Throughout the third part, $f \in \mathcal{C}_{0}$ satisfies property $(\mathcal{P}_{1})$: there exist $x_{0} \in [0,1]$, $s \in ]0,1[$ and $c_{1} \in ]0, +\infty[$, such that for all $(j, k) \in \mathcal{I}$, $$|c_{j,k}(f)| \leq c_{1} (2^{-j} + |k 2^{-j} - x_{0}|)^{s}$$ We suppose furthermore that $\omega_{f}$ satisfies property $(\mathcal{P}_{2})$: for all integer $N \geq 1$, there exists $c_{4}(N) > 0$ such that for all $h \in ]0,1]$, $\omega_{f}(h) \leq c_{4}(N)(1 + |\log_{2} h|)^{-N}$. Deduce from the above that $\alpha_{f}(x_{0}) \geq s$. One may distinguish the cases $n_{0} \geq n_{1}$ and $n_{0} < n_{1}$.
Throughout the third part, $f \in \mathcal{C}_{0}$ satisfies property $(\mathcal{P}_{1})$: there exist $x_{0} \in [0,1]$, $s \in ]0,1[$ and $c_{1} \in ]0, +\infty[$, such that for all $(j, k) \in \mathcal{I}$,
$$|c_{j,k}(f)| \leq c_{1} (2^{-j} + |k 2^{-j} - x_{0}|)^{s}$$
We suppose furthermore that $\omega_{f}$ satisfies property $(\mathcal{P}_{2})$: for all integer $N \geq 1$, there exists $c_{4}(N) > 0$ such that for all $h \in ]0,1]$, $\omega_{f}(h) \leq c_{4}(N)(1 + |\log_{2} h|)^{-N}$.
Deduce from the above that $\alpha_{f}(x_{0}) \geq s$.
One may distinguish the cases $n_{0} \geq n_{1}$ and $n_{0} < n_{1}$.