In this part, $\lambda(t) = t$ for all $t \in \mathbb{R}^+$.
Show that if $E$ is not empty and if $\alpha$ is its infimum (we agree that $\alpha = -\infty$ if $E = \mathbb{R}$), then $Lf$ is of class $C^{\infty}$ on $]\alpha, +\infty[$ and express its successive derivatives using an integral.