A CR Poincaré inequality on the complex sphere (Q865326)

Let \(f\) be a smooth function on the unit sphere \( \mathbb{S}\) in \( \mathbb{C}^n\) and denote by \(\nabla_{\mathbb{C}}f\) its complex tangential gradient: this is the unique complex tangential vector field with the property that \(Xf(z) = \langle Xf(z),(\nabla_{\mathbb{C}}f)(z)\rangle \) for all complex tangential vector fields \(X\), where \( \langle\cdot, \cdot\rangle\) denotes the usual Hermitian Riemannian metric on complex-valued tangent vectors. It is proved that for \(1 \leq p <2n\) and \(q=2np/(2n-p),\) there exists a positive constant, depending only on \(n\) and \(p,\) such that \[ \| f-(f)_{\mathbb{S}}\| _{L^q(\mathbb{S})} \leq C \| \nabla_{\mathbb{C}}f\| _{L^p(\mathbb{S})} \] where, with \(d \sigma \) the normalized volume element on \(\mathbb{S}, (f)_{\mathbb{S}} \) is the integral mean \( \int_{\mathbb{S}} f \,d\sigma.\)