Welcome to our community

Be a part of something great, join today!

Problem of the Week #224 - Sep 13, 2016

Status
Not open for further replies.
  • Thread starter
  • Moderator
  • #1

Euge

MHB Global Moderator
Staff member
Jun 20, 2014
1,892
Here is this week's POTW:

-----
Let $(X,\mu)$ be a positive measure space. For $0 < p < \infty$, why is the mapping $\mathcal{L}^p(X,\mu) \to \mathcal{L}^1(X,\mu)$ sending $f$ to $\lvert f\rvert^p$, continuous?


-----


Remember to read the POTW submission guidelines to find out how to submit your answers!
 
  • Thread starter
  • Moderator
  • #2

Euge

MHB Global Moderator
Staff member
Jun 20, 2014
1,892
No one answered this week's problem. You can read my solution below.


For all pairs of nonnegative numbers $a, b$, $\lvert a^p - b^p\rvert \le \lvert a - b\rvert^p$ if $0 < p < 1$, and $\lvert a^p - b^p\rvert \le p(a^{p-1} + b^{p-1})\lvert a - b\rvert$ if $p \ge 1$. Thus, for all $f, g\in \mathcal{L}^p(X,\mu)$,

$$\| \lvert f\rvert^p - \lvert g\rvert^p \|_1 \le \|f - g\|_p^p \quad (0 < p < \infty)$$

and for $1\le p < \infty$,

$$\|\lvert f\rvert^p - \lvert g\rvert^p\|_1 \le p\int_X \lvert f\rvert^{p-1}\lvert f - g\rvert\, d\mu + p\int_X \lvert g\rvert^{p-1}\lvert f - g\vert\, d\mu \le p(\|f\|_p\|f - g\|_p + \|g\|_p\|f - g\|_p) = p(\|f\|_p + \|g\|_p)\|f - g\|_p$$

using Hölder's inequality in the penultimate step. These inequalities imply continuity of the map $f\mapsto \lvert f\rvert^p$ from $\mathcal{L}^p(X,\mu)$ to $\mathcal{L}^1(X,\mu)$.
 
Status
Not open for further replies.