Weighted Sobolev spaces, capacities and exceptional sets. (English) Zbl 1466.46031
Summary: We consider the weighted Sobolev space \(W_\omega^{m,p}(\Omega)\), where \(\Omega\) is an open subset of \(\mathbb R^n\), \(n\geq 2\), and \(\omega\) is a Muckenhoupt \(A_p\)-weight on \(\mathbb R^n\), \(1 \leq p < \infty \), \(m \in \mathbb{N}\). For the equalities \(W_\omega^{m,p}(\Omega\setminus E) = W_\omega^{m,p}(\Omega)\), \(\mathring{W}_\omega^{m,p}(\Omega\setminus E) =\mathring{W}_\omega^{m,p}(\Omega)\) to hold, conditions are obtained in terms of \(E\) as a set of zero \((p, m, \omega )\)-capacity, or an \(NC_{p,\omega}\)-set for the first equality. For the equality \(W^{m,p}(\Omega) = \mathring{W}^{m,p}(\Omega)\), the conditions are established for \(\mathbb R^n\setminus \Omega\) as a set of zero \((p, m, \omega)\)-capacity. Similar results are partially true for \(W_{p,\omega}^m(\Omega)\), \(L^m_{p,\omega}(\Omega)\).
MSC:
| 46E35 | Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems |
| 31C45 | Other generalizations (nonlinear potential theory, etc.) |
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