A characterization of product BMO by commutators

By Michaele L. Ferguson. 2001

113 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics

View Paper: https://www.semanticscholar.org/paper/59fd920675481c9e4b101c0816261aece1d81d52

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

Citations in the corpus

Bilinear forms on the Dirichlet space (2008, methodology): “However Ferguson, Lacey, and Terwilleger were able to extend methods of multivariable harmonic analysis and obtain a result for H(2) (D) [10], [11].”
Representation of bi-parameter singular integrals by dyadic operators (2011, unknown): “There is a wide body of more recent developments of which we here only mention the papers by Ferguson and Lacey [6], Lacey and Metcalfe [12] and Muscalu, Pipher, Tao and Thiele [13].”
New maximal functions and multiple weights for the multilinear Calderón-Zygmund theory (2009, unknown): “[21] and [35].”
FUNCTIONS OF BOUNDED MEAN OSCILLATION (2006, methodology): “Recently Lacey and Terwilleger [32] have obtained the analogue of the FergusonLacey characterization for BMO(R × R× R) in terms of third order commutators of the type [[[Mf ,H1], H2], H3], by extending to the tridisk the geometrical study done in [22], through a new and essential twist.”
Hankel Operators in Several Complex Variables and Product $BMO\zProd$ (2003, background): “In the case of more than one complex variable, there are different types of Hankel operators, and we only consider the so called little Hankel operators; little in the sense that the projection used in the definition is onto the smallest natural choice of subspaces of L2(Rn) to use.”
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].”
Multi-parameter paraproducts (2004, methodology): “This approach relied on the one dimensional BMO theory and also on Journé’s lemma [6] [4], but did not extend to prove the general d-parameter case.”