A characterization of product BMO by commutators
Copyright policy )A characterization of product BMO by commutators
Let b be a function on the plane. Let H_j, j=1,2, be the Hilbert transform acting on the j-th coordinate on the plane. We show that the operator norm of the double commutator [[ M_b, H_1], H_2] is equivalent to the Chang-Fefferman BMO norm of b. Here, M_b denotes the operator which is multiplication by b. This result extends a well known theorem of Nehari on weak factorization in the Hardy space H^1 to the same theorem on H^1 of a product domain. The product setting is more delicate because of the presence of a two parameter family of dilations. The method of proof depends upon (a) A dyadic decomposition of product BMO by wavelets (b) a prior estimate of Ferguson and Sadosky involving rectangular BMO (c) and a careful control of certain measures related to those of Carleson.
21 pages. To appear in Acta Math. Appendix corrected
- Georgia Institute of Technology United States
- GEORGIA TECH RESEARCH CORPORATION United States
- Wayne State University United States
- Wayne State College United States
product domain, commutator, Mathematics - Operator Algebras, Hardy space, Other spaces of holomorphic functions of several complex variables (e.g., bounded mean oscillation (BMOA), vanishing mean oscillation (VMOA)), \(H^p\)-spaces, Hilbert transform, \(H^p\)-spaces, Nevanlinna spaces of functions in several complex variables, Banach algebras of differentiable or analytic functions, \(H^p\)-spaces, Mathematics - Classical Analysis and ODEs, Linear operators on function spaces (general), Classical Analysis and ODEs (math.CA), FOS: Mathematics, Commutators, derivations, elementary operators, etc., Operator Algebras (math.OA), product BMO
product domain, commutator, Mathematics - Operator Algebras, Hardy space, Other spaces of holomorphic functions of several complex variables (e.g., bounded mean oscillation (BMOA), vanishing mean oscillation (VMOA)), \(H^p\)-spaces, Hilbert transform, \(H^p\)-spaces, Nevanlinna spaces of functions in several complex variables, Banach algebras of differentiable or analytic functions, \(H^p\)-spaces, Mathematics - Classical Analysis and ODEs, Linear operators on function spaces (general), Classical Analysis and ODEs (math.CA), FOS: Mathematics, Commutators, derivations, elementary operators, etc., Operator Algebras (math.OA), product BMO
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