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

Kryštof, Václav

Commentationes Mathematicae Universitatis Carolinae, Tome 62 (2021) no. 4, pp. 445-455.

Résumé

The author proved in 2018 that if $ G $ is an open subset of a Hilbert space, $ f_1,f_2\colon G\to\mathbb{R} $ continuous functions and $ \omega $ a nontrivial modulus such that $ f_1\leq 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\leq f\leq 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} $).

Détail

DOI : 10.14712/1213-7243.2021.031

Classification : 26B25
Keywords: Ilmanen's lemma; $ C^{1, \omega} $ function; semiconvex function with general modulus