Weighted conformal invariance of Banach spaces of analytic functions

Abstract

We consider Banach spaces of analytic functions in the unit disc which satisfy a weighted conformal invariance property, that is, for a fixed α>0 and every conformal automorphism φ of the disc, f→f∘φ(φ′)α defines a bounded linear operator on the space in question, and the family of all such operators is uniformly bounded in operator norm. Many common examples of Banach spaces of analytic functions like Korenblum growth classes, Hardy spaces, standard weighted Bergman and certain Besov spaces satisfy this condition. The aim of the paper is to develop a general approach to the study of such spaces based on this property alone. We consider polynomial approximation, duality and complex interpolation, we identify the largest and the smallest as well as the “unique” Hilbert space satisfying this property for a given α>0. We investigate the weighted conformal invariance of the space of derivatives, or anti-derivatives with the induced norm, and arrive at the surprising conclusion that they depend entirely on the properties of the (modified) Cesàro operator acting on the original space. Finally, we prove that this last result implies a John-Nirenberg type estimate for analytic functions g with the property that the integration operator f→∫0zf(t)g′(t)dt is bounded on a Banach space satisfying the weighted conformal invariance property.