If $0 < r < p < \infty$, then $\|f\|_r \leq (\frac{p}{p-r})^{1/r} \mu(X)^{1/r - 1/p} \|f\|_{p, w}$ (weak $L^p$ norm)

Question

Let $(X, \mu)$ be a finite measure space. If $0 < r < p < \infty$, prove that $||f||_r \leq (\frac{p}{p-r})^{1/r} \mu(X)^{1/r - 1/p} \|f\|_{p, w}$.

Here $\|f\|_{p, w}$ is the "weak $L^p$ norm" on $f$, equal to $\sup\limits_{k > 0} {k \mu(\lbrace x: |f(x)| > k \rbrace)^{1/p}}$. The problem comes with a hint to first show $\mu(\lbrace x: |f(x)| > t \rbrace) \leq \min\{\mu(X), t^-p ||f||_{p, w}\}$ and then use the layer cake principle.

I've figured out that inequality and I'm sure I could work out the rest if I knew what function to use for the layer cake principle, and what "version" of f I should be composing it with (i.e., $|f|, |f|^p, |f|^r$). I would appreciate some kind of hint toward finding that function, as my own attempts at doing so have failed.

Answer 1

Hint: if $\lambda(t)=\mu\left(\left\{x:|f(x)|>t\right\}\right)$, then $$\int_X|f|^r\,d\mu=r\int_0^{\infty}t^{r-1}\lambda(t)\,dt.$$ Then break this integral in two intervals, $[0,M]$ and $[M,\infty)$, and minimize the resulting function of $M$.