for the continuity in norm of the translation operator in the Musielak-Orlicz LM spaces. An application to the convergence in norm of approximate identities is given, whereby we prove density results of the smooth functions in LM , in both modular and norm topologies. These density results are then applied to obtain basic topological properties. 1 2 A. YOUSSFI AND Y. AHMIDA Define M : Ω × [0, ∞) → [0, ∞] by M (x, s) = sup t 0 {st − M (x, t)} for all s 0 and all x ∈ Ω. It can be checked that M ∈ φ. The Φ-function M is called the complementary function to M in the sense of Young. Given M ∈ φ, the Musielak-Orlicz space L M (Ω) consists of the set of all measurable functions u : Ω → R such that Ω M (x, |u(x)|/λ)dx < +∞ for some λ > 0. Equipped with the so-called Luxemburg norm [11, Theorem 7.7]). It is a particular case of the so-called modular function spaces, investigated by H. Nakano (see for instance [12]). We define E M (Ω) as the subset of L M (Ω) of all measurable functions u : Ω → R such thatDensity result of smooth functions in Musielak-Orlicz-Sobolev spaces with respect to the modular topology was claimed for the first time in [2] in Ω = R N and then for a bounded star-shape Lipschitz domain Ω in [3]. The authors assumed that the Φ-function M satisfies, among others, the log-Hölder continuity condition, that is to say there exists a constant A > 0 such that for all s 1,

Gossez's approximation theorems in Musielak–Orlicz–Sobolev spaces

“…Such a property in [16] guarantees that any element of W m L M (Ω) with compact support in Ω belongs to W m−1 E M (Ω). The embedding and approximate results obtained in [13] allowed Gossez to prove only the convergence of smooth functions with compact support only for |α| = m. Here, we extend the result to the more general setting of Musielak-Orlicz-Sobolev spaces and we enhance it by removing the cone property using [39,Lemma 4.1]. Our approach is based on the mean continuity of the translation operator on the set of bounded functions compactly supported in Ω.…”

“…Let M be an N -function whose complementary N -function M * satisfy the condition (M1). Then from [39,Theorem 1.4], the dual space of E M * is isomorphic to L M and the following weak- * topology σ(ΠL M , ΠE M * ) is well defined, thereby we define the space…”

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]. 5 and 1.6, then…”

“…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.…”