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Mat. Zametki, 2013, Volume 94, Issue 4, Pages 578–581 (Mi mz10327)  

This article is cited in 4 scientific papers (total in 4 papers)

On the Interpolation of Analytic Mappings

A. M. Savchuk, A. A. Shkalikov

M. V. Lomonosov Moscow State University

Abstract: Let $(E_0,E_1)$ and $(H_0,H_1)$ be two pairs of complex Banach spaces densely and continuously embedded into each other, $E_1\hookrightarrow E_0$ and $H_1\hookrightarrow H_0$ and also let $\|x\|_{E_0} \le \|x\|_{E_1}$. By $E_\theta=[E_0,E_1]_\theta$ and $H_\theta=[H_0,H_1]_\theta$ we denote the spaces obtained by the complex interpolation method for $\theta\in[0,1]$, and by $B_\theta (0,R)$ we denote an open ball of radius $R$ in the space $E_\theta$. Let $\Phi\colon B_0(0,R)\to H_0$ be an analytic mapping taking $B_1(0,R)$ into $H_1$, and let the estimates
$$ \|\Phi(x)\|_{H_\theta} \le C_\theta\|x\|_{H_\theta}\qquad for all\quad x\in B_\theta(0,R) $$
hold for $\theta = 0, 1$. Then, for all $\theta\in (0,1)$, the mapping $\Phi$ takes the ball $B_\theta(0,r)$ of radius $r\in(0,R)$ in the space $E_\theta$ into $H_\theta$ and
$$ \|\Phi(x)\|_{H_\theta}\le C_0^{1-\theta}C_1^\theta \frac{R}{R-r}\|x\|_{E_\theta}, \qquad x\in B_\theta(0,r). $$


Keywords: complex interpolation method, Banach space, homogenous analytic mapping, Lipschitz continuity.

DOI: https://doi.org/10.4213/mzm10327

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English version:
Mathematical Notes, 2013, 94:4, 547–550

Bibliographic databases:

UDC: 517.988.52+517.982.27
Received: 19.06.2013

Citation: A. M. Savchuk, A. A. Shkalikov, “On the Interpolation of Analytic Mappings”, Mat. Zametki, 94:4 (2013), 578–581; Math. Notes, 94:4 (2013), 547–550

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. A. M. Savchuk, A. A. Shkalikov, “Uniform stability of the inverse Sturm–Liouville problem with respect to the spectral function in the scale of Sobolev spaces”, Proc. Steklov Inst. Math., 283 (2013), 181–196  mathnet  crossref  crossref  mathscinet  isi  elib  elib
    2. T. Kappeler, A. M. Savchuk, P. Topalov, A. A. Shkalikov, “Interpolation of Nonlinear Maps”, Math. Notes, 96:6 (2014), 957–964  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    3. V. G. Zvyagin, V. P. Orlov, “On the Parabolic Problem of Motion of Thermoviscoelastic Media”, Math. Notes, 99:3 (2016), 465–469  mathnet  crossref  crossref  mathscinet  isi  elib
    4. A. G. Baskakov, D. M. Polyakov, “Spectral Properties of the Hill Operator”, Math. Notes, 99:4 (2016), 598–602  mathnet  crossref  crossref  mathscinet  isi  elib
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