Hello I have a question about Sobolev spaces.
Let $n \in \mathbb{N}$, $\alpha >\frac{n}{2}$, $c :\mathbb{R}^{n} \to \mathbb{R}$ be a $\alpha$-integrable function. I want to show the following: \begin{align*} {}^{\forall}f, {}^{\forall}g \in W^{1,2}(\mathbb{R}^{n}),\quad \int_{\mathbb{R}^{n}}|c(x)f(x)g(x)|\,dx<\infty\cdots(\ast) \end{align*}
My idea
Case1 $n \geq 2$,
\begin{align*} \int |c(x)f(x)g(x)|\,dx &\leq \left(\int |c(x)|^{\alpha}\,dx\right)^{\frac{1}{\alpha}} \left(\int|f(x)g(x)|^{\beta}\,dx\right)^{\frac{1}{\beta}} \quad \left( \frac{1}{\alpha}+\frac{1}{\beta}=1\right) \\ &\leq \left(\int |c(x)|^{\alpha}\,dx\right)^{\frac{1}{\alpha}} \left(\int\frac{|f(x)|^{2\beta}+|g(x)|^{2\beta}}{2}\,dx\right)^{\frac{1}{\beta}} \end{align*}
Since $f \in W^{1,2}(\mathbb{R}^{n})$, by Sobolev embedding theorem $f \in L^{2\beta}(\mathbb{R}^{n},dx)$
Case2 $n=1$, I feel that $(\ast)$ is not true in general. If you find a counterexample, please tell me.
Thank you in advance.