There is a Lemma that:
if $1 \leq p \leq +\infty,||f||_p$ for $ \alpha \in [p,+\infty],\rho > 1 $ satisfies $ (H_{\alpha}^{\rho}) $,with$ A_{\alpha}(\rho)=(1-1/{\rho})^{1-1/{\rho}} < 1 $。especially，if $ 0 <p < 1,||f||_p $for all $ \alpha \in [p,+\infty],\rho > 1 $ satisfies $ (H_{\alpha}^{\rho}) $,with $$ A_{\alpha}(\rho)=(1+(\rho - 1)^p / (\rho^p - 1))^{1/p}\wedge \rho(\rho - 1)/(\rho^p - 1)^{1/p} .$$ where$ W $ satisfies $ H_{\alpha}^{\rho} $ means that if there exists a constant $ A_{\alpha}(\rho) $,so that for all $ f \in \mathcal{F}, $ \begin{equation*} \left( \sum\limits_{k \in \mathbb{Z}}W(f_{\rho,k})^{\alpha} \right)^{1/ \alpha} \leq A_{\alpha}(\rho)W(f) \end{equation*} and $f_{\rho,k}=(f-\rho^k)^+ \wedge \rho^k(\rho - 1).$
Here is the proof for the case of $1 \leq p \leq +\infty$:
fix $ \rho > 1 $,write $$ ||f_k||_p^p=p\int_{0}^{\rho^{k+1}-\rho^k} t^{p-1}\mu(f-\rho^k \ge t)dt=p\int_{\rho^k}^{\rho^{k+1}} (s-\rho^k)^{p-1}\mu(f\ge s)ds. $$
if $ 1\le p \le +\infty $,it follows that \begin{align*} \sum\limits_k||f_k||_p^p &= p\sum\limits_k\int_{\rho^k}^{\rho^{k+1}} (s-\rho^k)^{p-1}\mu(f\ge s)ds\\ & \leq \left[ \sup_k \sup_{s\in [\rho^k,\rho^{k+1}]} \left( \frac{s-\rho^k}{s} \right)^{p-1} \right] \left[ p\sum_k\int_{\rho^k}^{\rho^{k+1}} s^{p-1}\mu(f\ge s)ds \right] & \le (1-1/\rho)^{p-1}||f||_p^p \end{align*}
Here is the question:What I can't solve is the second case with $0 < p < 1,$waiting for help~