Abstract: 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} ).
Keywords: Ilmanen's lemma; C^{1,\omega} function; semiconvex function with general modulus
DOI: DOI 10.14712/1213-7243.2021.031
AMS Subject Classification: 26B25