Morse-Sard theorem for d.c. functions and mappings on $\mathbb{R}^2$

Abstract

If f is a d.c. function on R 2 (i.e., f = f 1 - f 2 , where f 1 , f 2 are convex) and C is the set of all critical points off, then f(C) is a Lebesgue null set. This result was published by E. Landis in 1951 with a sketch of a proof which is based on the notion of planar variation of (discontinuous) functions on R 2 . We present a similar complete proof based on the well-known theory of BV functions and on a recent result of Ambrosio, Caselles, Masnou and Morel on sets with finite perimeter. Moreover, we generalize Landis' result to the case of a d.c. mapping f: R 2 → X, where X is a Banach space. Also results on Lipschitz BV 2 functions on R n are proved.