Interpolation of nonlinear maps
Kappeler, T. ; Savchuk, A. ; Shkalikov, A. ; Topalov, P.
In: Mathematical Notes, 2014, vol. 96, no. 5-6, p. 957-964
Zum persönliche Liste hinzufügen- Summary
- Let (X 0, X 1) and (Y 0, Y 1) be complex Banach couples and assume that X 1 ⊆ X 0 with norms satisfying ‖x‖X 0 ≤ c‖x‖X 1 for some c > 0. For any 0 < θ < 1, denote by X θ = [X 0, X 1] θ and Y θ = [Y 0, Y 1] the complex interpolation spaces and by B(r, X θ ), 0 ≤ θ ≤ 1, the open ball of radius r > 0 in X θ centered at zero. Then, for any analytic map Φ: B(r, X 0) → Y 0 + Y 1 such that Φ: B(r, X 0) → Y 0 and Φ: B(c −1 r, X 1) → Y 1 are continuous and bounded by constants M 0 and M 1, respectively, the restriction of Φ to B(c −θ r, X χ ), 0 < θ < 1, is shown to be a map with values in Y θ which is analytic and bounded by M 0 1 − θ M 1 θ .