This paper, published in *Compositio Mathematica* in 1984, by L. Caffarelli, R. Kohn, and L. Nirenberg, presents a detailed analysis of first-order interpolation inequalities with weights. The authors prove a theorem that establishes the existence of a positive constant \( C \) such that for a function \( u \in C_0^\infty(\mathbb{R}^n) \), the inequality \[ \|x^\gamma u\|_{L^r} \leq C \|x^\alpha |Du|_{L^p}^a \|x^\beta u\|_{L^s}^{1-a} \] holds if and only if certain conditions on the parameters \( p, q, r, \alpha, \beta, \sigma, \) and \( a \) are satisfied. The conditions include dimensional balance and constraints on the parameters \( \alpha \) and \( \sigma \). The proof is divided into several sections, addressing different cases such as \( n=1 \), \( \sigma=\alpha-1 \), and more general cases where \( n \geq 1 \) and \( \alpha \geq \sigma \geq \alpha-1 \). The authors also provide inequalities and auxiliary results that support the main theorem, including weighted Hardy-type inequalities and interpolation inequalities. The paper concludes with a detailed proof of the theorem in various scenarios, demonstrating the validity of the inequality under the specified conditions.This paper, published in *Compositio Mathematica* in 1984, by L. Caffarelli, R. Kohn, and L. Nirenberg, presents a detailed analysis of first-order interpolation inequalities with weights. The authors prove a theorem that establishes the existence of a positive constant \( C \) such that for a function \( u \in C_0^\infty(\mathbb{R}^n) \), the inequality \[ \|x^\gamma u\|_{L^r} \leq C \|x^\alpha |Du|_{L^p}^a \|x^\beta u\|_{L^s}^{1-a} \] holds if and only if certain conditions on the parameters \( p, q, r, \alpha, \beta, \sigma, \) and \( a \) are satisfied. The conditions include dimensional balance and constraints on the parameters \( \alpha \) and \( \sigma \). The proof is divided into several sections, addressing different cases such as \( n=1 \), \( \sigma=\alpha-1 \), and more general cases where \( n \geq 1 \) and \( \alpha \geq \sigma \geq \alpha-1 \). The authors also provide inequalities and auxiliary results that support the main theorem, including weighted Hardy-type inequalities and interpolation inequalities. The paper concludes with a detailed proof of the theorem in various scenarios, demonstrating the validity of the inequality under the specified conditions.