Invariant Subspaces in Certain Function Spaces on Euclidean Space.
In this paper we extend the result established for the euclidean space in [3] to the hyperbolic disk. This includes the reconstruction of a function defined in a fixed disk B(0,R) from its averages on disks of radii r1, r2 lying in B(0, R).
Let L¹(G)** be the second dual of the group algebra L¹(G) of a locally compact group G. We study the question of involutions on L¹(G)**. A new class of subamenable groups is introduced which is universal for all groups. There is no involution on L¹(G)** for a subamenable group G.
We study, in the context of doubling metric measure spaces, a class of BMO type functions defined by John and Nirenberg. In particular, we present a new version of the Calderón-Zygmund decomposition in metric spaces and use it to prove the corresponding John-Nirenberg inequality.