a weak $L^p$ embedding inequality

Question

For $1<p<\infty$ , if $f$ is a weak $L^p(\mathbb{R}^n)$ function , then prove that $f_\delta:=f.{\bf{1}}_{(|f|\geq\delta)}$ belongs to $L^q(\mathbb{R}^n)$ for $1\leq q<p$ and for all $\delta>0$ .

I started this way $$||f_\delta||_q^q=\int_{\{|f|\geq\delta\}}|f|^q\geq\delta^q\mu(\{|f|\geq\delta\})$$ where $\mu$ is the Lebesgue measure in $\mathbb{R}^n$ . It is well known that for weak $L^p$ one has $$[f]_p:=\sup_{\delta>0}\delta^p\mu(\{|f|>\delta\})<\infty$$ But from there I am unable to get the former inequality to be bounded . Any help is appreciated .

Answer 1

Hint: Use the Layer cake representation $$ \int |f|^q \, \mathrm d\mu = q \int_0^\infty t^{q-1} \mu\{|f| > t\} \, \mathrm dt. $$

Answer 2

Following gerw's hint : \begin{align*}||f_\delta||_q^q&=q\int_0^\infty t^{q-1}\mu\{|f_\delta|>t\}\,dt\\&=q\int_\delta^\infty t^p\mu\{|f|>t\}t^{q-p-1}\,dt\\&\leq q[f]_p\int_\delta^\infty t^{q-p-1}\,dt\\&=\frac{q[f]_p}{(p-q)\delta^{p-q}}\\&<\infty\end{align*}