Meromorphic functions and their derivatives: equivalence of norms

Abstract

For an inner function θ on the upper half-plane C + , we look at the star-invariant subspace K p θ := H P n θHP of the Hardy space H P . We characterize those 0 for which the differentiation operator f → f' provides an isomorphism between K θ p and a closed subspace of H P , with 1 < p < ∞. Namely, we show that such 0's are precisely the Blaschke products whose zero-set lies in some horizontal strip {a < 3z < b}, with 0 < a < b < oo, and splits into finitely many separated sequences. We also describe the case of a single separated sequence in terms of the left inverse to the differentiation map; the description involves coanalytic Toeplitz operators. While our main result provides a criterion for the H P -norms ∥f∥p and ∥f'∥p to be equivalent (written as ∥f∥p≈ ∥f' ∥p), where f ranges over a certain family of meromorphic functions with fixed poles, some other spaces Y that admit a similar estimate ∥f∥y ≈ ∥f'∥y under similar conditions are also pointed out.