PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 130, Number 2, Pages 405–412 S 0002-9939(01)06044-0 Article electronically published on May 25, 2001 ON THE COMPARISON OF THE SPACES L 1 BV (R n ) AND BV (R n ) YUDI SOEHARYADI (Communicated by Carmen C. Chicone) Abstract. The notion of L 1 -variation and the space L 1 BV arise in the study of regularity properties of solutions to perturbed conservation laws. In this article we show that this notion is equivalent to variation in the regular sense, and therefore the space L 1 BV is the same as the space BV in the sense of Cesari-Tonelli. We also point out some connection between the space L 1 BV and the Favard classes for translation semigroups. 1. Introduction We recently proposed measuring variation of functions utilizing the L 1 -norm [9]. For a measurable function on R n we define L 1 V ar(f ) = sup h=0 1 | h |  R n | f h − f | dx, where h ∈ R n and f h (x)= f (x + h). We also define L 1 BV (R n )= {f ∈ L 1 (R n ): L 1 V ar(f ) < ∞}. The norm ‖f ‖ L 1 BV = L 1 V ar(f )+ ‖f ‖ L 1 equips the space L 1 BV (R n ) with a Banach space structure. Note that the expression for L 1 V ar is similar to the concept of differential quotient used by Lions and Magenes [10] to study regularity problems in Hilbert spaces. The motivation to define variation in the L 1 -sense arises from measuring regu- larity of solutions to perturbed conservation laws of the form u t + div(f (u)) + g(u) H (x) =0,x ∈ R n ,t> 0, u(x, 0) = u 0 (x),x ∈ R n ; (1.1) see [9]. It is well known that solutions to this type of equation lose regularity due to occurence of shocks [13]. This Cauchy problem can be considered as an ordinary Received by the editors March 1, 2000 and, in revised form, June 12, 2000. 2000 Mathematics Subject Classification. Primary 46B99, 35D10, 47H20, 47D03. Key words and phrases. L 1 -variation, variation, total variation, essential variation, conserva- tion laws, perturbed conservation laws, m-dissipative operator, invariant set, Favard class. c 2001 American Mathematical Society 405 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use