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

摘要

The main result: for every sequence $\{\omega_m\}_{m=1}^\infty$ of positivenumbers ($\omega_m>0)$ there exists an isometric embedding $F:[0,1]\toL_1[0,1]$ 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 tothe complex plane which is an entire function.