Sharp Sobolev inequalities on the complex sphere

This paper is devoted to establish a class of sharp Sobolev inequalities on the unit complex sphere as follows: 1) Case $0<d<Q=2n+2$: for any $f\in C^\infty$ and $2\leq q \leq \frac{2Q}{Q-d}$, \begin{equation*} \|f\|_q^2\leq \frac{8(q-2)}{d(Q-d)} \frac{\Gamma^2((Q-d)/4+1)} {\Gamma^2((Q+d)/4)}\left( \int_{\mathbb{S}^{2n+1}} f\mathcal{A}_df d\xi -\frac{\Gamma^2((Q+d)/4)} {\Gamma^2((Q-d)/4)} \int_{\mathbb{S}^{2n+1}} |f|^2 d\xi\right) +\int_{\mathbb{S}^{2n+1}} |f|^2 d\xi; \end{equation*} 2) Case $d=Q$: for any $f\in C^\infty \cap\mathbb{R}\mathcal{P}$ and $2\leq q< +\infty$, \begin{equation*} \|f\|_q^2\leq \frac{q-2}{(n+1)!} \int_{\mathbb{S}^{2n+1}} f \mathcal{A}'_Q f d\xi +\int_{\mathbb{S}^{2n+1}} |f|^2 d\xi, \end{equation*} where $\mathcal{A}_d(0<d<Q)$ are the intertwining operator, $\mathcal{A}'_Q$ is the conditional intertwinor introduced in \cite{BFM2013}, and $d\xi$ is the normalized surface measure of $\mathbb{S}^{2n+1}$.