I know that
$$AC(a,b):=\left\{f \in C(a,b)|f(x) = f(c)+\int_c^x g(t) d \lambda(t),c \in (a,b), g \in L^1_{\text{loc}}(a,b)\right\}$$
$$AC[a,b]:=\left\{f \in C[a,b]|f(x) = f(c)+\int_a^x g(t) d \lambda(t), g \in L^1[a,b]\right\}.$$
Now I found the set $AC^1$, but without a definition what this set is. By looking at the context, I think it could be something like:
$$AC^1(a,b):=\left\{f \in C^1(a,b)|f(x) = f(c)+\int_c^x g(t) d \lambda(t),c \in (a,b), g \in L^1_{\text{loc}}(a,b)\right\}$$ or
$$AC^1[a,b]:=\left\{f \in C^1[a,b]|f(x) = f(c)+\int_a^x g(t) d \lambda(t), g \in L^1[a,b]\right\},$$ is this correct? Probably not, because we also want to have $f' \in L^1$ somehow. Therefore, I would be interested in an exaplanation what this set actually means?
The most natural way (to me) is to interpret the space $AC^1$ as the space of functions whose first derivative is in $AC$.
Remark: the class $AC$, as defined, is precisely the class of absolutely continuous functions. Googling "AC1" together with "absolutely continuous" brought up, for example, this paper in which $AC^{1}$ notation is explained consistently with my answer: the space of functions with absolutely continuous first derivative.