Let $f\in C^1(\mathbb T)=C^1(\mathbb R/\mathbb Z)$ be a function such that $$\hat f(k):=\int_{\mathbb T}f(x)e^{-2\pi ikx}\,dx=0,\qquad \forall k\in\{-N+1,\cdots,-1,0,1,\cdots, N-1\}.$$ Do we have $\|f\|_{L^\infty}\leq \frac CN \|f'\|_{L^1}$ for some $C>0$ indpendent of $f$ and $N$?

We have $\|f\|_{L^\infty}\leq \frac1{4N} \|f'\|_{L^\infty}$, see this MSE post for a clever proof. I just wonder, if we change $\|f'\|_{L^\infty}$ to $\|f'\|_{L^1}$, do we have the same inequality?

Maybe the decay $\frac1N$ is too fast to be true. If this is not ture, can we find a $C_N$ such that $$\|f\|_{L^\infty}\leq C_N \|f'\|_{L^1} \qquad \text{and }\ \ \ \ \lim_{N\to\infty}C_N=0?$$

Note that by the fundamental theorem of calculus we obtain $\|f\|_{L^\infty}\leq \|f'\|_{L^1}$. Also, by Cauchy inequality and the Plancherel identity, \begin{align*} \|f\|_{L^\infty}&\leq\sum_{|k|\geq N}|\hat f(k)|=\sum_{|k|\geq N}\frac{\left|\widehat{f'}(k)\right|}{2\pi |k|}\\ &\leq \frac1{2\pi}\left(\sum_{|k|\geq N}\left|\widehat{f'}(k)\right|^2\right)^{1/2}\left(\sum_{|k|\geq N}\frac1{|k|^2}\right)^{1/2}\\ &\leq \frac1{2\pi}\left\|f'\right\|_{L^2}\left(\frac2N\right)^{1/2}=\frac C{\sqrt N}\left\|f'\right\|_{L^2}. \end{align*}

Any help would be appreciated!