A new proof of an inequality of Littlewood and Paley

Abstract:

A fairly elementary new proof is presented of the inequality $(p \geq 2)$: \[ \int {{{\left | {h’} \right |}^p}{{\left ( {1 - \left | z \right |} \right )}^{p - 1}}dxdy \leq \left \| h \right \|_{{H^p}}^p} ,\quad f \in {H^p}.\] In addition, the inequality \[ \int {{{\left | h \right |}^{p - s}}{{\left | {h’} \right |}^s}{{(1 - \left | z \right |)}^{s - 1}}dxdy \leq \left \| h \right \|_{{H^p}}^p} \] is shown to hold for $h \in {H^p},p > 0$, if and only if $2 \leq s < p + 2$, generalizing the known case $s = 2$.