Clark measures and de Branges–Rovnyak spaces in several variables. Issue 2 (1st February 2023)

• Record Type: Journal Article
• Authors: Aleksandrov, Aleksei B.; Doubtsov, Evgueni
• Is Part Of: Complex variables and elliptic equations. Volume 68:Issue 2(2023)
• Page Start: 212
• Page End: 221
• Publication Date: 2023-02-01
• Subjects: Cauchy integrals -- Clark measures -- de Branges–Rovnyak spaces -- Hardy spaces -- Henkin measures -- inner functions; 32A35 -- 30J05 -- 31C10 -- 32A26 -- 46E22; Functions of complex variables -- Periodicals; Differential equations, Elliptic -- Periodicals; 515.905
• DOI: 10.1080/17476933.2021.1985480
• Languages: English
• ISSNs: 1747-6933

Abstract

Abstract : Let B n denote the unit ball of C n, n ≥ 1, and let D denote a finite product of B n j, j ≥ 1 . Given a non-constant holomorphic function b : D → B 1, we study the corresponding family σ α [ b ], α ∈ ∂ B 1, of Clark measures on the distinguished boundary ∂ D . We construct a natural unitary operator from the de Branges–Rovnyak space H ( b ) onto the Hardy space H 2 ( σ α ) . As an application, for D = B n and an inner function I : B n → B 1, we show that the property σ 1 [ I ] ≪ σ 1 [ b ] is directly related to the membership of an appropriate explicit function in H ( b ) .