Integral Inequality for Measurable Functions

[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 $C,m>0$.

 

[micromass]

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.