Interpolation of nonlinear maps

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.