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On equivalence of K- and J-methods for (n+1)-tuples of Banach spaces

100%
Asekritova I., Krugljak N.
Studia Mathematica
|
1997
|
tom 122
|
nr 2
99-116
EN
It is shown that the main results of the theory of real interpolation, i.e. the equivalence and reiteration theorems, can be extended from couples to a class of (n+1)-tuples of Banach spaces, which includes (n+1)-tuples of Banach function lattices, Sobolev and Besov spaces. As an application of our results, it is shown that Lions' problem on interpolation of subspaces and Semenov's problem on interpolation of subcouples have positive solutions when all spaces are Banach function lattices or their retracts. In general, these problems have negative solutions.
2
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The Lizorkin-Freitag formula for several weighted $L_{p}$ spaces and vector-valued interpolation

81%
Asekritova I., Krugljak N., Nikolova L.
Studia Mathematica
|
2005
|
tom 170
|
nr 3
227-239
EN
A complete description of the real interpolation space $L = (L_{p₀}(ω₀),...,L_{pₙ}(ωₙ))_{θ⃗,q}$ is given. An interesting feature of the result is that the whole measure space (Ω,μ) can be divided into disjoint pieces $Ω_{i}$ (i ∈ I) such that L is an $l_{q}$ sum of the restrictions of L to $Ω_{i}$, and L on each $Ω_{i}$ is a result of interpolation of just two weighted $L_{p}$ spaces. The proof is based on a generalization of some recent results of the first two authors concerning real interpolation of vector-valued spaces.
3
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Lions-Peetre reiteration formulas for triples and their applications

71%
Asekritova I., Krugljak N., Maligranda L., Nikolova L., Persson L.-E.
Studia Mathematica
|
2001
|
tom 145
|
nr 3
219-254
EN
We present, discuss and apply two reiteration theorems for triples of quasi-Banach function lattices. Some interpolation results for block-Lorentz spaces and triples of weighted $L_{p}$-spaces are proved. By using these results and a wavelet theory approach we calculate (θ,q)-spaces for triples of smooth function spaces (such as Besov spaces, Sobolev spaces, etc.). In contrast to the case of couples, for which even the scale of Besov spaces is not stable under interpolation, for triples we obtain stability in the frame of Besov spaces based on Lorentz spaces. Moreover, by using the results and ideas of this paper, we can extend the Stein-Weiss interpolation theorem known for $L_{p}(μ)$-spaces with change of measures to Lorentz spaces with change of measures. In particular, the results obtained show that for some problems in analysis the three-space real interpolation approach is really more useful than the usual real interpolation between couples.
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