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Denseness of C0(RN) in the Generalized Sobolev Spaces

WM,P(X)(RN)
  • Stefan Samko
Part of the International Society for Analysis, Applications and Computation book series (ISAA, volume 5)

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 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

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Copyright information

© Springer Science+Business Media Dordrecht 2000

Authors and Affiliations

  • Stefan Samko

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