click to see the picture of one related page from the paper
this is the link of the whole paper
What I cannot really clearly understands is the content bellow: the shortcut of the inequality
If you don’t want to see the pictures,here is the main description: Let $$N_{a,b}^{\rho}(f)=\left(\sum\limits_{k \in \mathbb{Z}}\rho^{bk}\mu(f \ge \rho^k)^{b/a}\right)^{1/b},\rho > 1,a,b >0,\\f \ \text{is measurable}, \mu \ \text{is the Lebesgue measure}.$$ Then there is the inequality $$ N_{r,u}^{\rho}(f) \leq \left( N_{s,v}^{\rho}(f) \right)^{\nu}\left(N_{t,w}^{\rho}(f) \right)^{1-\nu},\\ \text{with}\ \frac{1}{r}=\frac{\nu}{s}+\frac{1-\nu}{t}, \frac{1}{u}=\frac{\nu}{v}+\frac{1-\nu}{w},\nu \in (0,1). $$
This can be used to show that $$ ||f||_{a,b^{‘}} \leq 2^{\frac{2}{b}}||f||_{a,b},\ for \ \ 0<b\leq b^{‘}\leq \infty $$ Mark:make sure that you don’t mix the symbol v with $\nu$.
I’ll be sincerely appreciated if anyone can solve the problem.