Sobolev embedding theorem, inequalities

Question

I have a question about an inequality.

Let $n \in \mathbb{N}, \alpha >n$ and $b:\mathbb{R}^{n} \to \mathbb{R}$ be a $\alpha$-integrable function. (i.e. $b \in L^{\alpha}(\mathbb{R}^{n})$). I want to prove the following inequality:

For any $\epsilon$, there exists a constant $A_{\epsilon}$ such that \begin{align*} \int_{\mathbb{R}^{n}}|b|^{2}|f|^{2}dx \leq \epsilon \int_{\mathbb{R}^{n}}|\nabla f|^{2}dx+A_{\epsilon} \int_{\mathbb{R}^{n}}|f|^{2}dx,\quad f \in W^{1,2}(\mathbb{R}^{n})\cdots(\ast) \end{align*} where $W^{1,2}(\mathbb{R}^{n})$ is the $(1,2)$-Sobolev space on $\mathbb{R}^{n}$.

My attempt

I can partially prove this inequality by using the Sobolev embedding theorem:

\begin{align*} \int |b|^{2}|f|^{2}dx &\leq \left( \int|b|^{4}|f|^{2}dx \right)^{1/2}\left(\int|f|^{2}dx \right)^{1/2} \\ &=\| |b|^{2}|f| \|_{L^{2}} \|f\|_{L^{2}} \end{align*}

When $n=4$ and $\alpha=8$, by Hölder inequality

\begin{align*} \int |b|^{2}|f|^{2}dx &\leq \| |b|^{2}|f| \|_{L^{2}} \|f\|_{L^{2}} \\ &\leq \|b\|_{L^{\alpha}}^{2} \|f\|_{L^{q}} \|f\|_{L^{2}},\quad \left(\frac{1}{\alpha/2}+\frac{1}{q}=\frac{1}{2} \right) \end{align*}

Since $q=\frac{2 \alpha }{\alpha-4}=4 \in [2,\frac{2n}{n-2}]$, by Sobolev embedding theorem

\begin{align*} \int |b|^{2}|f|^{2}dx &\leq \|b\|_{L^{\alpha}}^{2} \|f\|_{L^{q}} \|f\|_{L^{2}} \\ &\leq \|b\|_{L^{\alpha}}^{2}\left( C \|f\|_{W^{1,2}} \right) \|f\|_{L^{2}} \\ &= C\|b\|_{L^{\alpha}}^{2}\|f\|_{L^{2}} \|f\|_{W^{1,2}} \\ &\leq \epsilon \|f\|_{W^{1,2}}^{2}+\frac{C\|b\|_{L^{\alpha}}^{4}}{4 \epsilon}\|f\|_{L^{2}}^{2} \\ &= \epsilon \int_{\mathbb{R}^{n}}|\nabla f|^{2}dx+\left(\epsilon+\frac{C\|b\|_{L^{\alpha}}^{4}}{4 \epsilon}\right)\int_{\mathbb{R}^{n}}|f|^{2}dx \end{align*}

The inequality $(\ast)$ holds in general? Thank you in advance.

ADD

When $n\geq3$, SOLVED. SEE BELOW ANSWER.

When $n=2$, by Hölder inequality $(p=\frac{\alpha}{2}, q=\frac{\alpha}{\alpha-2})$ and Sobolev embedding theorem \begin{align*} \int_{E_{M}}|b|^{2}|f|^{2} &\leq \left(\int|b|^{\alpha}\right)^{2/\alpha}\|f\|_{L^{2q}}^{2} \\ &\leq C\left(\int|b|^{\alpha}\right)^{2/\alpha}\|f\|_{W^{1,2}}^{2} \quad (2q=2+\frac{4}{\alpha-2} \geq 2) \end{align*}

Answer 1

Let $E_M = \{x\colon |b(x)|>M\}$ and take $M$ so large that $$ \int_{E_M} |b|^\alpha < \varepsilon. $$

Then you notice that by Holder ($p=n/(n-2)$, $q=2/n$) and then Sobolev ($2^* = 2n/(n-2)$) inequalities: $$ \int_{E_M} |f|^2|b|^2 \le ||f||^2_{L^{2^*}(E_M)} ||b||^2_{L^n(E_M)} \le C ||\nabla f||^2_2 \left(\int_{E_M} |b|^\alpha\right)^{2/n} \le C \varepsilon^{2/n} ||\nabla f||^2_2 $$ On the other hand $$ \int_{\mathbb R^n \setminus E_M} |f|^2|b|^2 \le M^2 ||f||_2^2 $$