Equivalence of Norms

Aug 2010
26
0
Germany
I'm somewhat stuck with the simple proof of the following:

Let the (Lebesgue-)measure of some \(\displaystyle \Omega\) be finite and \(\displaystyle 1 \le p \le q \le \infty\).
Then for all \(\displaystyle u \in L^q(\Omega)\) it is also true that \(\displaystyle u \in L^p(\Omega)\), whereby
\(\displaystyle ||u||_p \le \text{meas}(\Omega)^{\frac{1}{p}-\frac{1}{q}}||u||_q\);
for \(\displaystyle q=\infty\) set \(\displaystyle \frac{1}{q}:=0\).

Proof: If \(\displaystyle q=\infty\), then \(\displaystyle ||u||^p_p=\int_{\Omega}|u(x)|^p\,dx \le \text{meas}(\Omega) \sup_{\Omega \setminus N}|u|^p\), where N ist a large enough null set.

If \(\displaystyle q<\infty\), then the Hölder inequality should help, but I'm somewhat confused by the suitable choice of exponents.