Interpolation and duality in spaces of pseudocontinuable functions

Abstract

Given an inner function $$\theta $$ θ on the unit disk, let $$K^p_\theta :=H^p\cap \theta {\overline{z}}\overline{H^p}$$ K θ p : = H p ∩ θ z ¯ H p ¯ be the associated star-invariant subspace of the Hardy space $$H^p$$ H p . Also, we put $$K_{*\theta }:=K^2_\theta \cap \mathrm{BMO}$$ K ∗ θ : = K θ 2 ∩ BMO . Assuming that $$B=B_{{\mathcal {Z}}}$$ B = B Z is an interpolating Blaschke product with zeros $${\mathcal {Z}}=\{z_j\}$$ Z = { z j } , we characterize, for a number of smoothness classes X, the sequences of values $${\mathcal {W}}=\{w_j\}$$ W = { w j } such that the interpolation problem $$f\big |_{{\mathcal {Z}}}={\mathcal {W}}$$ f | Z = W has a solution f in $$K^2_B\cap X$$ K B 2 ∩ X . Turning to the case of a general inner function $$\theta $$ θ , we further establish a non-duality relation between $$K^1_\theta $$ K θ 1 and $$K_{*\theta }$$ K ∗ θ . Namely, we prove that the latter space is properly contained in the dual of the former, unless $$\theta $$ θ is a finite Blaschke product. From this we derive an amusing non-interpolation result for functions in $$K_{*B}$$ K ∗ B , with $$B=B_{{\mathcal {Z}}}$$ B = B Z as above.