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

## TL;DR

The paper constructs isometric embeddings of the interval into L1[0,1] that are nowhere differentiable as a whole but have infinitely differentiable, entire, and analytically extendable slices, with controlled derivatives.

## Contribution

It introduces a novel class of Lipschitz maps into L1[0,1] that are nowhere differentiable globally but infinitely differentiable and entire on each slice, with prescribed derivative bounds.

## Key findings

- Existence of such embeddings for any positive sequence
- Slices are infinitely differentiable and extendable as entire functions
- Global map is nowhere differentiable

## 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.