Compact Sobolev imbeddings on finite measure spaces
β Scribed by James G. Hooton
- Publisher
- Elsevier Science
- Year
- 1981
- Tongue
- English
- Weight
- 543 KB
- Volume
- 83
- Category
- Article
- ISSN
- 0022-247X
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## Abstract Let Ξ© be an open subset of β^__n__^ and let __p__ β [1, __n__). We prove that the measure of nonβcompactness of the Sobolev embedding __W__^__k,p__^~0~(Ξ©) β __L__^__p__\*^(Ξ©) is equal to its norm. This means that the entropy numbers of this embedding are constant and equal to the norm.
It is proved that a weak\* compact subset A of scalar measures on a \_-algebra is weakly compact if and only if there exists a nonnegative scalar measure \* such that each measure in A is \*-continuous (such a measure \* is called a control measure for A). This result is then used to obtain a very g