Interpolation of operators when the extreme spaces are $L^{∞}$

Abstract

ABSTRACT. In this paper, equivalence between interpolation properties of linear operators and monotonicity conditions are studied, for a pair (X 0 , X 1 ) of rearrangement invariant quasi Banach This interpolation property has been extensively studied in its connection with many aspects concerning r.i. spaces, for instance, Boyd or Zippin's indexes, monotonicity conditions, boundedness of some suitable "maximal" operators and so on. Here we are concerned with the case B 0 = B 1 = L ∞ and particularly in connection with the monotonicity property (M) given in $ 1 and the boundedness of only one operator.In this direction the former result is contained in Calderon's paper