Interpolation Theorems for the Pairs of Spaces (L p , L ∞ ) and (L 1 L q )

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

“…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 [5] where it is showed that both properties, say linear interpolation and monotonicity, are equivalent in the case of A 0 = A 1 = L 1 . Later on, Lorentz and Shimogaki [10] extended this result to the case A 0 = A 1 = L p with p > 1. The technique used by them consists on a linearization process of the L p case.Sharpley, Maligranda and other autors (see [11] and references quoted there) studied the case A 0 = Λ(X), A 1 = M (X) (see definitions in $ 2) and B 0 = B 1 = L ∞ or B 0 = Λ(Y ), B 1 = M (Y ) relating the interpolation properties with the boundedness of only one "maximal" operator ([18, theorem 4.7], [11, theorem 4.5]).…”

On the K-divisibility constant for some special finite-dimensional Banach couples

“…Then we can apply the theorem and proof of Lorentz and Shimogaki [19] to construct the required operator T . 2…”

“…In addition to its other good properties, (H 0 , H 1 ) is known, as shown in [2], to be an exact Calderón couple. (L 2 , L ∞ ) is also a Calderón couple [19] and the optimal decomposition for obtaining its K -functional exactly is quite simple to describe. But it turns out, perhaps rather surprisingly, that neither of these couples are exactly K -divisible in general, and one can even find two-dimensional versions of each of these couples for which exact K -divisibility does not hold.…”

On the interpolation constant for Orlicz spaces

“…also [15] for the direct proof). Moreover, G. G. Lorentz and T. Shimogaki [13,Theorem 7] observed that for the function…”