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

Authors: D. Pavlica and L. Zajicek

Issue: Volume 55 (2006), Issue 3, 1195-1208

Abstract: If $f$ is a d.c. function on $\mathbb{R}^2$ (i.e., $f = f_1 - f_2$, where $f_1$, $f_2$ are convex) and $C$ is the set of all critical points of $f$, 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 $\mathbb{R}^2$. 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 $f : \mathbb{R}^2 \to X$, where $X$ is a Banach space. Also results on Lipschitz $BV_2$ functions on $\mathbb{R}^n$ are proved.