Abstract. For $1 \leq p < \infty $, we denote by $\AC ^p [0, 1]$ the space of all absolutely continuous functions on the interval $[0, 1]$ whose derivatives belong to $L^p [0, 1]$. Under the assumption that ${\rm AC}^p [0, 1]$ is equipped with the norm $\| f \|_\sigma = |f(0)| + \| f' \|_L^p$ or $\| f \|_m = \max\{ |f(0)|, \| f' \|_L^p \}$, we characterize the surjective linear isometries on ${\rm AC}^p [0, 1]$.

DOI: 10.14232/actasm-012-327-3

AMS Subject Classification (1991): 46B04; 46E15

Keyword(s): linear isometry, absolutely continuous function

Received September 24, 2012, and in revised form December 3, 2012. (Registered under 77/2012.)