Differences of two semiconvex functions on the real line

Václav Kryštof; Luděk Zajíček

Commentationes Mathematicae Universitatis Carolinae (2016)

  • Volume: 57, Issue: 1, page 21-37
  • ISSN: 0010-2628

Abstract

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It is proved that real functions on 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 ℝ (i.e. those locally Lipschitz functions on ℝ for which f + ' ( x ) = lim t → x + f + ' ( t ) and f - ' ( x ) = lim t → x - f - ' ( t ) for each x ). Further, for each modulus ω , we characterize the class D S C ω of functions on ℝ which can be written as f = g - h , where g and h are semiconvex with modulus C ω (for some C > 0 ) using a new notion of [ ω ] -variation. We prove that f ∈ D S C ω if and only if f is continuous and there exists D > 0 such that f + ' has locally finite [ D ω ] -variation. This result is proved via a generalization of the classical Jordan decomposition theorem which characterizes the differences of two ω -nondecreasing functions (defined by the inequality f ( y ) ≥ f ( x ) - ω ( y - x ) for y > x ) on [ a , b ] as functions with finite [ 2 ω ] -variation. The research was motivated by a recent article by J. Duda and L. Zajíček on Gâteaux differentiability of semiconvex functions, in which surfaces described by differences of two semiconvex functions naturally appear.

How to cite

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Kryštof, Václav, and Zajíček, Luděk. "Differences of two semiconvex functions on the real line." Commentationes Mathematicae Universitatis Carolinae 57.1 (2016): 21-37. <http://eudml.org/doc/276796>.

@article{Kryštof2016,
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 $f^\{\prime \}_+(x) = \lim _\{t \rightarrow x+\} f^\{\prime \}_+(t)$ and $f^\{\prime \}_-(x) = \lim _\{t \rightarrow x-\} f^\{\prime \}_-(t)$ 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 $f^\{\prime \}_+$ 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 $f(y) \ge f(x)- \omega (y-x)$ 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. Zajíček on Gâteaux differentiability of semiconvex functions, in which surfaces described by differences of two semiconvex functions naturally appear.},
author = {Kryštof, Václav, Zajíček, Luděk},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {semiconvex function with general modulus; difference of two semiconvex functions; $\omega $-nondecreasing function; $[\omega ]$-variation; regulated function},
language = {eng},
number = {1},
pages = {21-37},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Differences of two semiconvex functions on the real line},
url = {http://eudml.org/doc/276796},
volume = {57},
year = {2016},
}

TY - JOUR
AU - Kryštof, Václav
AU - Zajíček, Luděk
TI - Differences of two semiconvex functions on the real line
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2016
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 57
IS - 1
SP - 21
EP - 37
AB - 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 $f^{\prime }_+(x) = \lim _{t \rightarrow x+} f^{\prime }_+(t)$ and $f^{\prime }_-(x) = \lim _{t \rightarrow x-} f^{\prime }_-(t)$ 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 $f^{\prime }_+$ 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 $f(y) \ge f(x)- \omega (y-x)$ 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. Zajíček on Gâteaux differentiability of semiconvex functions, in which surfaces described by differences of two semiconvex functions naturally appear.
LA - eng
KW - semiconvex function with general modulus; difference of two semiconvex functions; $\omega $-nondecreasing function; $[\omega ]$-variation; regulated function
UR - http://eudml.org/doc/276796
ER -

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