Characterization of the variable exponent Sobolev norm without derivatives

Abstract

The norm in classical Sobolev spaces can be expressed as a difference quotient. This expression can be used to generalize the space to the fractional smoothness case. Since the difference quotient is based on shifting the function, it cannot be generalized to the variable exponent case. In its place, we introduce a smoothed difference quotient and show that it can be used to characterize the variable exponent Sobolev space.

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

“…Instead of this, Diening and Hästö [16] creatively replaced the different quotient in the definition of the trace space by the sharp averaging operator M ♯ B (see, for example, [25, (1.3)] or (1.4) below). Motivated by the work [16], under the assumptions that the variable exponent p(·) satisfies the local log-Hölder continuity condition, the log-Hölder decay condition (at infinity) and p − ∈ (1, ∞) with p − as in (1.2), Hästö and Ribeiro [25,Theorem 4.1] obtained a new characterization of the variable exponent Sobolev space in terms of the sharp averaging operator, which is re-stated as Theorem 1.A below.…”

“…In this article, motivated by [16,17,25], we obtain a new characterization of the Musielak-Orlicz-Sobolev space on R n , including the classical Orlicz-Sobolev space, the weighted Sobolev space and the variable exponent Sobolev space, in terms of sharp ball averaging functions. Even in the special case, namely, the variable exponent Sobolev space, the obtained result in this article improves the corresponding result obtained in [25,Theorem 4.1] via weakening the assumption f ∈ L 1 (R n ) into f ∈ L 1 loc (R n ), which positively confirms a conjecture proposed by Hästö and Ribeiro in [25,Remark 4.1].…”

“…Moreover, from (1.6) and the properties of Musielak-Orlicz functions, we further deduce that (1.7) holds true. Comparing with [25,Theorem 4.1], instead of f ∈ L 1 (R n ), we now only need to assume that f ∈ L 1 loc (R n ) in Theorem 1.10. To overcome the difficulty causing by this weaker assumption, in the proof of Theorem 1.10, we flexibly use the boundedness of the Hardy-Littlewood maximal operator on the Musielak-Orlicz space L Φ (R n ), which is assumed to hold true, and the subtle growth properties of Musielak-Orlicz functions obtained in Lemma 2.2 below.…”

Characterization of generalized Orlicz spaces

“…It is not difficult to see that a direct generalization of the difference quotient to the nontranslation invariant generalized Orlicz case is not possible [24,Section 1]. For instance Besov and Triebel-Lizorkin spaces in this context have been defined using Fourier theoretic approach [1,15].…”

“…For instance Besov and Triebel-Lizorkin spaces in this context have been defined using Fourier theoretic approach [1,15]. Hästö and Ribeiro [24], following [13], adopted a more direct approach with a smoothed difference quotient expressed by means of the sharp averaging operator Theorem 1.1. Let Ω ⊂ R n be an open set, let ϕ ∈ Φ(Ω) satisfy Assumptions (A), (aInc), and (aDec), and let (ψ ε ) ε be a family of functions satisfying (1.1) and (1.2).…”