Vectorial Hankel operators, Carleson embeddings, and notions of BMOA

Abstract

Let {textrm{BMOA}_{mathcal{N}mathcal{P}}left(mathcal{L}right)} denote the space of {mathcal{L}}-valued analytic functions {phi} for which the Hankel operator {Gamma_phi} is {H^2left(mathcal{H}right)}-bounded. Obtaining concrete characterizations of {textrm{BMOA}_{mathcal{N}mathcal{P}}left(mathcal{L}right)} has proven to be notoriously hard. Let {D^alpha} denote fractional differentiation. Motivated originally by control theory, we characterize {H^2left(mathcal{H}right)}-boundedness of {D^alphaGamma_phi}, where {alpha > 0}, in terms of a natural anti-analytic Carleson embedding condition. We obtain three notable corollaries: The first is that {textrm{BMOA}_{mathcal{N}mathcal{P}}left(mathcal{L}right)} is not characterized by said embedding condition. The second is that when we add an adjoint embedding condition, we obtain a sufficient but not necessary condition for boundedness of {Gamma_phi}. The third is that there exists a bounded analytic function for which the associated anti-analytic Carleson embedding is unbounded. As a consequence, boundedness of an analytic Carleson embedding does not imply that the anti-analytic ditto is bounded. This answers a question by Nazarov, Pisier, Treil, and Volberg.