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 \(Xk,p,a\) 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 \((k,1/p,a)\) and bridges integrability and regularity across the Lebesgue Hölder spectrum. As a consequence we obtain a generalized CKN inequality on bounded punctured domains \(Ω^n\{0\}\); the dependence of the constant on \(Ω\) 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 \(Ω\). Several corollaries, including a unified Hardy--Sobolev inequality, follow from the same interpolation mechanism.