L1 Compactness of Bounded BV Sets

Details L1 Compactness of Bounded BV Sets L1 Compactness of Bounded BV Sets

2002-12-15

arxiv.org/abs/math/0212198v1

Mathematics

Functional Analysis

3 pages

Scientific paper

Functions, uniformly bounded in $BV$ norm in some bounded open set $U$ in $R^n$, are compact in $L_1(U)$. This result is known when $U$ has Lipschitz boundary [EG Th. 4 p. 176], [G 1.19 Th. p. 17], [Z 5.34 Cor. p. 227]; the proof for general $U$ here, after identifying the operator theoretic definition of bounded $BV$ norm with that of the Tonelli variation, appeals to the standard compactness criterion in $L_1$ [DS 21 TH. p. 301] [Y, p. 275] (For completeness, these two auxiliary results are also presented).

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