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

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 $d\ge 4$, 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’'.