Let $a(s,t)$ be a non-negative measurable function on $(-\infty,\infty)\times \left[0,\right. \left.\infty\right)$ such that (a.e.) $$a(s,t) \leq 1,\,\,when \, 0<s<t$$ $$ \sup_{t>0}\left(\int_{-\infty}^0 +\int_t^\infty a(s,t)^{p'} ds\right)^{1/p'}=b<\infty.$$ Then there is a constant $c_0(p,b)$ such that if $\phi\geq 0$ with $$\int_{-\infty}^\infty \phi(s)^p ds \leq 1,$$ then $$\int_0^\infty e^{-F(t)} dt\leq c_0,$$ where $$F(t) =t-\left(\int_{-\infty}^\infty a(s,t)\phi(s) ds\right)^{p'}.$$
Above is the statement of Lemma which is proved by Adams' in his paper http://www.jstor.org/stable/1971445
In his proof he writes two things: 1) $\int_{-\infty}^{\infty} |E_\lambda| e^{-\lambda} d\lambda=\int_0^\infty e^{-F(t)} dt$, where $E_\lambda=\{t\geq 0: F(t)\leq \lambda\}$
2) By Using Holder's inequality, if $t\in E_\lambda$ then $$t-\lambda\leq \left([b^{p'}+t]^{1/p'}[1-L(t)^p]^{1/p}+bL(t)\right)^{p'}$$ where $L(t)=\left(\int_t^\infty \phi(s)^p ds\right)^{1/p}.$
Could anyone help me in proving above statements?