On general Caffarelli-Kohn-Nirenberg type inequalities involving non-doubling weights

TL;DR

This paper extends Caffarelli-Kohn-Nirenberg inequalities to include non-doubling weights, unifying critical and non-critical cases and clarifying their underlying relationships.

Contribution

It introduces a new framework that incorporates non-doubling weights into Caffarelli-Kohn-Nirenberg inequalities, revealing connections between different cases.

Findings

Unified treatment of critical and non-critical inequalities

Inclusion of non-doubling weights broadens applicability

Clarification of relationships between inequality cases

Abstract

We will establish the Caffarelli-Kohn-Nirenberg type inequalities with non-doubling weights being permitted. The classical Caffarelli-Kohn-Nirenberg type inequalities are categorized into non-critical and critical cases, and it is known that there is some kind of mysterious relationship between them. Interestingly the new framework in this treatise allows them to be integrated and reveals the meaning of mysterious relationships.