An introduction to composition operators in Sobolev spaces

Abstract

This is an updated version of notes for a series of lectures, given in Padova University in January 2018. We propose a survey on composition operators in classical Sobolev spaces. We mention results obtained in 2019, on the continuity of such operators. NotationN denotes the set of all positive integers, including 0. Z denotes the set of all integers. For x ∈ R n , |x| denotes its euclidean norm.If E, F are topological spaces, then E ֒→ F means that E ⊆ F , as sets, and the natural mapping E → F is continuous. supported functions on Ω, endowed with its natural topology, see [1, 1.56Let E be a subset of L 1,loc (R n ). We say that a function f ∈ L 1,loc (R n ) belongs locally to E if ϕf ∈ E for all ϕ ∈ D(R n ) ; in case E is endowed with a norm, we say that a functionIn all the paper, "ball" means "ball with non zero radius" (we exclude balls reduced to one point).