Part 9: A Program That Generalizes All Inequalities and Incorporates Them into Functions, While Simultaneously Optimizing Them via the Function C
Year: 2026
Type: preprint
Abstract: We develop conditional nonlinear refinements of Sobolev andGagliardo–Nirenberg interpolation inequalities on bounded Lipschitzdomains. Positive norm-gap terms are controlled either by quantitative growth and geometric convexity of the nonlinearity or by uniformcomparability of the underlying norms. In the critical regime westudy finite-family logarithmic factors defined through Sobolev-normratios, especially the minimum-ratio functional Amin and a hybridweighted-mean family Aλ,p. We establish ordering and parameterdependence results and a uniform error bound for approximatingAmin by negative-order power means. Fourier examples rule outlogarithmic exponents below 1/2 for the unrestricted two-dimensionalcritical Sobolev class. All estimates are stated with explicit hypotheses and follow from classical interpolation and critical logarithmicembeddings; no new unrestricted Sobolev embedding is claimed.
Author: Tuan, Tran
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7223556853