TITLE: Cesàro operator is bounded for $1<p<\infty$
QUESTION [7 upvotes]: The Cesàro operator $T\colon \ell_{p}\to\ell_{p}$ is defined by $(Tx)_{k}=\frac{1}{k}\sum_{j=1}^{k}x_{j},\: k\in\mathbb{N}$,
where $x=(x_{k})_{k=1}^{\infty}$ Show that $T$ is bounded if $1<p<\infty$.
I can do it for $p=\infty$, but not when it is between $1$ and $\infty$.
Thank you.

REPLY [10 votes]: Using Hardy inequality one may see that
$$
\Vert T(x)\Vert_p=
\left(\sum\limits_{k=1}^\infty \left|\frac{1}{k}\sum\limits_{j=1}^k x_j\right|^p\right)^{1/p}\leq
\left(\sum\limits_{k=1}^\infty \left(\frac{1}{k}\sum\limits_{j=1}^k |x_j|\right)^p\right)^{1/p}\leq
$$
$$
\left(\left(\frac{p}{p-1}\right)^p\sum\limits_{k=1}^\infty |x_j|^p\right)^{1/p}=
\frac{p}{p-1}\left(\sum\limits_{k=1}^\infty |x_j|^p\right)^{1/p}=
\frac{p}{p-1}\Vert x\Vert_p
$$
This means that
$$
\Vert T\Vert\leq\frac{p}{p-1}
$$