On the autonomous Nemytskii operator between Sobolev spaces in the critical and supercritical cases: Well-definedness and higher-order chain rule

Abstract

This paper has two aims. The first aim is to improve and complete the main result obtained in Dinca and Isaia (2013) [24,25], (2014) which provides sufficient conditions that assure the well-definedness and validity of the higher-order chain rule for autonomous Nemytskii operators between two arbitrary Sobolev spaces. The second aim is to prove that the above mentioned sufficient conditions are also necessary in the critical and supercritical cases. To meet this aim, a heuristic approach is used.Since the necessity of the above mentioned sufficient conditions has already been proved in Isaia (2015) for the subcritical case, a study covering all three cases is now complete and an open question formulated by Bourdaud and Moussai (2019) receives now a full answer.