real analysis - Taking the limit $lim_{prightarrow infty} left( frac{|f|_infty}{|f|_p}right)^p$

Taking the limit
$$\lim_{p\rightarrow \infty} \left( \frac{\|f\|_\infty}{\|f\|_p}\right)^p$$

First I think the expression after taking the limit will depend on the function $f$.

In my attempt, because it is in the form $``1^\infty"$, I tried to use L'Hopital's rule. And we can calculate the limit (assuming the integrals are defined and finite, I just want to see what the limit might look like).
\begin{align*}
\lim_{p\rightarrow \infty} \left(\frac{\|\nabla u \|_\infty}{\|\nabla u\|_p}\right)^{p} &= \lim_{p\rightarrow \infty} \exp\left( p\log\left(\frac{\|\nabla u \|_\infty}{\|\nabla u\|_p}\right)\right)\\
&=\lim_{p\rightarrow \infty} \exp\left( \frac{\log\left(\frac{\|\nabla u \|_\infty}{\|\nabla u\|_p}\right)}{\frac{1}{p}}\right)\\
&=\lim_{p\rightarrow \infty} \exp\left( \frac{\frac{d}{dp} \left[-\log\left(\frac{\|\nabla u \|_p}{\|\nabla u\|_\infty}\right)\right]}{\frac{-1}{p^2}}\right)\\
&=\lim_{p\rightarrow \infty} \exp\left( \frac{\left(\frac{\|\nabla u \|_\infty}{\|\nabla u\|_p}\right)\frac{\frac{d}{dp}\left[\|\nabla u\|_p \right]}{\|\nabla u \|_\infty}\}}{\frac{1}{p^2}}\right)\\
\end{align*}
where

\begin{align*}
\frac{d}{dp}\left[\|\nabla u\|_p \right] &= \frac{d}{dp}\left[\left(\int_{\mathbb R^N} |\nabla u |^p dx \right)^{1/p} \right] \\
&=\frac{d}{dp}\left[\exp\left(\frac{1}{p} \log\left(\int_{\mathbb R^N} |\nabla u |^p dx \right)\right)\right] \\
&=\|\nabla u \|_p \left\{\frac{-1}{p^2}\log\left(\int_{\mathbb R^N} |\nabla u |^p dx \right) + \frac{1}{p} \frac{1}{\int_{\mathbb R^N} |\nabla u |^p dx }\int_{\mathbb R^N} |\nabla u|^p \log(|\nabla u |) dx \right\}
\end{align*}

Putting it back into the limit we get
$$\lim_{p\rightarrow \infty} \exp \left(-\log\left(\int_{\mathbb R^N} |\nabla u |^p dx \right) + p \frac{1}{\int_{\mathbb R^N} |\nabla u |^p dx }\int_{\mathbb R^N} |\nabla u|^p \log(|\nabla u |) dx \right)$$
which simplifies to
$$\lim_{p\rightarrow \infty} \frac{\exp \left(p\frac{\int_{\mathbb R^N} |\nabla u|^p \log(|\nabla u |) dx}{\int_{\mathbb R^N} |\nabla u |^p dx } \right)}{\int_{\mathbb R^N} |\nabla u |^p dx }$$

And I am stuck. Is this a correct approach?

Thank you very much!

Answer

Suppose $f\in L^\infty([0,1])$ and $\|f\|_\infty> 0$. Let $E = \{x:|f(x)|=\|f\|_\infty\}.$ Then

$$\lim_{p\to \infty} \left( \frac{\|f\|_\infty}{\|f\|_p}\right)^p = \frac{1}{m(E)}.$$

(If $m(E)=0,$ the conclusion is that the limit is $\infty.$)

Proof: Let $M= \|f\|_\infty$. Then the expression equals

$$\frac{M^p}{M^p\cdot m(E) + M^p\int_{[0,1]\setminus E}|f/M|^p}.$$

Cancel the $M^p$ terms and then apply the dominated convergence theorem to see the integral in the denominator $\to 0.$ That gives the result.