The structure of reverse Hölder classes on metric measure spaces

This work extends some Euclidean results on the structure of the reverse Hölder classes to metric measure spaces with a doubling measure satisfying an annular decay condition. The equivalence between the variants used to define the Muckenhoupt class A∞A∞, and some standard correspondences between reverse Hölder classes and Muckenhoupt classes are utilized. It is established that the logarithm of the reverse Hölder classes is an open and convex subset of the space BMOBMO, and that the logarithm of the intersection of the reverse Hölder classes is a closed and convex subset of the space BMOBMO. As an intermediary result, we also establish that the intersection of the reverse Hölder classes consists precisely of the pointwise multipliers of the Muckenhoupt class A∞A∞.