Analogs of Wiener's ergodic theorems for semisimple Lie groups. II
[Part I by \textit{A. Nevo} and \textit{E. M. Stein} in Ann. Math. (2) 145, 565-595 (1997; Zbl 0884.43004).] Let \((X,{\mathcal B},m)\) be a standard Borel space with a Borel measurable ergodic \(G\)-action preserving the probability measure \(m\). If \(\nu_t\), \(t\in \mathbb{R}_+\), is a one-parameter family of probability measures on \(G\) such that the map \(t\to\nu_t\) is continuous when the space of probability measures is equipped with the weak-\(*\)-topology, then \(\nu_t\) is a pointwise ergodic family in \(L^p(X)\) if, for every \(f\in L^p(X)\), \[ \lim_{t\to\infty}\pi(\nu_t)f(x)=\int_Xf dm, \] where \(\pi(\nu_t)f(x):=\int_G f(g^{-1}x) d\nu_t(g)\) and the convergence is pointwise almost everywhere and in the \(L^p\) norm. The authors prove that if \(G\) is a connected semisimple group with finite center and no compact factors and if \(\nu_t\) is the family of ball averages on \(G\) defined by \[ \nu_t={1\over{m_G(B_t)}}\int_{B_t}\delta_g dm_G(g), \] where \(m_G\) is the Haar measure on \(G\), \(B_t\) is the ball of radius \(t\) with respect to the \(G\)-invariant Riemannian metric on \(G/K\), and \(\delta_g\) is the Dirac measure at \(g\), then \(\nu_t\) is a pointwise ergodic family in \(L^p(X)\) for every ergodic \(G\)-space \((X,{\mathcal B},m)\) as above. Moreover, again for every ergodic \((X,{\mathcal B},m)\), the family of ball averages satisfies the strong maximal inequality in \(L^p(X)\), \(1<p\leq\infty\), namely, \[ \left\|\sup_{t\geq 0}|\pi(\nu_t)f|\right\|_p\leq C_p\|f\|_p. \] The authors also show that for semisimple Lie groups satisfying property T, the convergence of the ball averages to the ergodic mean is at an exponential rate, namely there exists a positive constant \(b(G)\) such that for every \(\theta<b(G)\) and for all \(r\) and \(u\) such that \(0<u<1\) and \(2p=2r+pr\), then for almost every \(x\in X\) \[ \left|\pi(\nu_t)f(x)-\int_Gf dm\right|\leq B(x,f)\exp\left(-{{u\theta t}\over{4}}\right), \] where \(\|B(\cdot,f)\|_r\leq B\|f\|_p\) for all \(f\in L^p(X)\). The authors also generalize the above results to general radial averages and to actions with a spectral gap (where the group acting does not necessarily have property T).
scientific article; zbMATH DE number 1484733
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