Vectorial Hankel operators, Carleson embeddings, and notions of BMOA

Let (Formula presented.) denote the space of (Formula presented.)-valued analytic functions (Formula presented.) for which the Hankel operator (Formula presented.) is (Formula presented.)-bounded. Obtaining concrete characterizations of (Formula presented.) has proven to be notoriously hard. Let (Formula presented.) denote fractional differentiation. Motivated originally by control theory, we characterize (Formula presented.)-boundedness of (Formula presented.), where (Formula presented.), in terms of a natural anti-analytic Carleson embedding condition. We obtain three notable corollaries: The first is that (Formula presented.) 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 (Formula presented.). 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.