I want to show that the norms $\Vert\cdot\Vert_{a,p}$ on $C_b(\mathbb{R})$ defined as below are not equivalent: $$ \Vert f\Vert_{a,p}:=\left(\int_{\mathbb{R}}e^{-ax^2}|f(x)|^p \mathrm{d}x\right)^{1\over p}. $$
To prove this, I've set bounded functions $f_{b,c}(x):=e^{-bx^2}\cdot |x|^c$ where $b,c>0$, such that $$\Vert f_{b,c}\Vert_{a,p}=\cdots=(a+pb)^{-{pc+1\over 2p}}\cdot\left[\left({pc-1\over 2}\right)!\right]^{1\over p},$$ where $t!=\int_{\mathbb{R}_+}e^{-y}y^t\mathrm{d}y$. Therefore, (Case I) when $p=p^{\prime}$, $a\neq a^{\prime}$, $$ {\Vert f_{b,c}\Vert_{a^{\prime},p^{\prime}}\over\Vert f_{b,c}\Vert_{a,p}}=\left({a+pb\over a^{\prime}+pb}\right)^{pc+1\over 2p}\to 0\text{ OR}+\infty,\text{ as } c\to +\infty, $$ which shows that these two norms are not equivalent; (Case II) when $p\neq p^{\prime}$, I struggled with how to prove the non-equivalence. Can I use the conclusion of $L^p$ norms? Or can I tackle this similar to Case I?