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Article

MSC: 31B15, 46E35
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Keywords:
Sobolev space; metric measure space; Hajłasz-Sobolev space; Musielak-Orlicz space; capacity; variable exponent; zero boundary values
Summary:
We define and study Musielak-Orlicz-Sobolev spaces with zero boundary values on any metric space endowed with a Borel regular measure. We extend many classical results, including completeness, lattice properties and removable sets, to Musielak-Orlicz-Sobolev spaces on metric measure spaces. We give sufficient conditions which guarantee that a Sobolev function can be approximated by Lipschitz continuous functions vanishing outside an open set. These conditions are based on Hardy type inequalities.
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