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Keywords:
Ilmanen's lemma; $ C^{1,\omega} $ function; semiconvex function with general modulus
Ilmanen's lemma; $ C^{1,\omega} $ function; semiconvex function with general modulus
Summary:
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} $).
References:
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