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 $\mathbb{ S}^{ 2n+1}$.
Categories: math.AP