Images of nowhere differentiable Lipschitz maps of [0,1] into L₁[0,1]

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.