where $m$ denotes the Lebesgue measure.

We define $$\|f\|_{B^{1/p}_{p,p},\mathbb{D}}:=\|(1-|z|^2)^2f''\|_{L^p(\mathbb{D},\mu)}.$$

The next result can also be found in [@peller Appendix 2.6].

For any $p_0>1$ there is a constant $C>0$ such that for all holomorphic $f$ with $f(0)=f'(0)=0$ $$C^{-1}\|f\|_{B^{1/p}_{p,p},\mathbb{D}}\leq \|f\|_{B^{1/p}_{p,p}(S^1)}\leq C\|f\|_{B^{1/p}_{p,p},\mathbb{D}}, \quad \forall p\in [1,p_0].$$

We remark that the condition $f(0)=f'(0)=0$ plays no role once going to the extrapolation space because we can write any $f=f_0+g_0$ where $f_0(0)=f'_0(0)=0$ and $g_0=-f'(0)z-f(0)$ satisfies that $H_{\bar{g}_0}$ is finite rank.

For a holomorphic $f\in \cap_{q>p} B^{1/q}_{q,q}(S^1)$ we define $F_f\in \cap_{q>p}L^q(0,\infty)$ as the decreasing rearrangement of the function $(1-|z|^2)^2f''$ on $\mathbb{D}$ with respect to the measure $\mu$. We also define $\Phi_f\in \cap_{q>p}L^q(0,\infty)$ as the decreasing rearrangement of the function $S^1\times {{\mathbb N}}\ni (\theta,n)\mapsto W_n*f(\mathrm{e}^{i\theta})$ with respect to the product measure $\nu$ on $S^1\times {{\mathbb N}}$. It is follows from the well known fact that $L^q$-norms of functions coincides with the $L^q(0,\infty)$-norm of its decreasing rearrangement that for $q>p$ $$\begin{aligned}
\|f\|_{B^{1/q}_{q,q}(S^1)}&=\|W_n*f\|_{L^q(S^1\times {{\mathbb N}},\nu)}=\|\Phi_f\|_{L^q(0,\infty)}\quad \mbox{and}\\
\|f\|_{B^{1/q}_{q,q},\mathbb{D}}&=\|(1-|z|^2)^2f''\|_{L^p(\mathbb{D},\mu)}=\|F_f\|_{L^q(0,\infty)}.\end{aligned}$$

\[Thm1\] Let $p\geq 1$ and $\psi$ be a function as in Corollary \[equivalencescor\]. Assume that $f$ is holomorphic. Then the following assertions are equivalent:

1.  $\limsup_{h\searrow 0} \left(\frac{1}{\psi(\mathrm{e}^{\frac{1}{h}})} \int_{\textbf{D}} |f''(z)|^{p+h} (1-|z|^2)^{2p+2h-2} dz\right)^{\frac{p}{p+h}} <\infty;$

2.  $\limsup_{t\to \infty} \frac1{\psi(t)} \int_0^t F_f(s)^p ds < \infty;$

3.  