Interpolation with a parameter function. (English) Zbl 0619.46064

Persson, Lars Erik

Math. Scand. 59, 199-222 (1986).

The (Lions-Peetre) real interpolation spaces \(\bar A{}_{\theta,q}\) are defined by using the function norm \(\Phi (\phi)=(\int^{\infty}_{0}(\phi (t)/t^{\theta})^ qdt/t)^{1/q}\). By replacing \(t^{\theta}\) by a more general (parameter) function \(\rho =\rho (t)\) we obtain the spaces \(\bar A{}_{\rho,q}\). In this paper we shall point out the fact that most of the classical (and some new) theorems for the spaces \(\bar A{}_{\theta,q}\) can be formulated also for the more general spaces \(\bar A{}_{\rho,q}\). Sometimes we only need to adjust some recent results to the present situation but sometimes we must give separate proofs of our statements. Every result is given in a form which is very adjusted to immediate applications. This paper can be seen as a follow-up and unification of several results of this kind in the literature.

MSC:

46M35	Abstract interpolation of topological vector spaces
46E30	Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)

Keywords:

parameter function; K-functional; quasi-Banach spaces; real interpolation