Endpoint radial embedding

Determine whether the radial ball-mass Sobolev space E^{p,q}_{b,rad}(ℝ^N) is continuously embedded in L^{2^{p,q}_{b,*}}(ℝ^N) for general admissible parameter triples (p,q,b).

Background

Theorem 1 proves compact or continuous radial embeddings only for exponents strictly above the critical exponent 2{p,q}_{b,*}, while the sharpness theorem rules out exponents below it. The endpoint is known to behave differently in two special cases: it holds for q = 1 and fails for q = 2, b = N−1. Its status for general parameters is unresolved.

References

Theorem \ref{Theorem 1} leaves open the embedding at $s = 2{p,q}_{b,\ast}$. It holds when $q = 1$ Lemma 5.1 and fails when $q = 2$ and $b = N - 1$ Theorem 4. What happens for general $(p,q,b)$?

— A unifying zero-mass equation  (2609.26585 - Perera, 22 Sep 2026) in Section 6, item 3; Section 2, discussion following Theorem \ref{Theorem 1}