In mathematics, Nemytskii operators are a class of nonlinear operators on Lp spaces with good continuity and boundedness properties. They take their name from the mathematician Viktor Vladimirovich Nemytskii. Definition Let Ω be a domain (an open and connected set) in n-dimensional Euclidean space. A function f : Ω × Rm → R is said to satisfy the Carathéodory conditions if f(x, u) is a continuous function of u for almost all x ∈ Ω; f(x, u) is a measurable function of x for all u ∈ Rm. Given a function f satisfying the Carathéodory conditions and a function u : Ω → Rm, define a new function F(u) : Ω → R by The function F is called a Nemytskii operator. Boundedness theorem Let Ω be a domain, let 1 < p < +∞ and let g ∈ Lq(Ω; R), with Suppose that f satisfies the Carathéodory conditions and that, for some constant C and all x and u, Then the Nemytskii operator F as defined above is a bounded and continuous map from Lp(Ω; Rm) into Lq(Ω; R). References (Section 10.3.4) Operator theory