Sobolev inequalities for Musielak-Orlicz spaces. (English) Zbl 1392.46037
Summary: Our aim in this paper is to deal with Sobolev’s embeddings for Musielak-Orlicz-Sobolev functions in \(W^{1,\Phi }_0(\Omega )\) for \(\Omega \subset \mathbb {R}^N\), as extensions of P. Harjulehto and P. Hästö [Publ. Mat., Barc. 52, No. 2, 347–363 (2008; Zbl 1163.46022)], P. A. Hästö [Math. Res. Lett. 16, No. 2–3, 263–278 (2009; Zbl 1184.46033)] and P. Hästö et al. [Glasg. Math. J. 52, No. 2, 227–240 (2010; Zbl 1206.46035)]. Here \(\Phi \) is a function such that \(\phi (x,t)=t^{-1} \Phi (x,t)\) is uniformly almost increasing positive function of \(t > 0\).
MSC:
| 46E35 | Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems |
| 46E30 | Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) |
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