Abstract
Abstract In this article the authors study complex interpolation of Sobolev-Morrey spaces and their generalizations, Lizorkin-Triebel-Morrey spaces. Both scales are considered on bounded domains. Under certain conditions on the parameters the outcome belongs to the scale of the so-called diamond spaces.
Highlights
Introduction and Main ResultsOne of the most popular formulas in interpolation theory is given by [Lp0(Rd), Lp1(Rd)]Θ = Lp(Rd), (1.1) where ≤ p0 < p1 ∞, Θ and 1 p := 1−Θ p0 +
In addition we introduce the diamond spaces
Our treatment of the complex interpolation of Lizorkin-Triebel-Morrey spaces will be reduced to the calculation of a closure of some intersections by means of a formula due to Shestakov [49], [50]
Summary
All in all the general picture concerning complex interpolation of Lizorkin-Triebel-Morrey spaces seems to be more complicated than expected. In the Lemmas 3.12 and 3.13 we characterize this space via differences, which is very important for us We use this characterization to prove an embedding property on the intersection of Lizorkin-Triebel-Morrey spaces in Lemma 3.17 below. Our treatment of the complex interpolation of Lizorkin-Triebel-Morrey spaces will be reduced to the calculation of a closure of some intersections by means of a formula due to Shestakov [49], [50]. Denote positive constants that depend only on the fixed parameters d, s, u, p, q and probably on auxiliary functions Unless otherwise stated their values may vary from line to line.
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