an:00936753
Zbl 0859.46022
Haj??asz, Piotr
Sobolev spaces on an arbitrary metric space
EN
Potential Anal. 5, No. 4, 403-415 (1996).
00033510
1996
j
46E35 28A80
metric space; finite diameter; positive Borel measure; imbedding theorems; Muckenhoupt weight
Summary: We define the Sobolev space \(W^{1,p}\) for \(1<p\leq\infty\) on an arbitrary metric space with finite diameter and equipped with a 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.