Optimal Couples of Rearrangement Invariant Spaces for Generalized Maximal Operators

The optimal couples of rearrangement invariant spaces for boundedness of a generalized maximal operator, associated with a quasiconcave function, have been characterized in terms of certain indices connected with rearrangement invariant spaces and quasiconcave functions.

New classes of rearrangement-invariant spaces appearing in extreme cases of weak interpolation

We study weak type interpolation for ultrasymmetric spacesL?,Ei.e., having the norm??(t)f*(t)?E˜, where?(t)is any quasiconcave function andE˜is arbitrary rearrangement-invariant space with respect to the measuredt/t. When spacesL?,Eare not “too close” to the endpoint spaces of interpolation (in the sense of Boyd), the optimal interpolation theorem was stated in [13]. The case of “too close” spaces was studied in [15] with results which are optimal, but only among ultrasymmetric spaces. In this paper we find better interpolation results, involving new types of rearrangement-invariant spaces,A?,b,EandB?,b,E, which are described and investigated in detail.