The real interpolation method on couples of intersections

Suppose that (X ₀, X ₁) is a Banach couple, X ₀ ∩ X ₁ is dense in X ₀ and X ₁, (X₀,X₁)_θq (0 < θ < 1, 1 ≤ q < ∞) are the spaces of the real interpolation method, ψ ∈ (X ₀ ∩ X ₁), ψ ≠ 0, is a linear functional, N = Ker ψ, and N _i stands for N with the norm inherited from X _i (i = 0, 1). The following theorem is proved: the norms of the spaces (N₀,N₁)_θ,q and (X₀,X₁)_θ,q are equivalent on N if and only if θ ? (0, α) ∪ (β_∞, α₀ ∪ (β₀, α_∞) ∪ (β, 1), where α, β, α₀, β₀, α_∞, and β _∞ are the dilation indices of the function k(t)=K(t,ψ;X ₀ ^* ,X ₁ ^* ).