Composition followed by differentiation between bergman and hardy spaces

Abstract

Let Φ be an analytic self-map of the disc, and let Hp denote the Hardy space. The operator DCΦ is defined for functions analytic in the disc by DCΦ(f) = (f ˚ Φ)'. We show that compactness and boundedness of the map DCΦ : Hp → Hq, p, q ≥ 1, are equivalent to the conditions Φ' ∈ Hq and ||Φ||∞ < 1. For α > -1 and p ≥ 1, Apα denotes the weighted Bergman space. In the case 1 ≤ p ≤ q, DCΦ : Apα → Aqβ is bounded if and only if a related measure obeys a Carleson-type condition. Compactness is characterized by the analogous little-oh condition. For 1 ≤ q < p, inequality is used to show that boundedness and compactness are equivalent to an integrability condition on a weighted integral. © 2005 Rocky Mountain Mathematics Consortium.