Limit cases of reiteration theorems

Abstract

We consider real interpolation methods defined by means of slowly varying functions and rearrangement invariant spaces, for which we present a collection of reiteration theorems for interpolation and extrapolation spaces. As an application we obtain interpolation formulas for Lorentz-Karamata type spaces, for Zygmund spaces LlogL, L exp and for the grand and small Lebesgue spaces.

Characterization of interpolation between Grand, small or classical Lebesgue spaces

“…We have chosen a direct method for the proof of our results by computing the Kfunctionals. In some part of the manuscript (for instance Theorem 5.1) we can adopt an alternative proof as using limiting reiteration theorems [1,10,11,13]. Although, we observe that it is not possible to get the result without computations.…”

Limiting reiteration for real interpolation with logarithmic functions

“…were established in [6,9,[11][12][13]15,17,21] for the parameters 0 ≤ θ 0 < θ 1 ≤ 1, and 0 ≤ θ ≤ 1; similar theorems were proven in [18,20] for the spaces (X θ 0 ,b 0 ,E 0 , X θ 1 ,b 1 ,E 1 ) θ,b,E . In [1] sharp reiteration theorems were proven when 0 ≤ θ 0 < θ 1 ≤ 1 and θ ∈ {0, 1}.…”

“…The investigation of interpolation spaces for limiting values and corresponding reiteration theorems was motivated by certain questions in function spaces (see, e.g., [14] and references therein). Recently in [18][19][20] the scale X θ,b,E defined by means of a slowly varying function b and a rearrangement-invariant Banach function space E was introduced and investigated. So, on the one hand, the results of [18][19][20] are wider than in papers dealing with the scale X θ,q,b because there only the case E = L q is considered.…”

“…Recently in [18][19][20] the scale X θ,b,E defined by means of a slowly varying function b and a rearrangement-invariant Banach function space E was introduced and investigated. So, on the one hand, the results of [18][19][20] are wider than in papers dealing with the scale X θ,q,b because there only the case E = L q is considered. On the other hand, the scale X θ,b,E does not include the spaces X θ,q,b for q < 1, because the corresponding L q -spaces are not Banach spaces.…”

New Young Inequalities and Applications

“…The papers by Evans and Opic [18] and also Evans, Opic and Pick [19], where the authors study logarithmic interpolation methods, inspired the appearance of the some other limiting methods defined by means of slowly varying functions and rearrangement invariant (r.i.) spaces. These have been studied by T. Signes and one the the present authors in [20,21,22,23,24, 25] and allow to produce limit spaces that are not in the classical real interpolation scale.On the other hand, several papers that study bilinear interpolation theorems have been recently published. See for example Mastylo [39] or Cobos and Segurado [13] where we can find bilinear interpolation theorems for logarithmic methods.…”