an:01179585 Zbl 0973.46070 Nursultanov, E. D. A multi-parameter interpolation functor and the Lorentz space $$L_{p\vec q}$$, $$\vec q=(q_1,\dots,q_n)$$ EN Funct. Anal. Appl. 31, No. 2, 136-138 (1997); translation from Funkts. Anal. Prilozh. 31, No. 2, 79-82 (1997). 0016-2663 1573-8485 1997
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46M35 46E30 real interpolation method; Marcinkiewicz theorem; reiteration theorem; many-parameter interpolation method; bilinear interpolation theorem Summary: The real interpolation method, which stems from the basic Marcinkiewicz theorem, was introduced by Lons and Peetre. It is described by the functor $\Phi_{\theta q}(\varphi)= \left( \int^\infty_0 \bigl(t^{- \theta}\varphi (t)\bigr)^q {dt\over t}\right)^{1/q}.$ \textit{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 \textit{V. I. Demitriev}, and \textit{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. 0162.44502; 0432.46067