##
**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.

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.

Reviewer: A.Iozzi (College Park)

### MSC:

43A80 | Analysis on other specific Lie groups |

28D15 | General groups of measure-preserving transformations |

22D40 | Ergodic theory on groups |