2003/10/21 by Michael T. Lacey, Michael T Lacey, Lacey, Michael T +2 · 1 citation
Mathematics · #Advanced Harmonic Analysis Research #Classical Analysis and ODEs (math.CA) #Differential Equations and Boundary Problems #FOS: Mathematics #Holomorphic and Operator Theory #Operator Algebras (math.OA) #math.CA #math.OA
paper · pdf · doi:10.48550/arxiv.math/0310348
22 pages, 13 references. Paper to appear in Houston Journal of Math. Small changes to the manuscript
openalex publication_date 2003/10/21 · arxiv created 2007/04/05 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
H2\zProd denotes the Hardy space of square integrable functions analytic in each variable separately. Let P\ominus be the natural projection of L2\zProd onto \z8H2\zProd. A Hankel operator with symbol b is the linear operator from H2\zProd to \z8H2\zProd given by Hb \zvf=P\ominus b \zvf. We show that \md0 \norm Hb ..≃ \norm P⊕b.BMO\zProd., \emd where the right hand norm is S.-Y. Chang and R. Fefferman product BMO. This fact has well known equivalences in terms of commutators and the weak factorization of H1\zProd. In the case of two complex variables, this is due to Ferguson and Lacey \citeMR1961195. While the current proof is inductive, and one can take the one complex variable case as the basis step, it is heavily influenced by the methods of Ferguson and Lacey. The induction is carried out with a particular form of a lemma due to Journé \citeMR87g:42028, which occurs implicitly in the work of J. Pipher \citeMR88a:42019.