Further generalized versions of Ilmanen’s lemma on insertion of $C^{1,\omega }$ or $C_{\text{loc}}^{1,\omega }$ functions

Kryštof, Václav. "Further generalized versions of Ilmanen’s lemma on insertion of $C^{1,\omega }$ or $C^{1,\omega }_{\text{\rm loc}}$ functions." Commentationes Mathematicae Universitatis Carolinae 62.4 (2021): 445-455. <http://eudml.org/doc/298292>.

@article{Kryštof2021,
abstract = {The author proved in 2018 that if $ G $ is an open subset of a Hilbert space, $ f_1,f_2\colon G\rightarrow \mathbb \{R\} $ continuous functions and $ \omega $ a nontrivial modulus such that $ f_1\le f_2 $, $ f_1 $ is locally semiconvex with modulus $ \omega $ and $ f_2 $ is locally semiconcave with modulus $ \omega $, then there exists $ f\in C^\{1,\omega \}_\{\text\{loc\}\}(G) $ such that $ f_1\le f\le f_2 $. This is a generalization of Ilmanen’s lemma (which deals with linear modulus and functions on an open subset of $ \mathbb \{R\}^\{n\} $). Here we extend the mentioned result from Hilbert spaces to some superreflexive spaces, in particular to $ L^p $ spaces, $ p\in [2,\infty ) $. We also prove a “global" version of Ilmanen’s lemma (where a $ C^\{1,\omega \} $ function is inserted between functions on an interval $ I\subset \mathbb \{R\} $).},
author = {Kryštof, Václav},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {Ilmanen’s lemma; $ C^\{1,\omega \} $ function; semiconvex function with general modulus},
language = {eng},
number = {4},
pages = {445-455},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Further generalized versions of Ilmanen’s lemma on insertion of $C^\{1,\omega \}$ or $C^\{1,\omega \}_\{\text\{\rm loc\}\}$ functions},
url = {http://eudml.org/doc/298292},
volume = {62},
year = {2021},
}

TY - JOUR
AU - Kryštof, Václav
TI - Further generalized versions of Ilmanen’s lemma on insertion of $C^{1,\omega }$ or $C^{1,\omega }_{\text{\rm loc}}$ functions
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2021
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 62
IS - 4
SP - 445
EP - 455
AB - The author proved in 2018 that if $ G $ is an open subset of a Hilbert space, $ f_1,f_2\colon G\rightarrow \mathbb {R} $ continuous functions and $ \omega $ a nontrivial modulus such that $ f_1\le f_2 $, $ f_1 $ is locally semiconvex with modulus $ \omega $ and $ f_2 $ is locally semiconcave with modulus $ \omega $, then there exists $ f\in C^{1,\omega }_{\text{loc}}(G) $ such that $ f_1\le f\le f_2 $. This is a generalization of Ilmanen’s lemma (which deals with linear modulus and functions on an open subset of $ \mathbb {R}^{n} $). Here we extend the mentioned result from Hilbert spaces to some superreflexive spaces, in particular to $ L^p $ spaces, $ p\in [2,\infty ) $. We also prove a “global" version of Ilmanen’s lemma (where a $ C^{1,\omega } $ function is inserted between functions on an interval $ I\subset \mathbb {R} $).
LA - eng
KW - Ilmanen’s lemma; $ C^{1,\omega } $ function; semiconvex function with general modulus
UR - http://eudml.org/doc/298292
ER -