Denseness of C0(RN) in the Generalized Sobolev Spaces

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


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.


Compact Support Variable Order Convolution Operator Uniform Boundedness Variable Exponent 
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© Springer Science+Business Media Dordrecht 2000

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  • Stefan Samko

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