I am trying to understand the various inclusions of Lorentz and $L^p$ spaces under certain conditions and under certain measure spaces. I am trying to understand why $L^p$ is in $L^{(p,\infty)}$? Here the norms are given as $\|f\|_p=(\int |f|^pdx)^{1/p}$ and $\|f\|_{L^{(p,\infty)}}=\sup_{t>0}t^{1/p}f^{**}(t)$.
Attempt: I need to show that the latter norm is "bounded" by the former (upto a scalar). since $t^{1/p}$ is increasing, so we have to have that $\|ft^{1/p}\|_p$ is larger than $\|f^* t^{1/p}\|_p$ by the Polya inequality. however this gives no information on the maximal function $f^{**}$ so we cannot proceed any further.
Here $f^*$ is the decreasing re-arrangement and $f^{**}(y)=\int_0^yf^* (x)dx$.
We have $d_{f}=d_{f^{\ast}}$ and $\displaystyle\int|f|^{p}=\int_{0}^{\infty}f^{\ast}(u)^{p}du$.
For any $t>0$, $\displaystyle\int_{0}^{\infty}f^{\ast}(u)^{p}du\geq\int_{0}^{t}f^{\ast}(u)^{p}du\geq t\left(\int_{0}^{t}f^{\ast}(u)du\right)^{p}=tf^{\ast\ast}(t)^{p}$.