A note on 𝐶^{1,𝛼}-smooth approximation of Lipschitz functions

Abstract

We show that on super-reflexive spaces a Moreau-Yosida type of regularisation by infimal convolution together with a known insertion-type theorem (a variant of Ilmanen’s lemma) easily give an approximation of a Lipschitz function by a C 1 , α C^{1,\alpha } -smooth Lipschitz function with the same Lipschitz constant. This is a generalisation of the well-known theorem of J.-M. Lasry and P.-L. Lions from Hilbert spaces. It also gives a new self-contained and probably simpler proof of the Lasry-Lions theorem.