Let $1<p<\infty$ and $f \in L^p(\mathbb{T})$. It is known that both Fourier sums of $f$ and Cesàro sums of the Fourier sums of $f$ converges in $L^p(\mathbb{T})$ to $f$, i.e.: \begin{align*} \bigg\|\sum_{k=-N}^N \hat{f}(k) e_k-f\bigg\|_p &\to 0, N\to\infty \\ \bigg\|\frac{1}{N+1}\sum_{n=0}^N\sum_{k=-n}^n \hat{f}(k) e_k-f\bigg\|_p &\to 0, N\to\infty \end{align*} where $\hat{f}$ is the (discrete) Fourier transform of $f$ and $(e_n)_{n\in\mathbb{Z}} = (t\mapsto e^{int})_{n\in\mathbb{Z}}$ is the Fourier base.
I'm wondering if we can somehow guarantee that the first or the second sum has better convergence rate under suitable conditions of regularity.
It follows an example of what I have in mind. Suppose that $f \in C^1(\mathbb{T})$. Let $(F_N)_{N\in\mathbb{N}}$ be the Fejér kernel and for $t\in \mathbb{T}$ let $\tau_t(f) = x\mapsto f(x-t)$. If $\delta >0$ we have that: \begin{align*} \bigg\|\frac{1}{N+1}\sum_{n=0}^N\sum_{k=-n}^n \hat{f}(k) e_k-f\bigg\|_p &=\|F_N*f-f\|_p \\ &\le \sup_{t\in[-\delta,\delta]}\|\tau_t(f)-f\|_p + \frac{\|f\|_p}{\pi N}\Big(\frac{\sin(N\delta/2)}{\sin(\delta/2)}\Big)^2 \\ &\le \|f'\|_\infty\delta + \frac{\|f\|_p}{\pi N}\Big(\frac{1}{\sin(\delta/2)}\Big)^2 \end{align*} Then, if we choose $\delta = \delta_N = N^{-1/3}$ we obtain \begin{align*} \bigg\|\frac{1}{N+1}\sum_{n=0}^N\sum_{k=-n}^n \hat{f}(k) e_k-f\bigg\|_p &\le \|f'\|_\infty N^{-1/3} + \frac{\|f\|_p}{\pi N}\Big(\frac{1}{\sin(1/(2N^{1/3}))}\Big)^2 \\ &\le C N^{-1/3} \end{align*} On the other hand, if we also suppose that $1\le p\le 2$ then \begin{align*} \bigg\|\sum_{k=-N}^N \hat{f}(k) e_k-f\bigg\|_p &\le \bigg\|\sum_{k=-N}^N \hat{f}(k) e_k-f\bigg\|_2 = \bigg\|\sum_{|k|\ge N+1} \hat{f}(k) e_k\bigg\|_2 \\ &= \bigg(\sum_{|k|\ge N+1} |\hat{f}(k)|^2\bigg)^{1/2} \le \frac{1}{N+1} \bigg(\sum_{k \in \mathbb{Z}} n^2 |\hat{f}(k)|^2\bigg)^{1/2} \\ &\le \tilde{C}N^{-1}. \end{align*} So, if $f \in C^1(\mathbb{T})$ and $1\le p \le 2$ we somehow manage to get a better estimate for the rate of convergence of the Fourier sums of $f$ than the one we got for the Cesàro means. However, I don't know if these estimates are sharp in any sense and if $p>2$ I don't know how to treat the Fourier sums. So:
Are this estimates sharp in any sense? (i.e. can we do better than this under the stated conditions or there are functions that realize these bounds?) What about the case $p>2$? Are there other techniques to get estimates like these? Where can I find something in this sense in the literature?