This question is on a point in D.R. Adams paper "A Sharp Inequality of J. Moser for Higher Order Derivatives". Precisely the lemma says:  

Given $a(s,t)$ be a non negative measureable function on $(-\infty,\infty)\times [0,\infty)$ such that
$$
a(s,t)\leq 1\;\text{ when }\;0<s<t\label{1}\tag{1}
$$ and
$$
\sup\_{t>0}\left(\int\_{-\infty}^0+\int\_t^{\infty} a(s,t)^q ds\right)^{\frac{1}{q}}=b<\infty\label{2}\tag{2}
$$
Then there is a constant $c\_0=c\_0(p,b)$ such that if $\phi\geq 0$ and
$$
\int\_{\mathbb{R}}\phi(s)^p ds\leq 1\label{3}\tag{3}
$$
then
$$
\int\_{0}^{\infty} e^{-F(t)} dt\leq c\_0
$$
where
$F(t)=t-\left(\int\_{\mathbb{R}}a(s,t)\phi(s) ds\right)^q$ and
$q=\frac{p}{p-1}$.


1. He claims that
$$
\int\_{0}^{\infty}e^{-F(t)}dt = \int\_{\infty}^{\infty}|E\_{\lambda}|e^{-\lambda}d\lambda
$$ where $E\_{\lambda}:=\{t\geq 0:F(t)\leq\lambda\}$. I don't see how to prove it. I was thinking of the Layer cake representation but I couldn't prove it as the integral in R.H.S is over $\mathbb{R}$ not in $(0,\infty)$.
2. Furthermore he claims that if $t\in E\_{\lambda}$ then
$$
t-\lambda\leq((b^q+t)^{\frac{1}{q}}(1-L(t)^p)^{\frac{1}{p}}+bL(t))^q
$$
where $L(t)=\left(\int\_s^{\infty}\phi(s)^p ds\right)^{\frac{1}{p}}$ (here the 1st term of RHS comes out from using Holder inequality and equations \eqref{1}, \eqref{2}, but the last portion i.e $+bL(t)$ is not coming rather some weaker estimate is coming.


Any help would be very much helpful to understand the thing going on.



Doubts in first lemma in the paper of Adams regarding sharp Moser inequality
For the second, rewrite $F(t) \leq \lambda$ as
$$t - \lambda \leq \left(\int\_{\mathbb{R}} a(s,\,t)\phi(s)\,ds\right)^q,$$
which reduces the problem to showing that
$$\int\_{\mathbb{R}} a(s,\,t)\phi(s)\,ds \leq (b^q + t)^{1/q}(1-L^p(t))^{1/p} + bL(t).$$
To verify this inequality write the left side as
$$\int\_{-\infty}^t a(s,\,t)\phi(s)\,ds + \int\_t^{\infty}a(s,\,t)\phi(s)\,ds,$$
and apply Holder's inequality to each term. The second term is clearly bounded by $bL(t)$. For the first, use that $\int\_{-\infty}^t \phi^p \leq 1-L^p(t)$ (this uses $\int\_{\mathbb{R}}\phi^p \leq 1$) and that $\int\_{-\infty}^t a^q(s,\,t) = \int\_{-\infty}^0 a^q + \int\_0^t a^q \leq b^q + t$ (using the definition of $b$ for the first, and the bound of $1$ on $a$ in the relevant interval for the second).


