I'm trying to follow the proof of the Poincaré inequality given in Brezis' book titled "Functional Analysis, Sobolev Spaces and Partial Differential Equations". For those who have the book it is on page 218.
For those who don't have the book the statement is the following:
Suppose $I$ is a bounded interval of $\mathbb{R}$. Then there exists a constant $C$ (depending only on $p$ and $|I| < \infty$) such that
\begin{equation}
\lVert u \rVert_{W^{1,p}(I)} \leq C \lVert u' \rVert_{L^p(I)}
\end{equation}
for all $u \in W^{1,p}_0(I)$.
The proof goes as follows.
Let $u \in W^{1,p}_0(I)$. We establish that $|u(x)| \leq \lVert u' \rVert_{L^1(I)}$ via the fundamental theorem of calculus and thus $\lVert u \rVert_{L^{\infty}(I)} \leq \lVert u' \rVert_{L^1(I)}$. The inequality then follows via Holder's inequality.
It is the last sentence that I am struggling to understand. My initial thought was this working:
\begin{align}
\lVert u \rVert_{W^{1,p}(I)}^p &= \lVert u \rVert_{L^p(I)}^p + \lVert u' \rVert_{L^p(I)}^p \\
&\leq K\lVert u \rVert_{L^{\infty}(I)}^p + \lVert u' \rVert_{L^p(I)}^p \\
&\leq K\lVert u' \rVert_{L^{1}(I)}^p + \lVert u' \rVert_{L^p(I)}^p \\
&\leq CK\lVert u' \rVert_{L^p(I)}^p +\lVert u' \rVert_{L^p(I)}^p \\
&= (CK+1)\lVert u' \rVert_{L^p(I)}^p
\end{align}
where the constant $C$ is the one obtained from using the fact that the interval is bounded and applying Holder's inequality accordingly. It is the constant $K$ I am unsure of. By the calculation done in the proof we get that $u \in L^{\infty}(I)$ and so there exists a constant such that the $p$-norm is less than a constant times the $\infty$-norm BUT this constant depends on $u$ which it shouldn't. That makes me think I am misunderstanding how to apply Hölder's inequality.
Any help would be appreciated.
Best Answer
Maybe I am missing something: $$ ||u||_{L^p(I)}^p=\int_I |u|^p \leq ||\,|u|^p \,||_{L^{\infty}(I)} \int_I 1=\mu(I)||u||_{L^{\infty}(I)}^p=K||u||_{L^{\infty}(I)}^p $$ So the constant depends only on the intervall $I$.
Proceeding you get $$ ||u||_{L^{\infty}(I)}^p \leq ||u'||_{L^{1}(I)}^p\leq_{Jensen} \mu(I)^{p-1}||u'||_{L^{p}(I)}^p $$ Jensens inqequality also gives you no constant depending on $u$.
A small comment on Hoelders: We view $|u|^p$ as an $L^1$-function instead of $u$ as an $L^p$ function when applying Hoelders inequality.