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Analogs of Wiener’s ergodic theorems for semisimple Lie groups. II. (English) Zbl 0978.22006
[Part I by A. Nevo and 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).

MSC:
 22D40 Ergodic theory on groups 22E30 Analysis on real and complex Lie groups 28D10 One-parameter continuous families of measure-preserving transformations 43A10 Measure algebras on groups, semigroups, etc. 43A90 Harmonic analysis and spherical functions
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