Local ergodic theorems for \(K\)-spherical averages on the Heisenberg group (Q1574457)

Let \(H^n=\mathbb{C}^n \times\mathbb{R}\) denote the Heisenberg group. Let \((H^n,K)\) be a Gelfand pair, where \(K\) is a subgroup of the unitary group \(U(n)\). Given a point \(w\in\mathbb{C}^n\) there is a measure \(\mu_{K\cdot w}\) which is supported on the \(K\)-orbit \(K\cdot w\) through \(w\) such that \[ f*\mu_{K\cdot w}(z,t) =\int_K f\bigl((z,t) (k\cdot w,0)^{-1} \bigr)dk, \] where \(dk\) is the normalized Haar measure on \(K\). The aim of this paper is to prove local ergodic theorems for the sphere and ball averages of the measure \(\mu_{K\cdot x}\) supported on orbits through real points \(x\in\mathbb{R}^n\). We define the ball averages \(\nu_r\) and the averages \(\sigma_r\) of the measure \(\mu_{K\cdot x}\) by \(\nu_r= \int_{|x|\leq r}\mu_{K\cdot x}dx\) and \(\sigma_r= \int_{|x |=r} \mu_{K\cdot x}d \mu_r^{n-1}\), where \(\mu_r^{n-1}\) is the normalized surface measure on the sphere of radius \(r\) in \(\mathbb{R}^n\). Theorem 1: The ball averages \(\nu_r\) are a local ergodic family in \(L^p\) for all \(1<p< \infty\). Theorem 2: Let \(n\geq 3\) and \(p>{n \over n-1}\). Then the sphere averages \(\sigma_r\) are a local ergodic family in \(L^p\). The author also considers averages over annuli in the case of the reduced Heisenberg group and he shows that if the function has zero mean value then the maximal function associated to the annulus averages has a better behaviour than the spherical maximal function.