Interpolation Space - The Reiteration Theorem

The Reiteration Theorem

An intermediate space X of the compatible couple (X₀, X₁) is said to be of class θ if

with continuous injections. Beside all real interpolation spaces (X₀, X₁)_θ, q with parameter θ and 1 ≤ q ≤ ∞, the complex interpolation space (X₀, X₁)_θ is an intermediate space of class θ of the compatible couple (X₀, X₁).

The reiteration theorems says, in essence, that interpolating with a parameter θ behaves, in some way, like forming a convex combination a = (1 - θ) x₀ + θ x₁ : taking a further convex combination of two convex combinations gives another convex combination.

Theorem. Let A₀, A₁ be intermediate spaces of the compatible couple (X₀, X₁), of class θ₀ and θ₁ respectively, with 0 < θ₀, θ₁ < 1 and θ₀ ≠ θ₁. When 0 < θ < 1 and 1 ≤ q ≤ ∞, one has

It is notable that when interpolating with the real method between A₀ = (X₀, X₁)_θ0,q 0 and A₁ = (X₀, X₁)_θ1,q 1, only the values of θ₀ and θ₁ matter. Also, A₀ and A₁ can be complex interpolation spaces between X₀ and X₁, with parameters θ₀ and θ₁ respectively.

There is also a reiteration theorem for the complex method.

Theorem. Let (X₀, X₁) be a compatible couple of complex Banach spaces, and assume that X₀ ∩ X₁ is dense in X₀ and in X₁. Let A₀ = (X₀, X₁)_θ0 and A₁ = (X₀, X₁)_θ1, where 0 ≤ θ₀ ≤ θ₁ ≤ 1. Assume further that X₀ ∩ X₁ is dense in A₀ ∩ A₁. Then, for every θ ∈ ,

The density condition is always satisfied when X₀ ⊂ X₁ or X₁ ⊂ X₀.