Let $f \in C^{\alpha}(\mathbb{T})$ be a holder-continuous function, i.e. $$ |f(x) - f(y)| \le C|x-y|^{\alpha} $$ for some $0 < \alpha < 1$. We use $\|f\|_{C_\alpha}$ to denote the smallest positive constant that the above inequality hold.
Let $S_N(f)$ be the partial sum of the fourier series of $f$, and I want to show the following inequality $$ \|S_N(f) - f\|_\infty = \sup_x\left|\int_{-0.5}^{0.5} \frac{\sin(2N+1)\pi t}{\sin \pi t} (f(x-t)-f(x)) \mathrm{d} t\right| \lesssim \frac{\log N}{N^{\alpha}}\|f\|_{C_\alpha} $$ holds. In other words, I want to determine the precise rate of uniform convergence.
The rate can be found in https://en.wikipedia.org/wiki/Convergence_of_Fourier_series#Uniform_convergence.
My approach is to break the interval into two parts: $|x| < \delta$ and $\delta < |x| < 0.5$. In the first part we will apply the condition of Holder continuity, and in the second part we will apply Riemann-Lebesgue Lemma; finally I choose an appropriate $\delta$ to make the sum small. However, the obtained rate is slower than $N^{-\alpha}$. And I couldn't figure out how to get the $\log(N)$ term involved. Such issue also appears in another answer How to show for $\alpha\in (0,1)$, any $f\in C^\alpha([0,1]/{\sim})$ has a Fourier series $S_nf$ uniformly converging to $f$.
Any help/comment would be much appreciated. Thank you in advance!