Previous |  Up |  Next


We study the relationship between derivates and variational measures of additive functions defined on families of figures or bounded sets of finite perimeter. Our results, valid in all dimensions, include a generalization of Ward’s theorem, a necessary and sufficient condition for derivability, and full descriptive definitions of certain conditionally convergent integrals.
[B] B. Bongiorno: Essential variation. Measure Theory Oberwolfach 1981, Springer Lecture Notes Math., 945, 1981, pp. 187–193. MR 0675282
[BPS] B. Bongiorno, L. Di Piazza and V. Skvortsov: The essential variation of a function and some convergence theorems. Anal. Math. 22 (1996), no. 1, 3–12. DOI 10.1007/BF02342334 | MR 1384345
[BPT] B. Bongiorno, W.F. Pfeffer and B.S. Thomson: A full descriptive definition of the gage integral. Canadian Math. Bull. 39 (1996), no. 4, 390–401. DOI 10.4153/CMB-1996-047-x | MR 1426684
[BV] B. Bongiorno and P. Vetro: Su un teorema di F. Riesz. Atti Acc. Sci. Lettere e Arti Palermo (IV) 37 (1979), 3–13.
[Di] L. Di Piazza: A note on additive functions of intervals. Real Anal. Ex. 20(2) (1994–95), 815–818. MR 1348103
[EG] L.C. Evans and R.F. Gariepy: Measure Theory and Fine Properties of Functions. CRC Press, Boca Raton, 1992. MR 1158660
[Fe] H. Federer: Geometric Measure Theory. Springer-Verlag, New York, 1969. MR 0257325 | Zbl 0176.00801
[Gu] E. Giusti: Minimal Surfaces and Functions of Bounded Variation. Birkhäuser, Basel, 1984. MR 0775682 | Zbl 0545.49018
[P1] W.F. Pfeffer: The Gauss-Green theorem. Adv. Math. 87 (1991), 93–147. DOI 10.1016/0001-8708(91)90063-D | MR 1102966 | Zbl 0732.26013
[P3] W.F. Pfeffer: A descriptive definition of a variational integral and applications. Indiana Univ. Math. J. 40 (1991), 259–270. DOI 10.1512/iumj.1991.40.40011 | MR 1101229 | Zbl 0747.26010
[P] W.F. Pfeffer: The Riemann Approach to Integration. Cambridge Univ. Press, Cambridge, 1993. MR 1268404 | Zbl 0804.26005
[Sa] S. Saks: Theory of the Integral. Dover, New York, 1964. MR 0167578
[St] E.M. Stein: Singular Integrals and Differentiability Properties of Function. Princeton Univ. Press, Princeton, 1970. MR 0290095
[T] B. S. Thomson: Derivates of Interval Functions. Mem. Amer. Math. Soc., #452, Providence, 1991. MR 1078198 | Zbl 0734.26003
[V] A.I. Volpert: The spaces $BV$ and quasilinear equations. Math. USSR-SB. 2 (1967), 255–267. MR 0216338
Partner of
EuDML logo