The theorem of Carleson and Hunt shows that if $f\in L^p(\mathbb{T}), 1<p<\infty$, where $\mathbb{T}=\mathbb{R}/\mathbb{Z}$, the sequence of the partial sums of its Fourier series $(S_nf(x))_{n=0}^\infty$ converges for almost every $x\in \mathbb{T}.$ I met the following estimate on $(S_nf)$: $$|S_nf(x)|\le C_pn^{1/p}\|f\|_{L^p}$$ and the given reference is the classical book written by Zygmund "Trigonometric series". However, I don't find its proof in this book.
I am interested in this result. Can someone give a reference on its proof? Many thanks.
I don't have a reference on hand, but if $1 < p \le 2$, I can see a direct approach based on Hölder's and the Hausdorff–Young inequality.
I assume $\widehat{S_nf}(a) = 1_{[-n,n]}(a)\widehat f(a)$. Note $|S_nf(x)|\le \sum_{|a|\le n}|\widehat f(a)|\le (2n+1)^{1/p}\|\widehat f\|_{\ell^{p'}(\mathbb Z)}$. If $1< p \le 2$, $\|\widehat f\|_{\ell^{p'}(\mathbb Z)}\le \|f\|_{L^p(\mathbb T)}$ by Hausdorff–Young.