Necessity of the radial condition (R2) for higher fractional orders

Determine whether condition (R2), namely \(\beta(m+2r)<mb+r(n-2s)\), is necessary for the weighted Gagliardo–Nirenberg inequality of Grushin type in the class of functions radial in \(x\) (or radial in both \(x\) and \(y\)) when \(s\ge\tfrac12\), under the standing hypotheses (H0)–(H3) and the necessary condition (R1).

Background

The paper proves that radial symmetry in the $x$-variable enlarges the admissible range of the weight exponent $\beta$. In the radial setting, condition (R1) is shown to be necessary, while the pair (R1)–(R2) is sufficient for the weighted Gagliardo–Nirenberg inequality. When $s<\tfrac12$, the authors establish that (R1) alone is both necessary and sufficient, because (R1) implies (R2).

For $s\ge\tfrac12$, the paper establishes only a gap: (R1) remains necessary, whereas (R1) together with (R2) is sufficient. Consequently, it is unresolved whether (R2) represents a genuine obstruction or merely a technical condition arising from the radial Hardy–Sobolev argument, particularly at the endpoint corresponding to $g=\infty$.