A Unified Hölder Lebesgue Framework for Caffarelli Kohn Nirenberg Inequalities

We develop a unified Hölder Lebesgue scale $X^p$ and its weighted, higher order variants $X^{k,p,a}$ to extend the Caffarelli Kohn Nirenberg (CKN) inequality beyond the classical Lebesgue regime. Within this framework we prove a two parameter interpolation theorem that is continuous in the triplet $(k,1\mathrm{/}p,a)$ and bridges integrability and regularity across the Lebesgue Hölder spectrum. As a consequence we obtain a generalized CKN inequality on bounded punctured domains $\mathrm{\Omega }\subset {\mathbb{R}}^n\setminus \{0\}$; the dependence of the constant on $\mathrm{\Omega }$ is characterized precisely by the (non)integrability of the weights at the origin. At the critical endpoint $p=n$ we establish a localized, weighted Brezis Wainger type bound via Trudinger Moser together with a localized weighted Hardy lemma, yielding an endpoint CKN inequality with a logarithmic loss. Sharp constants are not pursued; rather, we prove existence of constants depending only on the structural parameters and coarse geometry of $\mathrm{\Omega }$. Several corollaries, including a unified Hardy–Sobolev inequality, follow from the same interpolation mechanism.