Real Interpolation with Logarithmic Functors and Reiteration

Abstract

Abstract. We present "reiteration theorems" with limiting values θ = 0 and θ = 1 for a real interpolation method involving broken-logarithmic functors. The resulting spaces lie outside of the original scale of spaces and to describe them new interpolation functors are introduced. For an ordered couple of (quasi-) Banach spaces similar results were presented without proofs by Doktorskii in [D].

Alternative characterisations of Lorentz-Karamata spaces

“…The family of all slowly varying functions is denoted by SV ; it includes not only powers of iterated logarithms and the broken logarithmic functions of [10], but also such functions as t → exp (|log t| a ), a ∈ (0, 1). (The last mentioned function has the interesting property that it tends to infinity more quickly than any positive power of the logarithmic function.)…”

Limiting reiteration for real interpolation with slowly varying functions

“…[8]) and Evans and Opic (cf. [14]) consider limiting cases of the reiteration problem with respect to the interpolation spaces X θ,q;A ≡ (X 0 , X 1 ) θ,q;A := {f ∈ X 0 + X 1 : f θ,q;A < ∞} , ( …”

Optimal embeddings and compact embeddings of Bessel-potential-type spaces

“…The family of all slowly varying functions includes not only powers of iterated logarithms and the broken logarithmic functions of [20], but also such functions as t → exp |log t| a , a ∈ (0, 1). (The last mentioned function has the interesting property that it tends to infinity more quickly than any positive power of the logarithmic function.…”