2016/09/03 by Xuan Thinh Duong, Duong, Xuan Thinh, Ji Li +5
Mathematics · #Advanced Harmonic Analysis Research #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Holomorphic and Operator Theory #Mathematical Analysis and Transform Methods
paper · pdf · doi:10.48550/arxiv.1609.00784
openalex publication_date 2016/09/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper, we provide a direct and constructive proof of weak factorization of h1(ℝ) (the predual of little BMO space bmo(ℝ×ℝ) studied by Cotlar-Sadosky and Ferguson-Sadosky), i.e., for every f∈ h1(ℝ×ℝ) there exist sequences \αjk\∈ℓ1 and functions gjk,hkj∈ L2(ℝ2) such that f=∑k=1^∞∑j=1^∞αkj( hkj H1H2 gkj - gkj H1H2 hkj) in the sense of h1(ℝ), where H1 and H2 are the Hilbert transforms on the first and second variable, respectively. Moreover, the norm ‖f‖h1(ℝ×ℝ) is given in terms of ‖gkj‖L2(ℝ2) and ‖hkj‖L2(ℝ2). By duality, this directly implies a lower bound on the norm of the commutator [b,H1H2] in terms of ‖b‖_\rm bmo(ℝ×ℝ). Our method bypasses the use of analyticity and the Fourier transform, and hence can be extended to the higher dimension case in an arbitrary n-parameter setting for the Riesz transforms.