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# A Short Proof of an Inequality of Littlewood and Paley

Miroslav Pavlović
Proceedings of the American Mathematical Society
Vol. 134, No. 12 (Dec., 2006), pp. 3625-3627
Stable URL: http://www.jstor.org/stable/4098198
Page Count: 3
A very short proof is given of the inequality $\int_{\left\vert z\right\vert < 1} \left\vert\nabla u(z)\right\vert^{p}(1 - \left\vert z\right\vert)^{p-1} dxdy \leq C_p \left(\frac{1}{2\pi} \int_0^{2\pi} \left\vert f(e^{it})\right\vert^{p}\;dt - \left\vert u(0)\right\vert^{p}\right)$, where p > 2, and u is the Poisson integral of $f \in L^{p}(\partial\mathbb{D})$, $\mathbb{D} = \{z : \left\vert z\right\vert < 1\}$.