A Unified Hölder Lebesgue Framework for Caffarelli Kohn Nirenberg Inequalities

Abstract: We develop a unified H\"older 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/p,a)) and bridges integrability and regularity across the Lebesgue H\"older spectrum. As a consequence we obtain a generalized CKN inequality on bounded punctured domains (\Omega\subset\mathbb{R}^n\setminus{0}); the dependence of the constant on (\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 (\Omega). Several corollaries, including a unified Hardy--Sobolev inequality, follow from the same interpolation mechanism.