Orbits and co-orbits of ultrasymmetric spaces in weak interpolation

Ultrasymmetric spaces form a large class of rearrangement-invariant spaces which are not only intermediate but also interpolation between Lorentz and Marcinkiewicz spaces with the same fundamental function. They include Lebesgue, Lorentz, Lorentz–Zygmund and many other classical spaces. At the same time they have rather simple analytical description, making them suitable for stating various interpolation properties, especially in “extreme” cases of weak interpolation. In the present paper we consider ultrasymmetric spaces which are so “close” to the endpoint spaces that the ratio of their fundamental functions is a slowly varying function b(t)∼b(t2), and find for them explicitly the upper and the lower optimal interpolation spaces near the “right” and near the “left” endpoints. In result we obtain four new types of rearrangement-invariant spaces (not ultrasymmetric) and study some other properties of them.