Let $f: \mathbb R \to \mathbb R$ be a measurable function. Define the perturbation operator $T_f$ on measurable functions $g: \mathbb R \to \mathbb R$ by
$$T_f (g)(x) := f(x + g(x)) - f(x).$$
Observe that $f$ is Lipschitz continuous if and only if $T_f$ is strong type $(\infty, \infty)$. Further, we have $f \in W^{1, 1}$ modulo constants if and only if $T_f$ is strong type $(\infty, 1)$, by a standard characterisation of Sobolev spaces.
Questions:
Is it true that $f$ is absolutely continuous if and only if $T_f$ is strong type $(1, 1)$?
Is it true that $f$ is of bounded variation if and only if $T_f$ is weak type $(\infty, 1)$?
Bonus Question: What property of $f$ corresponds to $T_f$ being of weak type $(1 ,1)$? Or strong type $(p, p)$ for $1 < p < \infty$?
