title: Nemytskii operator
text: # Nemytskii operator
  
  In mathematics, Nemytskii operators are a class of nonlinear operators on $L^p$ 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 × $R^m$ → 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 ∈ $R^m.$
  
  Given a function f satisfying the Carathéodory conditions and a function u : Ω → $R^m,$ define a new function F(u) : Ω → R by
  
      $F(u)(x)=f{\bigx,u(x){\big )}.$
  
  The function F is called a Nemytskii operator.
  
  ## Boundedness theorem
  
  Let Ω be a domain, let 1 < p < +∞ and let g ∈ $L^q(Ω;$ R), with
  
      ${\frac {1}{p}}+{\frac {1}{q}}=1.$
  
  Suppose that f satisfies the Carathéodory conditions and that, for some constant C and all x and u,
  
      ${\big |}f(x,u){\big |}\leq C|u|^{p-1}+g(x).$
  
  Then the Nemytskii operator F as defined above is a bounded and continuous map from $L^p(Ω;$ $R^m)$ into $L^q(Ω;$ R).
  
  