Radial Sobolev embeddings on spherically symmetric Riemannian manifolds

TL;DR

This paper characterizes Sobolev spaces of radial functions on spherically symmetric Riemannian manifolds, deriving sharp embeddings, decay estimates, and unifying classical results in Euclidean and hyperbolic spaces.

Contribution

It provides a sharp one-dimensional reduction for radial Sobolev spaces on manifolds, enabling optimal embeddings and decay estimates that extend classical Euclidean and hyperbolic results.

Findings

Sharp characterization of radial Sobolev spaces via weighted Sobolev spaces

Optimal Sobolev-type embeddings into weighted Lebesgue spaces

New decay estimates for radial Sobolev functions near origin and infinity

Abstract

We study Sobolev spaces of radial functions on spherically symmetric Riemannian manifolds. Using geodesic polar coordinates, we give a sharp one-dimensional reduction: a radial function belongs to the Sobolev space on the manifold if and only if its radial representation lies in an associated weighted Sobolev space on an interval, with weights determined explicitly by the metric. This characterization allows us to prove optimal Sobolev-type embeddings for radial functions into weighted Lebesgue spaces on both bounded and unbounded spherically symmetric manifolds. As further consequences, we establish new radial lemmas and decay estimates that capture the precise behaviour of radial Sobolev functions near the origin and at infinity. Our results unify and extend the classical radial embeddings in Euclidean and hyperbolic spaces.