The case $ \mu(X)=\infty $ the inequality is true.
Suppose $ \mu(X) $ is finite, we have
Given $ p^{'}=\frac{p+r}{2} $,
$ \int_{X}|f|^{r}d\mu \leq \int_{X}|f|^{p^{'}}(\mu(X))^{1-r/p^{'}} \leq \int_{X}|f|^{p^{'}}(\mu(X))^{1-r/p} $ by Holder.
Now, let $ g=|f|^{p{'}} $, then $ w(y)=\mu(\{g>y\} \leq \frac{c_{0}}{y^{p/p{'}}} $
$ \int_{X}g d\mu = \int_{0}^{\infty}w(y)dy \leq c_{0}\int_{0}^{\infty}\frac{dy}{y^{p/p{'}}} $
Let $ c=c_{0}\int_{0}^{\infty}\frac{dy}{y^{p/p{'}}} $. DONE!!!
Posted after 5pm Friday:
The above answer is wrong. It should be:
$ \int_{X}g d\mu = \int_{0}^{\infty}w(y)dy \leq \int_{0}^{1}w(y)dy + \int_{1}^{\infty}w(y)dy \leq \mu(X) + c_{0}\int_{1}^{\infty}\frac{dy}{y^{p/p{'}}} $
Let $ c=\mu(X) + c_{0}\int_{1}^{\infty}\frac{dy}{y^{p/p{'}}} $.
