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How to define reasonably weighted Sobolev spaces. (English) Zbl 0557.46025
Examples are given showing that for some weight functions the usual Sobolev weight space \(W^{k,p}(\Omega;S)\) need not be well defined or need not be complete. A class \(B_ p(\Omega)\) of weight functions is introduced (roughly speaking, the 1/(1-p)-th powers of the weights should be locally integrable) for which the space mentioned has all the above properties and it is shown how to modify the definition if some of the weights do not belong to \(B_ p(\Omega)\). The same is done for the case of the space \(W_ 0^{k,p}(\Omega;S)\)- the closure of \(C^{\infty}_ 0(\Omega).\)
(Authors’ remark: Theorem 2.1 weakening the conditions on the weights is correct only if the dimension of \(\Omega\) is one or if the set \({\mathfrak M}_ 1\) contains all multi-indices of length one.)

46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
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