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A multi-parameter interpolation functor and the Lorentz space \(L_{p\vec q}\), \(\vec q=(q_1,\dots,q_n)\). (English. Russian original) Zbl 0973.46070
Funct. Anal. Appl. 31, No. 2, 136-138 (1997); translation from Funkts. Anal. Prilozh. 31, No. 2, 79-82 (1997).
Summary: The real interpolation method, which stems from the basic Marcinkiewicz theorem, was introduced by Lons and Peetre. It is described by the functor \[ \Phi_{\theta q}(\varphi)= \left( \int^\infty_0 \bigl(t^{- \theta}\varphi (t)\bigr)^q {dt\over t}\right)^{1/q}. \] J. Peetre [“A theory of interpolation of normed spaces”, Notes Math. 39 (1968; Zbl 0162.44502)] noticed that under some general conditions on \(\Phi\) this functor defines an interpolation method that shares many properties of the real method. The central result in this area is the reiteration theorem \[ (\overline A_{\Phi_1},\overline A_{\Phi_2})_F=\overline A_{F(\Phi_1, \Phi_2)}, \tag{1} \] which asserts that the interpolation problem for a couple \(A_{\Phi_1}\), \(A_{\Phi_2}\) can be reduced to the interpolation of the parameters \(\Phi_1\) and \(\Phi_2\) [see V. I. Demitriev, and V. I. Ovchinnikov, Dokl. Akad. Nauk SSSR 246, 794-797 (1979; Zbl 0432.46067)]. In the present paper, we introduce a functor \(\Phi_{\theta\vec q}\), \(\vec q=(q_1, \dots,q_n)\) that generates a many-parameter Lorentz space \(L_{p\vec q}\). We study interpolation properties of these spaces, which, according to (1), solve the reiteration problem for the corresponding method. The suggested many-parameter interpolation method permits one to describe some finer scales of the Besov spaces \(B^\alpha_{p\vec q}\) and to refine the bilinear interpolation theorem.
MSC:
46M35 Abstract interpolation of topological vector spaces
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
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References:
[1] J.-L. Lions, I-III, Ann. Scuola Norm. Sup. Pisa,13, 389–403 (1959); ibid,14, 317–331 (1960); J. Math. Pures Appl.,42, 195–203 (1963).
[2] J.-L. Lions and J. Peetre, Inst. Hautes Études Sci. Publ. Math.,19, 5–68 (1964). · Zbl 0148.11403
[3] J. Peetre, Lecture Notes, Brasilia, 1963 [Notas de Matematica, Vol. 39, 1968, pp. 1–68].
[4] V. I. Dmitriev and V. I. Ovchinnikov, Dokl. Akad. Nauk SSSR,246, No. 4, 794–797 (1979).
[5] Yu. A. Brudnyi and N. Ya. Kruglyuak, Dokl. Akad. Nauk SSSR,256, No. 1, 14–17 (1981).
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