On the interpolation of analytic mappings

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 ? E ₀ and H ₁ ? H ₀ and also let $\left\| x \right\|_{E_0 } \leqslant \left\| x \right\|_{E_1 } $ . By E _θ = [E ₀, E ₁]_θ and H _θ = [H ₀, H ₁]_θ we denote the spaces obtained by the complex interpolation method for θ ∈ [0, 1], and by B _θ(0,R) we denote an open ball of radius R in the space E _θ. Let Φ: B ₀(0,R) → H ₀ be an analytic mapping taking B ₁(0,R) into H ₁, and let the estimates $\left\| {\Phi (x)} \right\|_{H_\theta } \leqslant C_\theta \left\| x \right\|_{H_\theta } for allx \in B_\theta (0,R)$ hold for θ = 0, 1. Then, for all θ ∈ (0, 1), the mapping Φ takes the ball B _θ(0,r) of radius r ∈ (0,R) in the space E _θ into H _θ and $\left\| {\Phi (x)} \right\|_{H_\theta } \leqslant C_0^{1 - \theta } C_1^\theta \frac{R} {{R - r}}\left\| x \right\|_{E_\theta } ,x \in B_\theta (0,r). $ .     

On light scattering in transformer oil

GK transformer oil is studied using the methods of UV and visible spectroscopy, dynamic light scattering, small-angle light scattering, scanning electron microscopy, and element analysis. The effect exerted on the optical properties of transformer oil by colloidal, micelle-like, and solid particles formed in oil during its exploitation is investigated.