[Paper Review] Functions of bounded mean oscillation and Hankel operators on compact abelian groups

А. Р. Миротин, R.V. Dyba|arXiv (Cornell University)|Feb 22, 2019

Differential Equations and Boundary Problems4 references19 citations

TL;DR

This paper generalizes functions of bounded mean oscillation (BMO) and Hankel operators to compact abelian groups with linearly ordered character groups. Under the assumption that the dual group contains a minimal positive element, it establishes that a function belongs to BMO(G) or BMOA(G) if and only if the corresponding Hankel operator is bounded, extending classical results to this non-commutative harmonic analysis setting.

ABSTRACT

Generalization of functions of bounded mean oscillation and Hankel operators to the case of compact abelian groups with linearly ordered dual is considered. Spaces of functions of bounded mean oscillation and of bounded mean oscillation of analytic type on such groups are described in terms of boundedness of corresponding Hankel operators under the assumption that the dual group contains a minimal positive element.

Motivation & Objective

• To extend the theory of functions of bounded mean oscillation (BMO) and Hankel operators from the classical setting (e.g., the circle group) to compact abelian groups with linearly ordered character groups.
• To characterize the spaces BMO(G) and BMOA(G) in terms of boundedness of associated Hankel operators on such groups.
• To establish a generalization of Fefferman's duality theorem for BMO in this broader harmonic analysis framework.
• To provide a functional analytic characterization of BMO and BMOA spaces via operator-theoretic conditions on Hankel operators.
• To lay the foundation for a non-commutative extension of classical Hankel operator theory in the context of ordered abelian groups.

Proposed method

• Define BMO(G) as the space of functions expressible as f + ḡ, where f, g ∈ L∞(G), and ḡ is the harmonic conjugate of g.
• Define BMOA(G) as the intersection of BMO(G) with the Hardy space H¹(G), representing analytic-type BMO functions.
• Introduce the Hankel operator Hφ associated with a function φ ∈ L²(G), defined via the orthogonal projection onto H²₋(G).
• Use the orthogonal decomposition L²(G) = H²(G) ⊕ H²₋(G) and the projections P₊ and P₋ to define the Hankel operator Hφ = P₋φ.
• Leverage the existence of a minimal positive element in the character group X to ensure the existence of a well-defined positive cone X₊ and a linear order.
• Apply spectral theory and duality techniques to relate the boundedness of Hφ to the membership of φ in BMO(G) or BMOA(G).

Experimental results

Research questions

• RQ1 How can the concept of functions of bounded mean oscillation be generalized to compact abelian groups with linearly ordered character groups?
• RQ2 What is the operator-theoretic characterization of BMO(G) and BMOA(G) in terms of Hankel operators on such groups?
• RQ3 Under what conditions does the boundedness of a Hankel operator imply that the generating function lies in BMO(G) or BMOA(G)?
• RQ4 Can the classical duality between BMO and the space of functions with vanishing mean oscillation be extended to this non-commutative setting?
• RQ5 What role does the existence of a minimal positive element in the character group play in ensuring the equivalence between bounded Hankel operators and BMO functions?

Key findings

• The space BMO(G) consists precisely of functions φ ∈ L²(G) for which the associated Hankel operator Hφ is bounded.
• The space BMOA(G) is characterized as the set of functions φ ∈ L²(G) such that Hφ is bounded and φ ∈ H¹(G).
• A function φ ∈ L²(G) generates a bounded Hankel operator Hφ if and only if P₋φ ∈ BMO(G), where P₋ is the projection onto H²₋(G).
• If the dual group X contains a minimal positive element, then every bounded Hankel operator on G arises from a symbol in L∞(G), i.e., Hφ = Hg for some g ∈ L∞(G).
• The norm equivalence in BMO(G) is established via the operator norm of the associated Hankel operator, showing that the BMO norm is equivalent to the operator norm of Hφ.
• The proof relies on the existence of a minimal positive element in X to ensure the existence of a well-defined positive cone and to allow the use of spectral and duality arguments in the proof of the main theorem.