Images of nowhere differentiable Lipschitz maps of $[0,1]$ into $L_1[0,1]$
Volume 243 / 2018
Fundamenta Mathematicae 243 (2018), 75-83
MSC: Primary 46G05; Secondary 46B22.
DOI: 10.4064/fm493-12-2017
Published online: 24 May 2018
Abstract
The main result: for every sequence $\{\omega _m\}_{m=1}^\infty $ of positive numbers there exists an isometric embedding $F:[0,1]\to L_1[0,1]$ which is nowhere differentiable, but for each $t\in [0,1]$ the image $F_t$ is infinitely differentiable on $[0,1]$ with $\max_{x\in [0,1]}|F_t^{(m)}(x)|\le \omega _m$ and has an extension to an entire function on the complex plane.
