Let $-\infty<\alpha<\frac{n}{2}<\beta<+\infty$. Prove the validity of the following inequality: $\|g\|_{L^{1}(\mathbb{R}^{n})}\leq C\||x|^{\alpha}g(x)\|_{L^{2}(\mathbb{R}^{n})}^{\frac{\beta-n/2}{\beta-\alpha}}\||x|^{\beta}g(x)\|_{L^{2}(\mathbb{R}^{n})}^{\frac{n/2-\alpha}{\beta-\alpha}}$ for some constant $C=C(n,\alpha,\beta)$ independent of $g$
The author gives us a hint: First prove $\|g\|_{L^{1}}\leq C\||x|^{\alpha}g(x)\|_{L^{2}}+\||x|^{\beta}g(x)\|_{L^{2}}$ and then replace $g(x)$ by $g(\lambda x)$ for some suitable $\lambda>0$
According to the author's hint, I have proven the ultimate estimate by scaling and balance method, but how to prove the inequality in the hint?