Václav Kryštof, Luděk Zajíček
Differences of two semiconvex functions on the real line

Comment.Math.Univ.Carolin. 57,1 (2016) 21-37.

Abstract:It is proved that real functions on $\mathbb R$ which can be represented as the difference of two semiconvex functions with a general modulus (or of two lower $C^1$-functions, or of two strongly paraconvex functions) coincide with semismooth functions on $\mathbb R$ (i.e.\ those locally Lipschitz functions on $\mathbb R$ for which $f'_+(x) = \lim_{t \to x+} f'_+(t)$ and $f'_-(x) = \lim_{t \to x-} f'_-(t)$ for each $x$). Further, for each modulus $\omega$, we characterize the class $DSC_{\omega}$ of functions on $\mathbb R$ which can be written as $f=g-h$, where $g$ and $h$ are semiconvex with modulus $C\omega$ (for some $C>0$) using a new notion of $[\omega]$-variation. We prove that $f \in DSC_{\omega}$ if and only if $f$ is continuous and there exists $D>0$ such that $f'_+$ has locally finite $[D \omega]$-variation. This result is proved via a generalization of the classical Jordan decomposition theorem which characterizes the differences of two $\omega$-nondecreasing functions (defined by the inequality $f(y) \geq f(x)- \omega(y-x)$ for $y>x$) on $[a,b]$ as functions with finite $[2\omega]$-variation. The research was motivated by a recent article by J.~ Duda and L.~ Zaj\'\i\v cek on G\^ateaux differentiability of semiconvex functions, in which surfaces described by differences of two semiconvex functions naturally appear.

Keywords: semiconvex function with general modulus; difference of two semiconvex functions; $\omega$-nondecreasing function; $[\omega]$-variation; regulated function

DOI: DOI 10.14712/1213-7243.2015.153
AMS Subject Classification: 26A51 26B05 26A45 26A48