Sharp Sobolev inequalities on the complex sphere

TL;DR

This paper establishes sharp Sobolev inequalities on the complex sphere, involving intertwining operators and extending classical results to new cases with explicit constants.

Contribution

It introduces new sharp Sobolev inequalities on the complex sphere using intertwining operators and conditional intertwinors, expanding previous theoretical frameworks.

Findings

Derived sharp inequalities for different parameter ranges

Explicit constants involving Gamma functions are provided

Extended Sobolev inequalities to complex spheres with new operators

Abstract

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 $\mathbb{S}^{2n+1}$.