## Analogs of Wiener’s ergodic theorems for semisimple groups. I.(English)Zbl 0884.43004

The authors prove maximal and pointwise ergodic theorems for ball averages on simple real-rank one Lie groups which are analogs of N. Wiener’s maximal and pointwise ergodic theorems for ball averages on $$\mathbb{R}^n$$.
More precisely, let $$G$$ be a simple Lie group of real-rank one and let $$K$$ be a maximal compact with normalized Haar measure $$m_K$$. Let $$n(G)=dim_\mathbb{R}(G/K)$$. If $$X$$ is a standard Borel space with a Borel measurable $$G$$-action and a $$G$$-invariant probability measure $$\lambda$$, let $$\pi:G\to Iso(L^p(X))$$ be the induced isometric representation. If $$\nu_t$$ is a family of probability measures on $$G$$, define the corresponding Markov operators on $$L^p(X)$$, $$\pi(\nu_t)f(x)=\int_Gf(g^{-1}x)d\nu_t(g)$$ and the associated maximal function $$M_\nu f(x)=\sup_{t>0}|\pi(\nu_t)f(x)|$$.
Definition. Let $$t\to\nu_t$$ be an assignment of probability measures on $$G$$ which is continuous with respect to the weak-$$\ast$$-topology on $$C(G)^\ast$$.
(i) $$\nu_t$$ is a pointwise ergodic family in $$L^p$$ if for any Borel probability space $$X$$ and any $$f\in L^p(X)$$ one has $$\lim_{t\to\infty}\pi(\nu_t)f(x)=E_1f(x)$$, for a.e. $$x$$ and in the $$L^p$$ norm, where $$E_1$$ is the conditional expectation of $$f$$ with respect to the $$\sigma$$-algebra of $$G$$-invariant sets.
(ii) $$\nu_t$$ satisfies the local ergodic theorem in $$L^p$$ if for any Borel probability space $$X$$ and any $$f\in L^p(X)$$, one has $$\lim_{t\to0}\pi(\nu_t)f(x)=\pi(\nu_0)f(x)$$ pointwise for a.e. $$x$$ and in the $$L^p$$ norm.
The one-parameter families of probability measures considered in the paper are the following:
(1) The convolution $$\sigma_t=m_K*\delta_{a_t}*m_K$$, where $$A=\{a_t:t\in\mathbb{R}\}$$ is a one-parameter subgroup of hyperbolic elements so that $$G=KA_+K$$ is a Cartan decomposition.
(2) The absolutely continuous measure $$\beta_t$$ on $$G$$ with density $$vol(B_t)^{-1}\chi_{B_t}$$, where $$B_t=\{g\in G:d(gK,K)\leq t\}$$ with $$d$$ the $$G$$-invariant Riemannian metric on $$G$$. (Here $$\beta_0=m_K$$.)
Theorem. Let $$G$$ be as above. Then $$\beta_t$$ satisfies the strong maximal inequality $$|M_\beta f|_p\leq C_p(G)|f|_p$$, $$p>1$$.
Corollary. If $$G$$ is as above, $$\beta_t$$ is a pointwise ergodic family in $$L^p$$, $$1<p<\infty$$, and satisfies the local ergodic theorem.
The case $$p=1$$ is still an open problem.
The authors also obtain results for sphere averages which are a generalization of the results by A. Nevo [Duke Math. J. 76, 113-140 (1994; Zbl 0838.43013); 86, 239-259 (1997; Zbl 0869.43005)] for $$G=SO(1,n)$$, $$n>2$$.
Theorem. Let $$G$$ be as above with $$n(G)>2$$. If $$f\in L^p(X)$$ is non-negative and $$p>n(G)(n(G)-1)^{-1}$$, then $$M_\sigma f$$ is well-defined, Lebesgue measurable and satisfies $$|M_\sigma f|_p\leq C_p(G)|f|_p$$.
A proof that the range of $$p$$ is the best possible can be found in the papers of A. Nevo cited above. As before,
Corollary. Let $$G$$ be as above with $$n(G)>2$$. Then $$\sigma_t$$ is a pointwise ergodic family in $$L^p$$ and, for $$n(G)(n(G)-1)^{-1}<p<\infty$$, satisfies the local ergodic theorem.
Notice that the condition $$n(G)>2$$ excludes the group $$G=SO^o(1,2)$$ and its covering groups. It is not known whether for $$G=SL(2,\mathbb{R})$$, $$\sigma_t$$ is a pointwise ergodic family in $$L^p$$, $$2<p<\infty$$.
Some related results were previously obtained in some particular cases; a strong maximal inequality in $$L^p$$, $$p>1$$, was proven for $$X=G/K$$ by J. L. Clerc and E. M. Stein [Proc. Natl. Acad. Sci. USA 71, 3911-3912 (1974; Zbl 0296.43004)] for general semisimple Lie groups with finite center and no compact factors; results relative to sphere averages can be found in [E. M. Stein, Proc. Natl. Acad. Sci. USA 73, 2174-2175 (1976; Zbl 0332.42018)], [E. M. Stein and S. Wainger, Bull. Am. Math. Soc. 84, 1239-1295 (1978; Zbl 0393.42010)] and [J. Bourgain, J. Anal. Math. 47, 69-85 (1986; Zbl 0626.42012)] on $$L^p(\mathbb{R}^n)$$ and in [A. El. Kohen, J. Operator Theory 3, 41-56 (1980)], [A. Greenleaf, Indiana Univ. Math. J. 30, 519-537 (1981; Zbl 0517.42029)] and [C. D. Sogge and E. M. Stein, J. Anal. Math. 54, 165-188 (1990; Zbl 0695.42012)] on $$L^p(G/K)$$; sphere averaging results on $$\Gamma\backslash G$$ where $$\Gamma$$ is a lattice in $$G$$ were proven in [G. A. Margulis, Ph. D. Thesis, Moscow State University, 1970] and [D. Y. Kleinbock and G. A. Margulis, preprint, Yale University, November 1994] and by A. Katok for $$X$$ the unit tangent bundle of a compact Riemann surface.
The same authors, jointly with G. Margulis, are currently treating the general case of semisimple groups.

### MSC:

 43A80 Analysis on other specific Lie groups 28D15 General groups of measure-preserving transformations 22D40 Ergodic theory on groups
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