Let $0<q<p<\infty$. Let $E \subset \mathbb{R}^n$ be measurable with finite measure. Let $f$ be a measurable function on $\mathbb{R}^n$ such that the set $N = \sup_{\lambda >0} \lambda^p| \{x\in \mathbb{R}^n: |f(x)|>\lambda\}| <\infty$.
1- Prove that $\int_E |f|^q$ is finite.
2- Refine the argument of (1) to prove that $\int_E |f|^q \leq CN^{q/p} |E|^{1-q/p}$,
where $C$ is a constant that depends only on $n, p$ and $q$.
I want to ensure my argument for (1) is correct, and I could not verify the second one. Any help is appreciated. Here is my proof for point (1).
My proof
Define $E_\lambda = \{ x \in E: |f(x)| > \lambda\}$. Using Layer-Cake representation:
$$ \int_E |f|^q = q \int_0^\infty \lambda^{q-1} |E_\lambda| d\lambda = q\left( \int_0^1 \lambda^{q-1} |E_\lambda| d\lambda + \int_1^\infty \lambda^{q-1} |E_\lambda| d\lambda \right). $$
The first integral is finite,
$$ \int_0^1 \lambda^{q-1} |E_\lambda| d\lambda<\infty.$$
The second integral is estimated as follows:
$$\int_1^\infty \lambda^{q-1} |E_\lambda| d\lambda \leq \int_1^\infty \lambda^{q-(1+p)} \lambda^{p} |E_\lambda| d\lambda \leq \int_1^\infty \lambda^{q-(1+p)}\lambda^{p} |\{ x \in \mathbb{R}^n: |f(x)| > \lambda \}| d\lambda $$
$$\int_1^\infty \lambda^{q-1} |E_\lambda| d\lambda\leq N \int_1^\infty \lambda^{q-(1+p)} d\lambda < \infty.$$
Then
$$\int_E |f|^q < \infty.$$
Instead of splitting the integral at $t=1$, let us split it at some arbitrary $t>0$: $$ \int_E |f|^q = q\left( \int_0^t(..) d\lambda + \int_t^\infty (..) d \lambda \right) $$ Then using your calculations (but explicitly computing the finite integrals) gives $$ \int_E |f|^q \le q \left( \frac{ t^q }q|E| + N \frac{t^{q-p}}{p-q}\right) $$ Now, we minimize the right-hand side with respect to $t$. (It is a convex function in $t$.) Its derivative is zero at $$ t = \left( \frac N{|E|} \right)^{1/p}, $$ which results in $$ \int_E |f|^q \le \left( 1 + \frac q{p-q}\right) N^{q/p} |E|^{1-q/p}. $$