Let $(X,\mu)$ be a measure space and let $E$ be a subset of $X$ with $\mu(E\,)<\infty$ Assume that $f\in L^{p,\infty}(X,\mu)$ for some $0<p<\infty$.
Show that for $0<q<p$ we have $$\int_{E}|f(x)|^{q}\,d\mu(x)\le \frac{p}{p-q}\mu(E\,)^{1-\frac{q}{p}}\lVert f\,\rVert_{L^{p,\infty}}^{q}$$
I got some hint : Use $\mu(E\,\cap\{|f|>\alpha\})\leq \min(\mu(E\,) ,\alpha^{-p}\lVert f \rVert_{L^{p,\infty}}^p\,\,)$
Here is my working :
\begin{align*} \int_{E}|f(x)|^{q}\,d\mu(x)&=\int_{E\,\cap\{|f|\leq\alpha\}}|f(x)|^{q}\,d\mu(x)\,+ \int_{E\,\cap\{|f|>\alpha\}}|f(x)|^{q}\,d\mu(x)\\ &\leq\int_{E\,\cap\{|f|\leq\alpha\}}\alpha^{q}\,d\mu(x)\ + \int_{E\,\cap\{|f|>\alpha\}}|f(x)|^{q}\,d\mu(x)\\ &=\alpha^{q}\mu\bigg(E\,\cap\{|f|<\alpha\}\bigg) + \int_{E\,\cap\{|f|>\alpha\}}|f(x)|^{q}\,d\mu(x) \end{align*}
Then I get stuck with the problem and no ideal how to proceed.
Someone can give some direction for proceeding if have the time.Thanks for your reading.
\begin{align*} \int_{E}|f(x)|^{q}d\mu(x)&=q\int_{0}^{\infty}\alpha^{q-1}\mu(|f|>\alpha)d\alpha\\ &\leq q\int_{0}^{\infty}\alpha^{q-1}\min(\mu(E),\alpha^{-p}\|f\|_{L_{p,\infty}^{p}})d\alpha\\ &=q\left(\int_{0}^{\gamma}+\int_{\gamma}^{\infty}\right)\alpha^{q-1}\min\left(\mu(E),\alpha^{-p}\|f\|_{L^{p,\infty}}^{p}\right)d\alpha\\ &=\mu(E)\gamma^{q}+\|f\|_{L^{p,\infty}}^{p}\dfrac{q}{p-q}\gamma^{q-p}, \end{align*} where $\gamma$ is such that $\mu(E)=\gamma^{-p}\|f\|_{L^{p,\infty}}^{p}$, so whenever $\alpha\leq\gamma$, then $\alpha^{p}\leq\gamma^{p}=\mu(E)^{-1}\|f\|_{L^{p,\infty}}^{p}$, so $\min\left(\mu(E),\alpha^{-p}\|f\|_{L^{p,\infty}}^{p}\right)=\mu(E)$, and similar estimation goes for $\alpha\geq\gamma$.