# How slow/fast can L^pL^p norm grow?

This is actually an exercise in Rudin’s Real and Complex Analysis, $L^p$ spaces chapter. Could anyone help me out? Thanks in advance.

Motivation: It’s well known that if we have a function $f$ which belongs to $L^p(0,1)$ for all $p\ge 1$. Then $\lim_{p\rightarrow \infty}\|f\|_p=\|f\|_{\infty}$ (moreover, $\|f\|_p$ is increasing in $p$). This is true even if $\|f\|_{\infty}=\infty$.

Question: How slow (fast) can $\|f\|_p$ grow when $\|f\|_{\infty}=\infty$? More precisely, given any positive increasing function $\Phi$ with $\lim_{p\rightarrow \infty}\Phi(p)=\infty$, can we always find a function $f$ which belongs to $L^p(0,1)$ for all $p\ge 1$, and $\|f\|_{\infty}=\infty$, such that $\|f\|_p\le (\ge)\Phi(p)$ for large $p$?

I’ll answer the question of whether $\|f\|_p$ can grow arbitrarily slowly; the answer is yes. I’m fairly certain that it can grow arbitrarily quickly as well, and, though I haven’t given it much thought, I suspect a similar argument can be concocted. The problem doesn’t seem to rely on the interval $(0,1)$, so what I write below doesn’t either. If you really want to put everything in $(0,1)$, it is not hard to do so.
Let $\Phi$ be as you say. Here’s the idea: We choose disjoint sets $E_n$ with positive (but as yet undetermined) measure, and we require that $f=\sum_{n=1}^{\infty} c_n\chi_{E_n}$, where $\{c_n\}$ is some sequence of positive numbers increasing to infinity, and $\chi_{E_n}$ is the characteristic function of $E_n$. This ensures that $f\notin L^{\infty}$. Assuming (as we may) that $\Phi(p)$ is bounded away from $0$, we see that the quotient $c_n^p/\Phi(p)^p$ is bounded in $p$ for each fixed $n$. So we are free to choose our sets $E_n$ so small in measure that $(c_n/\Phi(p))^pm(E_n)<2^{-n}$, independently of $p$. To conclude, we simply observe that
This means $\|f\|_p<\Phi(p)$ for all $p$ (or, if you like, for all $p\in[a,\infty)$, where $\Phi(p)$ is bounded away from $0$ on $[a,\infty)$).