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

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

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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• http://mi.mathnet.ru/eng/mz/v94/i4/p578

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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
2. T. Kappeler, A. M. Savchuk, P. Topalov, A. A. Shkalikov, “Interpolation of Nonlinear Maps”, Math. Notes, 96:6 (2014), 957–964
3. V. G. Zvyagin, V. P. Orlov, “On the Parabolic Problem of Motion of Thermoviscoelastic Media”, Math. Notes, 99:3 (2016), 465–469
4. A. G. Baskakov, D. M. Polyakov, “Spectral Properties of the Hill Operator”, Math. Notes, 99:4 (2016), 598–602
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