Self-improving behaviour of inner functions as multipliers

Let X and Y be two spaces of analytic functions in the disk, with X ⊂ Y . For an inner function θ, it is sometimes true that whenever f ∈ X and f θ ∈ Y , the latter product must actually be in X. We discuss this phenomenon for various pairs of (analytic) smoothness classes X and Y . © 2006 Elsevier Inc. All rights reserved