# zbMATH — the first resource for mathematics

Radial geometric analysis on groups. (English) Zbl 1059.22006
Kotani, Motoko (ed.) et al., Discrete geometric analysis. Proceedings of the 1st JAMS symposium, Sendai, Japan, December 12–20, 2002. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-3351-0/pbk). Contemporary Mathematics 347, 221-244 (2004).
Radial geometric analysis on a group $$G$$ is the study of convolution operators on $$G$$ which are defined by a distance function, such as the ball, shell and sphere averages associated with an invariant admissible metric on $$G$$ . In the paper under review several general principles that may prove to hold for all radial averages on all locally compact second countable groups are formulated.
The evidence for their validity together with many open problems are presented. The three main themes are comparisons between the behavior of radial (and some other geometric) averages on groups along the following dichotomies: 1) Polynomial volume growth versus exponential volume growth. 2) Amenable groups versus non-amenable groups. 3) Isometry groups of Euclidean versus non-Euclidean symmetric spaces of higher rank.
The paper is focused on geometric analysis on connected Lie groups. The main facts discussed in the paper are:
1) The type of volume growth, namely polynomial or exponential growth, is detected in the behavior of the maximal function associated with shell averages.
2) There exists an exponential-maximal inequality in terms of ball averages, much stronger than the Hardy-Littlewood-Wiener maximal inequality which is connected with the existence of a spectral gap in the regular representation of a non-amenable group.
3) There exist natural product-type averages of very small support on non-Euclidean symmetric spaces of higher rank which behave very differently than their Euclidean analogs.
For the entire collection see [Zbl 1052.58002].
##### MSC:
 22D40 Ergodic theory on groups 22E30 Analysis on real and complex Lie groups 28D10 One-parameter continuous families of measure-preserving transformations 43A10 Measure algebras on groups, semigroups, etc. 43A90 Harmonic analysis and spherical functions