Pointwise ergodic theorems for radial averages on simple Lie groups. II
The present paper is a continuation of Part I [ibid. 76, 113-140 (1994; Zbl 0838.43013)], and we begin by briefly recalling the setup and the notation: -- \(G=G_n= SO^0 (n,1)\) is the group of orientation-preserving isometries of \(n\)-dimensional real hyperbolic space \(H^n\), \(n\geq 2\). -- \(K=\) a fixed maximal compact subgroup. \(m_K=\) Haar probability measure. -- \(A= \{a_t|t\in\mathbb{R}\}=\) a one-parameter group of hyperbolic translations such that \(G=KA_+K\) is a Cartan decomposition. -- \(\sigma_t=\) the bi-\(K\)-invariant probability measure on \(G\) given by \(\sigma_t= m_K* \delta_{a_t}* m_K\), where \(*\) denotes convolution. Note that \(\sigma_0=m_K\). -- \(\mu_t= 1/t\int^t_0 \sigma_sds\), the uniform average of \(\sigma_s\), \(0\leq s\leq t\). We define \(\mu_0= m_K\). -- \(M(G,K)=\) the commutative convolution algebra (of bi-\(K\)-invariant complex bounded Borel measures on \(G)\) generated by \(\sigma_t\), \(t\geq 0\). -- \((X,{\mathcal B}, \lambda)=\) a standard Borel space with a Borel measurable \(G\)-action which preserves the probability measure \(\lambda\). -- \(\pi(\nu) f(x)= \int_Gf(g^{-1}x)d \nu(g)=\) the Markov operator on \(L^p(X)\) corresponding to a probability measure \(\nu\) on \(G\). -- \(M_\mu f(x)= \sup_{t \geq 0} |\pi (\mu_t) f(x)|\), and \(M_\sigma f(x) =\sup_{t\geq 0}|\pi(\sigma_t) f(x) |\), maximal functions associated with the action of \(\sigma_t\) and \(\mu_t\) in \(L^p(X)\), \(1\leq p\leq \infty\). Finally, recall also the following definition. Definition. Let \(\nu_t\), \(t\geq 0\), be a one-parameter family of probability measures on \(G\). Assume that \(t \mapsto\nu_t\in M(G)\) is continuous in the \(w^*\)-topology of \(M(G)\) as the dual of \(C_0(G)\). Let \((X, {\mathcal B}, \lambda) \) denote a \(G\)-space as above. (1) \(\nu_t\) is called a pointwise ergodic family in \(L^p\) if, for any \(f\in L^p(X)\), \(\lim_{t\to\infty} \pi(\nu_t) f(x)= E_1(f)(x)\), where the convergence is pointwise almost everywhere and in the \(L^p\)-norm, and \(E_1\) is the conditional expectation of \(f\) with respect to the \(\sigma\)-algebra of \(G\)-invariant sets. (2) \(\nu_t\) is said to satisfy the local ergodic theorem in \(L^p\) if, for any \(f\in L^p(X)\), \(\lim_{t\to 0} \pi(\nu_t) f(x)= \pi (\nu_0) f(x)\), where the convergence is for almost every \(x\), and in the \(L^p\)-norm. We can now state the main results. Theorem 1. Let \(G=SO^0(n,1)\), where \(n\geq 3\) is odd. Then \(\sigma_t\) is a pointwise ergodic family in \(L^p\), for \(n/(n-1)<p<\infty\). -- As usual, the main ingredient in the proof is provided by the following theorem. Theorem 2. Let \(G=SO^0(n,1)\), where \(n\geq 3\) is odd. Then \(M_\sigma f\) is well defined and Lebesgue measurable for any nonnegative \(f\in L^p(X)\), \(p>n/(n-1)\), and the strong maximal inequality \(|M_\sigma f |_p \leq C_p(G) |f |_p\) holds. Furthermore, this result is the best possible without further assumptions on the action. -- Similar results hold for the local ergodic theorem and the local maximal inequality. The local results are formulated and proved in \(\S 5\).
scientific article; zbMATH DE number 997804
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