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Sobolev embedding theorems for spaces \(W^{k,p(x)}(\Omega)\). (English) Zbl 0995.46023
The well-known Sobolev embedding theorem in \({\mathbb{R}}^N\), \[ W^k_p (\Omega) \hookrightarrow L_q (\Omega) \] is extended to cases with variable integrability: \[ W^k_{p(x)} (\Omega) \hookrightarrow L_{q(x)} (\Omega). \] This applies also to compactness assertions if the domain \(\Omega\) is bounded.

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