Abstract
We define Sobolev space W 1,p for 1<p≤∞ on an arbitrary metric space with finite diameter and equipped with finite, positive Borel measure. In the Euclidean case it coincides with standard Sobolev space. Several classical imbedding theorems are special cases of general results which hold in the metric case. We apply our results to weighted Sobolev space with Muckenhoupt weight.
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This work is supported by KBN grant no. 2 1057 91 01
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Hajłasz, P. Sobolev spaces on an arbitrary metric space. Potential Anal 5, 403–415 (1996). https://doi.org/10.1007/BF00275475
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DOI: https://doi.org/10.1007/BF00275475