Criteria for Hardy space membership

Question

Assume that $p>2$ and let $H^p$ be the Hardy space of holomorphic functions in the unit disk $D$. It seems that $f\in H^p$ if and only if $$P(f):=\int_0^{2\pi}\left(\int_0^{1}|f'(re^{it})|^2(1-r)dr\right)^{p/2} dt<\infty$$ and moreover $$\|f\|^p_p\le C_p(|f(0)|^p+P(f))$$ but I cannot find the reference for this.

Answer 1

This actually holds for all $p>0$. The function \begin{equation} G[f](\zeta):=\Big( \int_0^1(1-r)|f'(r\zeta)|^2 dr \Big)^{\frac 12}\end{equation} is sometimes called Paley Littlewood $g$-function. The fact that $G[f] \in L^p(\partial \mathbb{D}, d\theta)$ if and only if $f\in H^p$ is the first part of Theorem 1.1 in this expository paper for example https://www.sciencedirect.com/science/article/pii/S0723086913000078