A inequality

1. Jul 13, 2013

amirmath

For what class of functions we have:
$$\int_{\Omega} [f(x)]^m dx \leq C\Bigr ( \int_{\Omega} f(x)dx\Bigr)^{m},$$
where $\Omega$ is open bounded and $f$ is measurable on $\Omega$ and $C,m>0$.

2. Jul 13, 2013

micromass

Staff Emeritus
For all $m$?

Well, take $f$ positive. You want $\|f\|_m\leq C^{1/m}\|f\|_1$, for all $m$. So by taking limits, we get

$$\|f\|_\infty = \lim_{m\rightarrow +\infty} \|f\|_m\leq \lim_{m\rightarrow +\infty}C^{1/m}\|f\|_1 = \|f\|_1$$

In particular, you want the sup-norm to exist. This already forces your function to be bounded a.e.

Last edited: Jul 13, 2013