A short proof of an inequality of Littlewood and Paley

A very short proof is given of the inequality

$\int_{\vert z\vert<1}\vert\nabla u(z)\vert^p (1-\vert z\vert)^{p-... ...rac 1{2\pi}\int_0^{2\pi} \vert f(e^{it})\vert^p\,dt -\vert u(0)\vert^p\right),$

where $p>2,$ and $u$ is the Poisson integral of $f\in L^p(\partial \mathbb{D}),$ $\mathbb{D}=\{z: \vert z\vert<1\}.$