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<p>Embedding theorems for symmetric functions without any boundary conditions 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 homogeneous Dirichlet conditions. 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 any boundary conditions for symmetric functions on geodesic balls or the entire hyperbolic space. Our main results include Theorems <InternalRef RefID="FPar4">1.2</InternalRef>, <InternalRef RefID="FPar5">1.3</InternalRef>, <InternalRef RefID="FPar9">1.4</InternalRef>, which establish weighted Sobolev embedding theorems, and Theorems <InternalRef RefID="FPar12">1.5</InternalRef> together with <InternalRef RefID="FPar13">1.6</InternalRef>, which present Adams-Trudinger-Moser type of embedding theorems. In particular, a key result is Theorem <InternalRef RefID="FPar3">1.1</InternalRef>, a highly nontrivial comparison 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 is 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.</p>

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

  • 期刊论文

  • 发表日期：

    2025年9月16日

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  • João Marcos Do Ó,
  • Guozhen Lu,
  • Raoní Ponciano

摘要

Embedding theorems for symmetric functions without any boundary conditions 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 homogeneous Dirichlet conditions. 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 any boundary conditions for symmetric functions on geodesic balls or the entire hyperbolic space. Our main results include Theorems 1.2, 1.3, 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, a highly nontrivial comparison 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 is 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.