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Nonclassical interpolation in spaces of smooth functions. (English) Zbl 0934.46038

Let \(\text{bmo}(\mathbb{R})\) be the inhomogeneous BMO-space on \(\mathbb{R}\), let \(J_\alpha\) be the Bessel-potential. Then \(\text{bmo}^\alpha(\mathbb{R})= J_\alpha\text{bmo}(\mathbb{R})\). Of interest is \(0<\alpha< 2\), where one has explicit descriptions. Similarly \(\text{bmo}(M)\) where \(M\) is an interval, circle, \(\mathbb{R}^+\) or \(\mathbb{R}\). The main aim of the paper is to prove \[ \langle L_\infty(M), L^k_\infty(M)\rangle^{\theta,2}\cap L_\infty(M)= \text{bmo}^{k\theta}(M), \] where \(k=1\) or \(k=2\), \(0<\theta<1\), and \(\langle\cdot, \cdot\rangle^{\theta,2}\) is a non-classical interpolation method. Applications to the boundedness of commutators are given.
Reviewer: H.Triebel (Jena)

MSC:

46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
46M35 Abstract interpolation of topological vector spaces
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