Estimates in Möbius invariant spaces of analytic functions

Abstract

We consider a class of spaces of analytic functions on the unit disc which are Möbius invariant and whose topology is essentially determined by a conformal invariant seminorm. Standard examples of such spaces are the Bloch space, BMOA, the Dirichlet spaces and their recent generalizations QK , which make the object of our interest. We prove a general inequality for the seminorms of dilated functions, radial growth estimates, embedding theorems in Lp -spaces on the unit disc, as well as integral estimates of exponentials of functions in such spaces. Finally, we discuss some properties of the inner-outer factorization for those QK spaces which are contained in the Nevanlinna class.