A sharp Sobolev inequality on the Caffarelli-Kohn-Nirenberg hyperbolic space

TL;DR

This paper explores a sharp Sobolev inequality on the Caffarelli-Kohn-Nirenberg hyperbolic space, revealing dimension-dependent sharpness properties and extending classical results to a new geometric setting.

Contribution

It establishes a new sharp Sobolev inequality on the Caffarelli-Kohn-Nirenberg hyperbolic space, highlighting dimension-dependent sharpness and extending classical Euclidean and hyperbolic results.

Findings

Sharp inequality holds for effective dimension n in [3,4)

Dimension 3 is critical for sharpness in the hyperbolic setting

Extension of classical Sobolev inequalities to Caffarelli-Kohn-Nirenberg hyperbolic space

Abstract

In the Euclidean space $\mathbb{R}^d$, the sharp classical Sobolev inequality is equivalent by conformal invariance to a Sobolev inequality on the hyperbolic space $\mathbb{H}^d$. This inequality is sharp in dimension $d\geq 4$, but it is not in dimension $d=3$ by results of Benguria, Frank and Loss, as well as Mancini and Sandeep. In this article, we investigate a similar phenomenon for the Caffarelli-Kohn-Nirenberg inequality and its hyperbolic analogue. In our setting, the condition for improving the inequality reads $n\in [3,4)$, where $n$ is an ``effective dimension''.