Norm-Additive Maps on the Positive Unit Sphere of $C^1([0,1])$

Abstract

Let $C^1(I)$ denote the space of all continuously differentiable real-valued functions on the closed unit interval $I=[0,1]$. For $1\le p\le \infty$, we equip $C^1(I)$ with the norm \[ \Vert{f}\Vert_{(p)} := \begin{cases} \left(|f(0)|^p+\Vert{f'}\Vert_{I}^p\right)^{1/p}, & 1\le p<\infty, \\ \max\{|f(0)|,\Vert{f'}\Vert_{I}\}, & p=\infty, \end{cases} \qquad (f\in C^1(I)), \] where $\Vert \cdot {\Vert }_I$ denotes the supremum norm on $I$. Let ${\mathcal{S}}_p^{+}:=\{f\in C^1(I):f(0)\ge 0,f^{\prime }\ge 0,\Vert f{\Vert }_{(p)}=1\}$ be the positive unit sphere of $C^1(I)$. A map $T:{\mathcal{S}}_p^{+}\to {\mathcal{S}}_p^{+}$ is called norm-additive if \[ \|T(f)+T(g)\|_{(p)}=\|f+g\|_{(p)} \qquad (f,g\in \mathcal{S}_p^+). \] For $1<p\le \infty$, we prove that every surjective norm-additive map $T:{\mathcal{S}}_p^{+}\to {\mathcal{S}}_p^{+}$ is of the form \[ T(f)(t) = f(0)+\int_0^t f'(τ(s))\,ds \] for all $f\in {\mathcal{S}}_p^{+}$ and $t\in I$, where $\tau :I\to I$ is a homeomorphism. Conversely, every map of this form is a surjective norm-additive map. Consequently, every such map extends to a surjective linear isometry from $(C^1(I),\Vert \cdot {\Vert }_{(p)})$ onto itself. For $p=1$, however, we construct a surjective norm-additive map on ${\mathcal{S}}_1^{+}$ which is not of the above form and admits no linear extension to $C^1(I)$, showing that the assumption $p>1$ is essential.