Fractional Sobolev-type embedding on CR sphere and Heisenberg group

This paper studies critical fractional Sobolev inequalities with lower-order terms on the standard CR sphere ${\mathbb{S}}^{2n+1}$. Let $Q=2n+2$, let $s\in (0,1)$, let $1<p<Q$, and let $p_s^{\ast }=\frac{Qp}{Q-sp}$. For the inequality $\mathrm{\parallel }u{\mathrm{\parallel }}_{L^{p_s^{\ast }}({\mathbb{S}}^{2n+1})}\le A[u{]}_{s,p}+B\mathrm{\parallel }u{\mathrm{\parallel }}_{L^p({\mathbb{S}}^{2n+1})}$, we prove that the admissible lower-order coefficients are exactly $[\mathrm{\mid }{\mathbb{S}}^{2n+1}{\mathrm{\mid }}^{-s\mathrm{/}Q},\mathrm{\infty })$. For the power-type inequality $\mathrm{\parallel }u{\mathrm{\parallel }}_{L^{p_s^{\ast }}({\mathbb{S}}^{2n+1})}^p\le A[u{]}_{s,p}^p+B\mathrm{\parallel }u{\mathrm{\parallel }}_{L^p({\mathbb{S}}^{2n+1})}^p$, we show that the admissible set is $[\mathrm{\mid }{\mathbb{S}}^{2n+1}{\mathrm{\mid }}^{-sp\mathrm{/}Q},\mathrm{\infty })$ when $1<p\le 2$, and $(\mathrm{\mid }{\mathbb{S}}^{2n+1}{\mathrm{\mid }}^{-sp\mathrm{/}Q},\mathrm{\infty })$ when $2<p<Q$. Via the Cayley transform, we derive the exact weighted counterpart on the Heisenberg group and prove that the corresponding admissible sets coincide with those on the sphere. We also show that nonlinear first-moment constraints do not improve the optimal lower-order coefficient, whereas finite-codimensional linear constraints excluding nonzero constants yield coercive inequalities.