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
j
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