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<p<\infty

(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:}

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