Sharp Sobolev and Adams-Trudinger-Moser inequalities for symmetric functions without boundary conditions on hyperbolic spaces

TL;DR

This paper establishes sharp Sobolev and Adams-Trudinger-Moser inequalities for symmetric radial functions on hyperbolic spaces without boundary conditions, expanding the understanding of embeddings in non-flat geometries.

Contribution

It introduces new weighted embedding theorems and comparison results for higher order derivatives of radial functions in hyperbolic spaces, without boundary zero assumptions.

Findings

Proved sharp weighted Sobolev embedding theorems for radial functions.

Established Adams-Trudinger-Moser type inequalities in hyperbolic spaces.

Developed novel radial lemmas and decay properties for higher order derivatives.

Abstract

Embedding theorems for symmetric functions without zero boundary condition have been studied on flat Riemannian manifolds, such as the Euclidean space. However, these theorems have only been established on hyperbolic spaces for functions with zero boundary condition. In this work, we focus on sharp Sobolev and Adams-Trudinger-Moser embeddings for radial functions in hyperbolic spaces, considering both bounded and unbounded domains. One of the main features of our approach is that we do not assume boundary zero condition for symmetric functions on geodesic balls or the entire hyperbolic space. Our main results include Theorems 1.2, 1.3, and 1.4, which establish weighted Sobolev embedding theorems, and Theorems 1.5 together with 1.6, which present Adams-Trudinger-Moser type of embedding theorems. In particular, a key result is Theorem 1.1 which is a highly nontrivial comparison result between norms of the higher order covariant derivatives and higher order derivatives of the radial functions. Higher order asymptotic behavior of radial functions on hyperbolic spaces are established to prove our main theorems. This approach includes novel radial lemmata and decay properties of higher order radial Sobolev functions defined in hyperbolic space.