Pointwise ergodic theorems for radial averages on simple Lie groups. I
Let \(G\) be a locally compact group. A family \(\nu_t\) of probability measures on \(G\) is said to be pointwise ergodic in \(L^p\) if, for any ergodic action \(\pi\) of \(G\) on a probability space \((X, {\mathcal B}, \lambda)\), and for all \(f\in L^p(X)\), \[ (\pi(\nu_t) f)(x)\to \int_X fd\lambda\qquad\text{a.e.} \] and in the \(L^p\)-norm. For \(G= \text{SO}_0(n, 1)\), \(K= \text{SO}(n)\), put \[ a_t= \exp\begin{pmatrix} 0 && t\\ & 0\\ t && 0\end{pmatrix}, \] and define \(\sigma_t= m_K* \delta_{a_t}* m_K\), where \(m_K\) is the normalized Haar measure on \(K\), and \[ \mu_t= {1\over t} \int^t_0 \sigma_s ds,\quad \sigma_0= m_K. \] As the main result, the author proves that the family \(\mu_t\) is pointwise ergodic in \(L^p\) for \(1< p< \infty\), and that, for \(n\geq 3\), the family \(\sigma_t\) is pointwise ergodic in \(L^2\). -- The proofs follow a method developed by Stein and Wainger. A maximal inequality, and the following Littlewood-Paley square function \[ R(f, x)^2= \int^\infty_1 t\Biggl|\Biggl({d\over dt} \pi(\sigma_t) f\Biggr)\Biggr|^2 dt \] are used. At this point, the author uses the spectral theory of the Gelfand pair \((G, K)\), more precisely, the theory of the commutative \(B^*\)-algebra \(A_\pi\) which is the closure in the operator norm of the algebra \(\{\pi(\mu)\mid \mu\in M(K\backslash G/K)\}\). The formula \[ \int_X R(f, x)^2 dx= \int^\infty_1 \int_{\text{Sp }A_\pi} t|\varphi_z(\sigma_t)|^2dm_f dt \] is proved, where \(m_f\) if the spectral measure of \(f\), and the \(\varphi_z\) are spherical functions of the Gelfand pair \((G, K)\).
scientific article; zbMATH DE number 727821
scientific article