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September 19, 2025

The Limiting Case of the Fractional Caffarelli–Kohn–Nirenberg Inequality in Dimension One

Key Points

The paper shows that the fractional Caffarelli–Kohn–Nirenberg inequality approaches Onofri’s inequality as the parameter converges.
As $ ext{parameter} o 1/2$, we observe significant implications for functions satisfying the extremal inequality.
An analysis of the weighted Liouville equation clarifies the nature of solutions relevant to the extremal function challenges.
The results provide insight into behaviors of inequalities in dimension one, suggesting avenues for further exploration.

Abstract

Abstract In this paper, we study the fractional Caffarelli–Kohn–Nirenberg (CKN) inequality in one dimension when the parameter converges (from the left) to its critical value 1/2, obtaining Onofri’s inequality in the unit disk as the limit. A difficulty that we encounter is the lack of an explicit expression for the extremal function at which the CKN inequality is attained, which we address by studying solutions of the weighted Liouville equation for the half-Laplacian in dimension one.

The Limiting Case of the Fractional Caffarelli–Kohn–Nirenberg Inequality in Dimension One

Key Points

Abstract

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