Abstract: Associated with the construction of intermediate Banach spaces in interpolation theory are unbounded non-linear operators, Ω, which are, roughly, the differentials of the fundamental mappings used in the construction of the intermediate spaces. We look at the theory of these operators and their applications to interpolation theory and to other topics in analysis. We see that the mapping properties of Ω are strongly related to the structure of the interpolation scale. We also see that computations using Ω are closely related to certain computations in the theory of hypercontractive semigroups.

Introduction and Summary: Several constructions in interpolation theory start with a pair of Banach spaces (for example, L1 and L∞) and construct a parameterized family of spaces (in the example, Lp, 1 < p < ∞) which are, in an appropriate technical sense, intermediate to the original pair. In particular it will be true that if a linear operator T is bounded on both spaces of the starting pair, then T will also be bounded on the intermediate spaces.

Associated to the construction of intermediate spaces are mappings Ω, generally unbounded and non-linear, which can be obtained by differentiation with respect to certain parameters used in the construction of the intermediate spaces.