|
This article is cited in 1 scientific paper (total in 1 paper)
Interpolation of Nonlinear Maps
T. Kappelera, A. M. Savchukb, P. Topalovc, A. A. Shkalikovb a Institut für Mathematik, Universität Zürich, Switzerland
b M. V. Lomonosov Moscow State University
c Northeastern University, USA
Abstract:
Let $(X_0, X_1)$ and $(Y_0, Y_1)$ be complex Banach couples and assume that $X_1\subseteq X_0$ with norms satisfying $\|x\|_{X_0} \le c\|x\|_{X_1}$ for some $c >\nobreak 0$. For any $0<\theta <1$, denote by $X_\theta = [X_0, X_1]_\theta$ and $Y_\theta = [Y_0, Y_1]_\theta$ the complex interpolation spaces and by $B(r, X_\theta)$, $0 \le \theta \le 1$, the open ball of radius $r>0$ in $X_\theta$ centered at zero. Then, for any analytic map $\Phi\colon B(r, X_0) \to Y_0+ Y_1$ such that $\Phi\colon B(r, X_0)\to Y_0$ and $\Phi\colon B(c^{-1}r, X_1)\to Y_1$ are continuous and bounded by constants $M_0$ and $M_1$, respectively, the restriction of $\Phi$ to $B(c^{-\theta}r, X_\theta)$, $0 < \theta <\nobreak 1$, is shown to be a map with values in $Y_\theta$ which is analytic and bounded by $ M_0^{1-\theta} M_1^\theta$.
Keywords:
interpolation, nonlinear maps, Banach couple, Hausdorff space.
Received: 23.09.2014
Citation:
T. Kappeler, A. M. Savchuk, P. Topalov, A. A. Shkalikov, “Interpolation of Nonlinear Maps”, Mat. Zametki, 96:6 (2014), 896–904; Math. Notes, 96:6 (2014), 957–964
Linking options:
https://www.mathnet.ru/eng/mzm10563https://doi.org/10.4213/mzm10563 https://www.mathnet.ru/eng/mzm/v96/i6/p896
|
Statistics & downloads: |
Abstract page: | 510 | Full-text PDF : | 176 | References: | 48 | First page: | 44 |
|