Two more counterexamples to the infinite dimensional Carleson embedding theorem

TL;DR

This paper constructs explicit and simple counterexamples to the infinite-dimensional Carleson embedding theorem, revealing limitations of the theorem and impacting related operator theory in control systems.

Contribution

It provides the first explicit construction of counterexamples and introduces a simple measure structure with implications for Hankel operators.

Findings

Explicit counterexamples to the theorem

A simple measure with tensor-square structure

Implications for Hankel-like operators in control theory

Abstract

The existence of a counterexample to the infinite-dimensional Carleson embedding theorem has been established by Nazarov, Pisier, Treil, and Volberg. We provide an explicit construction of such an example. We also obtain a non-constructive example of particularly simple form; the density function of the measure (with respect to a certain weighted area measure) is the tensor-square of a Hilbert space-valued analytic function. This special structure of the measure has implications for Hankel-like operators appearing in control theory.