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Density of \(C^{\infty}_0 (\mathbb R^n)\) in \(W^{1, p(x)} (\mathbb R^n)\) with assumption density of \(C^1 (\mathbb R^n)\). (English) Zbl 1330.46033

Summary: We present a sufficient condition for the density of \(C^{\infty}_0 (\mathbb R^n)\) in \(W^{1, p (x)} (\mathbb R^n)\) with the assumption that \(p(x)\) satisfies a condition such that \(C^1(\mathbb R^n)\) is dense in \(W^{1,p(x)}(\mathbb R^n)\). The origin of our work comes from a similar question of P. A. Hästö [Rev. Mat. Iberoam. 23, No. 1, 213–234 (2007; Zbl 1144.46031)] under the density of continuous or Hölder continuous functions whether it is possible to deduce the density of smooth functions.

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.)

Citations:

Zbl 1144.46031