Radial Sobolev embeddings on spherically symmetric Riemannian manifolds
João Marcos do Ó, Guozhen Lu, Raoní Ponciano
TL;DR
The paper establishes a sharp radial reduction for Sobolev spaces on spherically symmetric Riemannian manifolds: a radial function $u$ with $u(x)=v(d(x))$ belongs to $W^{k,p}(M)$ if and only if its radial profile $v$ lies in the one-dimensional weighted Sobolev space $W^{k,p}((0,R),\phi^{N-1})$, with weights dictated by the warping function $\phi$. It then proves precise embeddings of $W^{k,p}_{\mathrm{rad}}(M)$ into weighted Lebesgue spaces $L^q_{\phi^\theta}(M)$, including optimal ranges and radial lemmas that capture behavior near the origin and at infinity. The work also derives a decay lemma for the unbounded case, enabling compactness results and a unified treatment that extends Euclidean and hyperbolic radial embeddings to general spherically symmetric geometries. These results provide a robust framework for radial Sobolev embeddings on warped-product manifolds and open avenues for Adams-type, Trudinger–Moser-type, and fractional-analytic extensions in curved spaces.
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.