# Abelian Group Theory by R. Goebel, E. Walker

By R. Goebel, E. Walker

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4. 1 are made speciﬁcally in order to estimate certain spectral expressions that arise in the proof of the pointwise ergodic theorem. At issue is the estimate of π(∂βt ) , where ∂βt is a singular probability measure supported on the boundary of G t . We establish the required estimate for left-radial admissible averages in irreducible actions or for standard well-balanced averages in reducible actions. 1. 4. Finite-dimensional representations. 1. 8. 2. 1 for simple algebraic groups (in L 2 , say) is that convergence to the ergodic mean (namely, zero) holds for an arbitrary ergodic action when we consider the functions in the orthogonal complement of the space spanned by all ﬁnite-dimensional subrepresentations.

To establish the bound we use the spectral transfer principle to reduce the estimate to the regular representation. Finally, since we assume that the averages βt and hence ∂βt are left-radial, a favorable estimate can be deduced from the decay properties of the Harish-Chandra -function, provided the action is irreducible. When the action is reducible, however, the question of whether ∂βt is balanced among the simple factors of G inevitably comes up. For general families βt this problem seems to be out of reach by current techniques, so in the reducible case we assume further 38 CHAPTER 4 that the averages are standard and boundary-regular.

The exponential mean ergodic theorem in (L p , L r ) for p ≥ r ≥ 1, provided ( p, r ) = (1, 1) and ( p, r ) = (∞, ∞). 2. the exponential strong maximal inequality in (L p , L r ) for p > r ≥ 1. 3. the exponential pointwise ergodic theorem in (L p , L r ): for every f ∈ L p (X ), 1 < p < ∞, and almost every x ∈ X , f dμ ≤ B p,r ( f, x)e−ζ t , π X (λt ) f (x) − X where ζ = ζ p,r > 0 and r < p. 7. 1. 5, we note that the action of G + induced from the -action is indeed often (but perhaps not always) irreducible.