Reiteration theorems for the K -interpolation method in limiting cases

Abstract

Sharp reiteration theorems for the K-interpolation method in limiting cases are proved using two-sided estimates of the K-functional. As an application, sharp mapping properties of the Riesz potential are derived in a limiting case.

Limiting reiteration for real interpolation with logarithmic functions

“…. , α n ) let's denote λ A (t) := λ α 1 1 · · · · · λ α n n . All functions λ k and λ A are slowly varying functions in the sense of [21, Definition 2.1].…”

“…For this purpose we modified the technique of [17]. In the case θ = 1 we use apposite reiteration formulae of [1]. In the case θ = 0 it is not necessary because the proof of Theorem 1 is valid for θ = 0, too.…”

“…The definition of the real interpolation method X θ,q,b = (X 0 , X 1 ) θ,q,b involving slowly varying function b (see [ 1 of the main parameter. Particularly important is the case when b is a (broken-) logarithmic function.…”

New limiting real interpolation methods and their connection with the methods associated to the unit square

“…However, to get a meaningful definition they need to insert powers of the function g(t) = 1 + | log t| (see, for example, [18] or [19]) or, more generally, certain slowly varying function (see [1], [22]). We do not use here any weight of these kinds but the suitable term with the supremum which corresponds to the norm of the Gagliardo completion of A j in A 0 + A 1 (j = 0, 1) (see [3]).…”

“…Let (Ω, μ) be a σ-finite measure space. If μ(Ω) = 1, then we are in the ordered case with L ∞ → L 1 , and it is shown in [23] that (L ∞ , L 1 ) 1,1;K = L log L = L 1 (log L) 1 .…”