L1 Compactness of Bounded BV Sets

Isidore Fleischer

paper

L1 Compactness of Bounded BV Sets

arXiv:math/0212198

Abstract

Functions, uniformly bounded in $B V$ 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 $B V$ 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).

3 pages