By R.L. Lipsman

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This unavoidable assumption is reﬂected in the following deﬁnitions. 17. Balanced and well-balanced averages. Let G = G(1) · · · G(N ) be an almost direct product of N noncompact compactly generated subgroups. For a set I of indices I ⊂ [1, N ], let J denote its complement and G(I ) = i∈I G(i). Let G t be an increasing family of sets contained in G. 1. G t will be called balanced if for every I satisfying 0 < |I | < N and every compact set Q contained in G(I ), m G (G t ∩ G(J ) · Q) lim = 0. t→∞ m G (G t ) 2.

3. The estimate π(βt ) L 20 (X ) ≤ C exp(−θt) implies an exponential strong maximal inequality for any sequence of operators exp( 12 θ tk )βtk in L 20 , where tk is a sequence such that the sum of the norms converges. Repeat the argup ment in L 0 (X ). 4. Now distribute exp( 14 θn) equally spaced points in the interval [n, n + 1]. Then approximate π(βt ) f by π(βtn ) f using the closest point tn to t in the sequence tk . Estimate the difference using the exponential strong maximal inequality for the entire sequence βtk , and the local H¨older regularity of the family βt , applied when f is a bounded function to the points t and tk .

An increasing sequence of bounded Borel subsets G t , t ∈ N+ , on an lcsc totally disconnected group G will be called admissible if it is coarsely admissible and there exist t0 > 0 and a compact open subgroup K 0 such that for t ≥ t0 , K0Gt K0 = Gt . 7) Let us note the following regarding admissibility. 11. 1. 8). However, this argument fails for S-algebraic groups which have a totally disconnected simple component, and so we have required coarse admissibility explicitly in the deﬁnition. 2. 6) is of course equivalent to the function log m G (G t ) being uniformly locally Lipschitz-continuous for sufﬁciently large t.