Vectorial Hankel operators, Carleson embeddings, and notions of $BMOA$

Abstract

We consider operators of the type $D^{\alpha }:H^2(\mathcal{H})\to H^2(\mathcal{H})$, where $D^{\alpha }$ denotes a fractional differentiation operator, and ${\Gamma }_{\varphi }$ is a Hankel operator. For $\alpha >0$, we characterize boundedness in terms of a natural anti-analytic Carleson embedding condition. We obtain three notable corollaries. The first is that our main result does not extend to $\alpha =0$, i.e. Nehari-Page BMOA is not characterized by the natural anti-analytic Carleson embedding condition. The second is that when we add an adjoint embedding condition, we obtain a sufficient but not necessary condition for boundedness of ${\Gamma }_{\varphi }$. 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.

Accepted for publication in Geometric and Functional Analysis (GAFA)

References in corpus (3)

• A polynomially bounded operator on Hilbert space which is not similar to a contraction
• Two more counterexamples to the infinite dimensional Carleson embedding theorem
• On the characterization of Triebel--Lizorkin type spaces of analytic functions

Cited by in corpus (2)

• On the characterization of Triebel--Lizorkin type spaces of analytic functions
• Two more counterexamples to the infinite dimensional Carleson embedding theorem