Let $1\le p\le q <\infty$ and$$\|f\|_{m_{q}^{p}}:= \sup_{a\in\mathbb{R}^d,R\in(0,1)}|B(a,R)|^{1/p-1/q}\left(\int_{B(a,R)}|f(y)|^p dy\right)^{1/p}.$$
If we take $f(x)=|x|^{-d/q}$, then$$\|f\|_{m_{q}^{p}}= \sup_{R\in(0,1)}|B(0,R)|^{1/p-1/q}\left(\int_{B(0,R)}|y|^{-dp/q} dy\right)^{1/p}=\cdots.$$
The equality above was taken from an paper I read. Is this equality correct?It is clear from the definition that the following inequality is correct.$$\|f\|_{m_{q}^{p}}\geq \sup_{R\in(0,1)}|B(0,R)|^{1/p-1/q}\left(\int_{B(0,R)}|y|^{-dp/q} dy\right)^{1/p}=\cdots.$$But unfortunately I could not understand how this inequality turned into the above equality?
