Characterizations of even-order Musielak–Orlicz–Sobolev spaces via ball averages and their derivatives

• DOI: 10.1007/s13324-019-00342-6
• Publication Date: 2019-10-19T15:39:07Z
• Authors: Tao Ma; Zhenyu Yang
• Keywords: Birnbaum–Orlicz space; Ball (mathematics); Smoothness; Function space; Sobolev inequality

In this paper, the authors present some new characterizations of the Musielak–Orlicz–Sobolev spaces with even smoothness order via ball averages and their derivatives on the radius. Consequently, as special examples of the Musielak–Orlicz–Sobolev spaces studied in this paper, the corresponding characterizations for some weighted Sobolev spaces, Orlicz–Sobolev spaces and variable Sobolev spaces are also obtained. Since these characterizations depend only on ball averages and their derivatives on the radius, they provide some possible ways to introduce the corresponding function spaces on any metric measure space.