By Reiter H., Stegeman J.D.

The necessity for a brand new variation of Hans Reiter's monograph 'Classical harmonic research and in the neighborhood compact teams' has been felt for a very long time. His unique e-book has had a substantial impression on numerous components of harmonic research. Its kind and association distinguish it from different books on related issues. Remarkably, after greater than thirty years, the e-book has misplaced little or no of its mathematical relevance. Radical Banach algebras and amenable groups—to point out purely these—are flourishing topics of study these days. For either matters, and for a number of others, the publication comprises helpful details.

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The main motivation to consider the exponential strong maximal inequality in (L p (μ), L r (μ)) is that it implies the exponential pointwise ergodic theorem in (L p (μ), L r (μ)), as well as exponentially fast norm convergence to the ergodic mean. We recall, in comparison, that the ordinary strong maximal inequality implies pointwise convergence almost everywhere only after we have also established the existence of a dense subspace where almost sure pointwise convergence holds. In addition, convergence in norm requires a separate further argument.

In the second, we can take an ergodic action with a spectral gap for G but such that one of the factors admits an asymptotically invariant sequence of unit vectors of zero integral. 2. 2 for a one-parameter H¨older-admissible family acting in L 20 (X ) is the exponential decay of the operator norms: π X0 (βt ) ≤ Ce−θt . 11 below that this estimate holds for totally weakmixing actions under the strong spectral gap assumption or when the action has a spectral gap and the averages are well balanced.

Weak-mixing if the unitary representation in L 20 (X ) does not contain nontrivial ﬁnite-dimensional subrepresentations. 2. totally weak-mixing if, when G = G(1)G(2) · · · G(N ) is an almost direct product of N normal subgroups, the only ﬁnite-dimensional subrepresentation that G(i) admits is the trivial one (possibly with multiplicity greater than 1), or equivalently, if in the space orthogonal to the G(i)-invariants no ﬁnite-dimensional representations of G(i) occur, 1 ≤ i ≤ N . We will also apply below the obvious generalizations of these notions to general unitary representations.