A subset of Caffarelli–Kohn–Nirenberg inequalities in the hyperbolic space $${{\mathbb {H}}}^N$$ H N

Abstract

We prove a subset of inequalities of Caffarelli-Kohn-Nirenberg type in the hyperbolic space H N , N ≥ 2, based on invariance with respect to a certain nonlinear scaling group, and study existence of corresponding minimizers. Earlier results concerning the MoserTrudinger inequality are now interpreted in terms of CKN inequalities on the Poincaré disk.

Hardy and Caffarelli-Kohn-Nirenberg inequalities with nonradial weights

“…Because of their important roles in many areas of modern mathematics such as geometric analysis, partial differential equations, spectral theory, etc, the Caffarelli-Kohn-Nirenberg inequalities have been intensively investigated in many settings in the literature. See [10,12,13,14,15,17,21,26,33,34,39,40,42,45,47]. It is also worth mentioning that Caffarelli-Kohn-Nirenberg inequality is one of the most interesting inequalities in partial differential equations.…”