We prove an endpoint weak-type maximal inequality for the spherical maximal operator applied to radial funcions on symmetric spaces of constant curvature and dimension n≥ 2. More explicitly, in the Lorentz space associated with the... more
We prove an endpoint weak-type maximal inequality for the spherical maximal operator applied to radial funcions on symmetric spaces of constant curvature and dimension n≥ 2. More explicitly, in the Lorentz space associated with the natural isometry-invariant measure, we show that, for every radial function f, ∥ Mf∥ n′,∞≤ Cn∥ f∥ n′,1,n′=n/n-1. The proof uses only geometric arguments and volume estimates, and applies uniformly in every dimension.
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Let G be a connected semi-simple Lie group withfinite center and no compact factors, and (X, B ,m )a G-space with σ-finite G-invariant mea- sure m. For eachprobability measure µ on G consider the operator π(µ ): L2(X) → L2(X), given by... more
Let G be a connected semi-simple Lie group withfinite center and no compact factors, and (X, B ,m )a G-space with σ-finite G-invariant mea- sure m. For eachprobability measure µ on G consider the operator π(µ ): L2(X) → L2(X), given by π(µ)f = � G π(g)fdµ(g). The explicit spectral es- timates ("quantitative property T") of M. Cowling (Co2) and R. Howe (H) (see also (H-T)(Li)(Mo)(Oh)) are used to obtain explicit estimates of � π0(µ)� , where π0 is the representation on the space orthogonal to the space of G-invariant functions, provided π0 has a spectral gap. In particular, for actions of Kazhdan groups, the norm estimate is uniform and does not depend on the action. The norm estimates can be viewed as a spectral transfer principle, analogous to the transfer principle for amenable groups (see (W)(Ca) (C-W)(Hz1)(Co3)(Co4)). The spectral estimates are used to derive exponential-maximal inequalities for natural families of aver- ages on the group, as well as pointwise ergodic theorems in Lp for these averages. The pointwise convergence of the averages to the ergodic mean is exponentially fast with an explicit rate. This phenomenon in the case of bi-K-invariant mea- sures was established in (M-N-S), and here we discuss non-radial averages, which may be absolutely continuous, singular or discrete. Some other topics discussed are averages supported on lattice points, almost orthogonality, and best possible estimates of convolution norms and exponential rate of convergence to the ergodic mean.
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The present paper establishes upper and lower bounds on the speed of approximation in a wide range of natural Diophantine approximation problems. The upper and lower bounds coincide in many cases, giving rise to optimal results in... more
The present paper establishes upper and lower bounds on the speed of approximation in a wide range of natural Diophantine approximation problems. The upper and lower bounds coincide in many cases, giving rise to optimal results in Diophantine approximation which were inaccessible previously. Our approach proceeds by establishing, more generally, upper and lower bounds for the rate of distribution of dense orbits of a lattice subgroup $\Gamma $ in a connected Lie (or algebraic) group $G$, acting on suitable homogeneous spaces $G/H$. The upper bound is derived using a quantitative duality principle for homogeneous spaces, reducing it to a rate of convergence in the mean ergodic theorem for a family of averaging operators supported on $H$ and acting on $G/\Gamma$. In particular, the quality of the upper bound on the rate of distribution we obtain is determined explicitly by the spectrum of $H$ in the automorphic representation on $L^2(\Gamma\setminus G)$. We show that the rate is best possible when the representation in question is tempered, and show that the latter condition holds in a wide range of examples.
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As is well-known, the real rank of a simple Lie group that acts conformally on a pseudo-Riemannian manifold is bounded by means of the signature of the manifold. We give a precise description of the action whenever the real rank of the... more
As is well-known, the real rank of a simple Lie group that acts conformally on a pseudo-Riemannian manifold is bounded by means of the signature of the manifold. We give a precise description of the action whenever the real rank of the group reaches that bound, assuming the action is minimal.
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We prove mean and pointwise ergodic theorems for general families of averages on a semisimple algebraic (or S-algebraic) group G, together with an explicit rate of convergence when the action has a spectral gap. Given any lattice in G, we... more
We prove mean and pointwise ergodic theorems for general families of averages on a semisimple algebraic (or S-algebraic) group G, together with an explicit rate of convergence when the action has a spectral gap. Given any lattice in G, we use the ergodic theorems for G to solve the lattice point counting problem for general domains in G, and prove mean and pointwise ergodic theorems for arbitrary measure-preserving actions of the lattice, together with explicit rates of convergence when a spectral gap is present. We also prove an equidistribution theorem in arbitrary isometric actions of the lattice. For the proof we develop a general method to derive ergodic theorems for actions of a locally compact group G, and of a lattice subgroup Gamma, provided certain natural spectral, geometric and regularity conditions are satisfied by the group G, the lattice Gamma, and the domains where the averages are supported. In particular, we establish the general principle that under these conditio...
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. We analyze the structure of a continuous (or Borel) action of a connected semi-simple Lie group G with finite center and real rank at least 2 on a compact metric (or Borel) space X, using the existence of a stationary measure as the... more
. We analyze the structure of a continuous (or Borel) action of a connected semi-simple Lie group G with finite center and real rank at least 2 on a compact metric (or Borel) space X, using the existence of a stationary measure as the basic tool. The main result has the following corollary: Let P be a minimal parabolic subgroup of G, and K a maximal compact subgroup. Let λ be a P-invariant probability measure on X, and assume the P-action on (X,λ) is mixing. Then either λ is invariant under G, or there exists a proper parabolic subgroup Q⊂G, and a measurable G-equivariant factor map ϕ:(X,ν)→(G/Q,m), where ν=∫ K kλdk and m is the K-invariant measure on G/Q. Furthermore, The extension has relatively G-invariant measure, namely (X,ν) is induced from a (mixing) probability measure preserving action of Q.
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... We want to thank Kevin Corlette, Eli Glasner, Shahar Mozes, Jonathan Poritz and Garrett Stuck for useful discussions and suggestions during the preparation of this paper. Our thanks go also to the referee for a careful reading and... more
... We want to thank Kevin Corlette, Eli Glasner, Shahar Mozes, Jonathan Poritz and Garrett Stuck for useful discussions and suggestions during the preparation of this paper. Our thanks go also to the referee for a careful reading and many constructive comments. 2. Preliminaries ...
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Page 1. Annals of Mathematics, 145 (1997), 565-595 Analogs of Wiener's ergodic theorems for semisimple groups I By AMOS NEVO and ELIAS M. STEIN* ... Page 2. 566 AMOS NEVO AND ELIAS M. STEIN Theorem 2. 11 supt>o Jir(ut)fjjjp <... more
Page 1. Annals of Mathematics, 145 (1997), 565-595 Analogs of Wiener's ergodic theorems for semisimple groups I By AMOS NEVO and ELIAS M. STEIN* ... Page 2. 566 AMOS NEVO AND ELIAS M. STEIN Theorem 2. 11 supt>o Jir(ut)fjjjp < Cp(G) jfjpP > n(G) provided n(G) > 2. ...
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LetH=Hn=Cn×R denote the Heisenberg group, and letσrdenote the normalized Lebesgue measure on the sphere {(z, 0):|z|=r}. Let (X, B, m) be a standard Borel probability space on whichHacts measurably and ergodically by measure preserving... more
LetH=Hn=Cn×R denote the Heisenberg group, and letσrdenote the normalized Lebesgue measure on the sphere {(z, 0):|z|=r}. Let (X, B, m) be a standard Borel probability space on whichHacts measurably and ergodically by measure preserving transformations, and letπ(σr) denote the operator canonically associated withσronLp(X). We prove maximal and pointwise ergodic theorems inLp, for radial averagesσron the Heisenberg groupHn,n&amp;gt;1. The results are best possible for actions of the reduced Heisenberg group. The method of proof is to use the spectral theory of the Banach algebra of radial measures on the group and decay estimates for its characters to establish maximal inequalities using spectral methods, in particular Littlewood–Paley–Stein square-functions and analytic interpolation.