Uniformly Continuous Superposition Operators in the Spaces of Differentiable Functions and Absolutely Continuous Functions

Abstract

Let I, J ⊂ ℝ be intervals. We prove that if a superposition operator H generated by a two place h : I × J → ℝ,

$$ H(\phi )(x): = h(x,\phi (x)), $$

maps the set C ^r(I, J) of all r-times continuously differentiable functions ϕ : I → J into the Banach space C ^r(I, ℝ) and is uniformly continuous with respect to C ^r-norm, then

$$ h(x,y) = a(x)y + b(x), x \in I, y \in J, $$

for some a, b ∈ C ^r(I, ℝ).

For the Banach space of absolutely continuous functions an analogous result is proved.