Equivalence of $L^p$ norm in a bounded domain

Question

For $1<p<q<\infty$ and a bounded domain $\Omega \subseteq \mathbb R^2$, is the following true? $$C_1\Vert f \Vert_{L^{p}(\Omega)} \leq \Vert f \Vert_{L^{q}(\Omega)}\leq C_2\Vert f \Vert_{L^{p}(\Omega)} $$ The first inequality is easy by simply using the Holder inequality and $C_1$ depends on the finite measure of $\Omega$. How to show the second inequality?

Answer 1

It is simply untrue. Let $\Omega$ be any bounded open set containing the origin and let $f(x) = \dfrac 1{|x|^{2/q}}$.