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

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

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