Interpolation functions and the Lions-Peetre interpolation construction

The generalization of the Lions-Peetre interpolation method of means considered in the present survey is less general than the generalizations known since the 1970s. However, our level of generalization is sufficient to encompass spaces that are most natural from the point of view of applications, like the Lorentz spaces, Orlicz spaces, and their analogues. The spaces phi(X-0, X-1)(p0, p1) considered here have three parameters: two positive numerical parameters p(0) and p(1) of equal standing, and a function parameter phi. For p(0) not equal p(1) these spaces can be regarded as analogues of Orlicz spaces under the real interpolation method. Embedding criteria are established for the family of spaces phi(X-0, X-1)(p0, p1), together with optimal interpolation theorems that refine all the known interpolation theorems for operators acting on couples of weighted spaces L-p and that extend these theorems beyond scales of spaces. The main specific feature is that the function parameter phi can be an arbitrary natural functional parameter in the interpolation.