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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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