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Interpolation of nonlinear maps


Kappeler, Thomas; Savchuk, A M; Shkalikov, A A; Topalov, P (2014). Interpolation of nonlinear maps. Academy of Sciences of the USSR Mathematical Notes, 96(5-6):896-904.

Abstract

Let (X0, X1) and (Y0, Y1) be complex Banach couples and assume that X1 ⊆ X0 with norms satisfying ‖x‖X0 ≤ c‖x‖X1 for some c > 0. For any 0 < θ < 1, denote by Xθ = [X0, X1]θ and Yθ = [Y0, Y1] the complex interpolation spaces and by B(r, Xθ), 0 ≤ θ ≤ 1, the open ball of radius r > 0 in Xθ centered at zero. Then, for any analytic map Φ: B(r, X0) → Y0 + Y1 such that Φ: B(r, X0) → Y0 and Φ: B(c−1r, X1) → Y1 are continuous and bounded by constants M0 and M1, respectively, the restriction of Φ to B(c−θr, Xχ), 0 < θ < 1, is shown to be a map with values in Yθ which is analytic and bounded by M0 1 − θM1 θ.

Abstract

Let (X0, X1) and (Y0, Y1) be complex Banach couples and assume that X1 ⊆ X0 with norms satisfying ‖x‖X0 ≤ c‖x‖X1 for some c > 0. For any 0 < θ < 1, denote by Xθ = [X0, X1]θ and Yθ = [Y0, Y1] the complex interpolation spaces and by B(r, Xθ), 0 ≤ θ ≤ 1, the open ball of radius r > 0 in Xθ centered at zero. Then, for any analytic map Φ: B(r, X0) → Y0 + Y1 such that Φ: B(r, X0) → Y0 and Φ: B(c−1r, X1) → Y1 are continuous and bounded by constants M0 and M1, respectively, the restriction of Φ to B(c−θr, Xχ), 0 < θ < 1, is shown to be a map with values in Yθ which is analytic and bounded by M0 1 − θM1 θ.

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Additional indexing

Item Type:Journal Article, not refereed, original work
Communities & Collections:07 Faculty of Science > Institute of Mathematics
Dewey Decimal Classification:510 Mathematics
Language:English
Date:2014
Deposited On:29 Jan 2015 16:30
Last Modified:05 Apr 2016 19:00
Publisher:MAIK Nauka/Interperiodica
ISSN:0001-4346
Publisher DOI:https://doi.org/10.1134/S0001434614110339

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