Hankel Operators in Several Complex Variables and Product $BMO\zProd$

By Michael T. Lacey, E. Terwilleger. 2003

57 citations recorded by the source corpus.

$H^2\zProd$ denotes the Hardy space of square integrable functions analytic in each variable separately. Let $P^{\ominus}$ be the natural projection of $L^2\zProd$ onto $\z8{H^2\zProd}$. A Hankel operator with symbol $b$ is the linear operator from $H^2\zProd$ to $\z8{H^2\zProd}$ given by $H_b \zvf=P^{\ominus}\bar b \zvf$. We show that \md0 \norm H_b ..\simeq \norm P^{\oplus}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 $H^1\zProd$. In the case of two complex variables, this is due to Ferguson and Lacey \cite{MR1961195}. 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 Journ\'e \cite{MR87g:42028}, which occurs implicitly in the work of J. Pipher \cite{MR88a:42019}.

Subfields: Applied Mathematics

View Paper: https://www.semanticscholar.org/paper/af6c2ccf630eb37cf1c5b904185423d7cf2036ac

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In Applied Mathematics

Citations in the corpus

Bilinear forms on the Dirichlet space (2008, methodology): “The approach in [8] is not well suited for analysis on the Hardy space of the polydisk, H2 (Dn): However Ferguson, Lacey, and Terwilleger were able to extend methods of multivariable harmonic analysis and obtain a result for H2 (Dn) [10], [ 11 ].”
Commutators in the two-weight setting (2015, background): “Several lines of investigation came out of this work: generalizations to spaces of homogeneous type, [41], multi-parameter extensions, [8,10,23,26], connections to factorization of function spaces, [1,4,6,13], div-curl lemmas, [5,22] and additional interpretations in operator theory [32, 34, 42].”