# Denseness of *C*_{0}^{∞}(*R*^{N}) in the Generalized Sobolev Spaces

^{M,P(X)}(R

^{N})

## Abstract

The spaces *L* ^{ p } ^{(x)} (Ω), Ω ⊆ *R* ^{ n }, with variable order *p*(*x*) were studied recently. We refer to the pioneer work by I.I. Sharapudinov [6] and the later papers by O.Ková\(\tilde c\)ik and J. Rákosník [2] and by the author [3]–[5]. In the paper [2] the Sobolev type spaces *W* ^{ m } ^{ p } ^{(x)} (Ω) were also studied. D.E.Edmunds and J. Rákosník [1] dealt with the problem of denseness of *C* ^{∞}-functions in *W* ^{ m } ^{ p(x)}( Ω) and proved this denseness under some special monotonicity-type condition on *p*(*x*). We prove that *C* _{0} ^{∞} (*R* ^{ n })is dense in *W* ^{ m } ^{ p } ^{(} ^{ x } ^{)}(*R* ^{ n }) without any monotonicity condition, requiring instead that *p*(*x*)is somewhat better than just continuous — satisfies the Dini-Lipschitz condition. For this purpose we prove the boundedness of the convolution operators \(\frac{1}{{{ \in ^n}}}\kappa (\frac{x}{ \in })*f\)
in the space *L* ^{ p } ^{(x)} uniform with respect to є. This is the main result, the above mentioned denseness being its consequence, in fact.

## Keywords

Compact Support Variable Order Convolution Operator Uniform Boundedness Variable Exponent## Preview

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## References

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