L. Caffarelli, R. Kohn, and L. Nirenberg proved first-order interpolation inequalities with weights in the journal *Compositio Mathematica* (1984). These inequalities generalize standard interpolation inequalities between functions and their first derivatives in various $ L^p $ norms on $ \mathbb{R}^n $, with each term weighted by a power of $ |x| $. The authors show that such inequalities hold under specific conditions on parameters $ p, q, r, \alpha, \beta, \sigma $, and $ a $, which ensure the norms in the inequality are finite. The key result is that the inequality holds if and only if certain dimensional balance and other conditions are satisfied. The authors also demonstrate that the constant in the inequality is bounded on compact subsets of parameter space where the conditions hold. The proof involves verifying necessity and sufficiency, using tools such as weighted Hardy-type inequalities and interpolation techniques. The results are extended to both radial and non-radial functions, and the sufficiency is established for various cases, including when $ \sigma < \alpha - 1 $. The paper concludes with a detailed analysis of the conditions under which the inequality holds and the boundedness of the constant.L. Caffarelli, R. Kohn, and L. Nirenberg proved first-order interpolation inequalities with weights in the journal *Compositio Mathematica* (1984). These inequalities generalize standard interpolation inequalities between functions and their first derivatives in various $ L^p $ norms on $ \mathbb{R}^n $, with each term weighted by a power of $ |x| $. The authors show that such inequalities hold under specific conditions on parameters $ p, q, r, \alpha, \beta, \sigma $, and $ a $, which ensure the norms in the inequality are finite. The key result is that the inequality holds if and only if certain dimensional balance and other conditions are satisfied. The authors also demonstrate that the constant in the inequality is bounded on compact subsets of parameter space where the conditions hold. The proof involves verifying necessity and sufficiency, using tools such as weighted Hardy-type inequalities and interpolation techniques. The results are extended to both radial and non-radial functions, and the sufficiency is established for various cases, including when $ \sigma < \alpha - 1 $. The paper concludes with a detailed analysis of the conditions under which the inequality holds and the boundedness of the constant.