3

Given $f$ is Lebesgue measurable on $(0,1)$ and not essentially bounded, is it true that to every positive function $\Phi$ on $(0,\infty)$ such that $\Phi(p)\to\infty$ as $p\to\infty$ one can find an $f$ such that $\|f\|_p\to\infty$ but $\|f\|_p\leq\Phi(p)$ for all sufficiently large $p$?

My gut instinct tells me "no", but I don't have any logical arguments to back me up. If anyone has a suggestion as to where to start when thinking about this problem, I would appreciate it. (This question is asked in Chapter 3 of Rudin's Real and Complex Analysis text, which is entitled "$L^p$ Spaces".

Michael Hardy
  • 1
PiTheMathemagician
  • 571
  • 1
    This is a great classical analysis question. Hints: You may assume without loss of generality $\Phi(p)$ is as regular as you like (continuous for example). Consider a step function on a disjoint partition of $(0,1)$ and choose your coefficients and the measure of the sets in the partition carefully. – Chris Janjigian Oct 02 '14 at 19:10

0 Answers0