Images of nowhere differentiable Lipschitz maps of $[0,1]$ into $L_1[0,1]$

Abstract

The main result: for every sequence $\{{\omega }_m{\}}_{m=1}^{\infty }$ of positive numbers (${\omega }_m>0)$ 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 bounds ${\max }_{x\in [0,1]}|F_t^{(m)}(x)|\le {\omega }_m$ and has an analytic extension to the complex plane which is an entire function.