Weighted Hardy-Type Inequalities for Differences and the Extension Problem for Spaces with Generalized Smoothness |
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Authors: | Burenkov V I; Evans W D |
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Institution: | School of Mathematics, University of Wales Cardiff Senghennydd Road, Cardiff CF2 4YH |
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Abstract: | It is well known that there are bounded domains Rn whose boundaries are not smooth enough for there to exist a bounded linear extensionfor the Sobolev space into , but the embedding is nevertheless compact. For the Lipboundaries (0<<1) studied in 3, 4], there does not existin general an extension operator of into but there is a bounded linear extension of into and the smoothness retained by thisextension is enough to ensure that the embedding is compact. It is natural to ask if this is typicalfor bounded domains which are such that is compact, that is, that there exists a boundedextension into a space of functions in Rn which enjoy adequatesmoothness. This is the question which originally motivatedthis paper. Specifically we study the extension by zerooperator on a space of functions with given generalizedsmoothness defined on a domain with an irregular boundary, anddetermine the target space with respect to which it is bounded. |
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