## Partial sum operator of fourier series and L^p boundedness

Suppose $f\in L^1(-\pi,\pi)$ , consider the fourier series of $f$

$\displaystyle f\sim \sum \limits_{n\in\mathbb{Z}}c_ne^{inx}$

and its partial sum $\displaystyle S_N(f)=\sum \limits_{-N}^Nc_ne^{inx}$

$\mathbf{Problem:}$ The following statements are equivalent

(a)   $||S_Nf||_p\leq c_p||f||_p$, $1

(b)  $||S_Nf-f||_p\to \infty$ as $N\to \infty$

(c)   $||S_Nf||_{p'}\leq ||f||_{p'}$, $\frac{1}{p}+\frac{1}{p'}=1$

$\mathbf{Proof:}$

(a)$\Rightarrow$(b). Since trigonometric polynomial are dense in $L^p$, then we can choose $g$ to be a trigonometric polynomial are sufficiently close to $f$ in $L^p$.

$\displaystyle ||S_Nf-f||_p\leq ||S_Nf-S_Ng||_p+||S_Ng-g||_p+||g-f||_p$

$\displaystyle = ||S_N(f-g)||_p+||S_Ng-g||_p+||g-f||_p$

$\displaystyle \leq c_p||f-g||_p+||S_Ng-g||_p+||g-f||_p\to \infty$

(b)$\Rightarrow$(a). Uniform boundedness principle.

(a)$\Rightarrow$(c) and (c)$\Rightarrow$(a) are quite easy.

$\text{Q.E.D}\hfill \square$

$\mathbf{Remark:}$