Possible Duplicate:
How slow/fast can $L^p$ norm grow?
Rudin asked:
Suppose $f$ is Lesbegue measurable in $(0,1)$ and not essentially bounded. Then we know $|f|_{p}\rightarrow \infty$ as $p\rightarrow \infty$. Is it true that to every positive function $\Phi$ on $(0,\infty)$ such that $\Phi(p)\rightarrow \infty$ as $p\rightarrow \infty$, then $|f|_{p}\rightarrow \infty$ but $|f|_{p}\le \Phi(p)$ for all sufficiently large $p$?
This is not possible for any $\Phi$ since $\Phi$ has to be monotone. But suppose $\Phi$ is monotone already, then can we always find $f$ such that $|f|_{p}\le \Phi(p)$? I am thinking about $\Phi=O(x^{\alpha}\log[x]^{\beta})$ type of monotone increasing function.
, and the problem is substantially easier.
– 2012-12-25