Weak Type Interpolation Near “Endpoint” Spaces

Let T be an operator of weak types (a, b) and (p, q), where a<p and b<q. The Marcinkiewicz interpolation theorem and its generalizations due to Boyd, Krein–Semenov and others show that T maps certain rearrangement invariant spaces E which are “not too close” to La or Lp into certain spaces F. In this paper we consider analogous results for such an operator T in the case where, on the contrary, E is close to Lp in the sense that its fundamental function is t1/p. For example, E can be a Lorentz space Lp, r for 1⩽r⩽∞. The corresponding range spaces F are explicitly described and shown to be optimal. When E=Lp, r, then in some cases F is a member of the class of Lorentz–Zygmund spaces Lq, r(log L)s which were introduced and studied by Bennett and Rudnick. But in general F is strictly smaller than the corresponding Lorentz–Zygmund space and belongs to an apparently new class of r.i. spaces. Necessary and sufficient conditiones are given for the members of this new class to coincide with Lorentz–Zygmund spaces. Certain results of this paper can be applied to give an alternative proof and generalization of the optimal form of the limiting case of the Sobolev embedding theorem due to Hansson and Brezis–Wainger.