Interpolation of Intersections Generated by a Linear Functional

Let (X ₀, X ₁) be a Banach couple such that X ₀ ∩ X ₁ is dense in X ₀ and X ₁. By (X ₀, X ₁)_θ,q , 0 < θ < 1, 1 ⩽ q < ∞, we denote the spaces of the real interpolation method. Let ψ be a nonzero linear functional defined on some linear space M ⊂ X ₀ + X ₁ and such that ψ ∈ (X ₀ ∩ X ₁)*, and let N = Ker ψ. We examine conditions under which the natural formula is valid. In particular, the results obtained here imply those due to Ivanov and Kalton on the comparison of the interpolation spaces (X ₀, X ₁)_θ,q and (N ₀, X ₁)_θ,q , where ψ ∈ X ₀ ^* and N ₀ = Ker ψ. By way of application, we consider a problem, posed by Krugljak, Maligranda, and Persson, on the interpolation of intersections generated by an integral functional defined on weighted L _p -spaces.__________Translated from Funktsional’nyi Analiz i Ego Prilozheniya, Vol. 39, No. 2, pp. 61–64, 2005Original Russian Text Copyright © by S. V. Astashkin