A characterization of productBMO by commutators

By S. Ferguson, Michael T. Lacey. 2002

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In this paper we establish a commutator estimate which allows one to concretely identify the product BMO space, BMO(R2+ x R2+), of A. Chang and R. Fefferman, as an operator space on L2(R2). The one-parameter analogue of this result is a well-known theorem of Nehari [8]. The novelty of this paper is that we discuss a situation governed by a twoparameter family of dilations, and so the spaces H 1 and BMO have a more complicated structure. Here R2+ denotes the upper half-plane and BMO(R2+ • R2+) is defined to be the dual of the real-variable Hardy space H 1 on the product domain R2+ x R2+. There are several equivalent ways to define this latter space, and the reader is referred to [5] for the various characterizations. We will be more interested in the biholomorphic analogue of H 1, which can be defined in terms of the boundary values of biholomorphic functions on R 2 • R2+ and will be denoted throughout by Hi(R2+ • cf. [10]. In one variable, the space L2(R) decomposes as the direct sum H 2 ( R ) | where H2(R) is defined as the boundary values of functions in H2(R2+) and H2(R) denotes the space of complex conjugate of functions in H2(R). The space L2(R2), therefore, decomposes as the direct sum of the four spaces H 2 ( R ) | H2(R)@H2(R) , H 2 ( R ) | and H 2 ( R ) | where the tensor products are the Hilbert space tensor products. Let P~-,• denote the orthogonal projection of L2(R 2) onto the holomorphic/anti-holomorphic subspaces, in the first and second variables, respectively, and let Hj denote the one-dimensional Hilbert transform in the j t h variable, j -1 , 2. In terms of the projections P+,• HI=P+,++P+,--P-,+-P-,and H2=P+,++P-,+-P+,--P_,_.

Subfields: Applied Mathematics

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In Applied Mathematics

Citations in the corpus

FLAG HARDY SPACES AND MARCINKIEWICZ MULTIPLIERS ON THE HEISENBERG GROUP (2012, background): “Such operators have been studied, for example, in [Gundy and Stein 1979; Fefferman and Stein 1982; Fefferman 1986; 1987; 1999; Chang 1979; Chang and Fefferman 1985; 1982; 1980; Journé 1985; 1986; Pipher 1986; Ferguson and Lacey 2002], where both the L p theory for 1 < p <∞ and H p theory for 0< p ≤ ”
Bi-parameter paraproducts (2003, methodology): “s into a set called R max. For d≥ 1 an integer, we denote by Rd max the set of all R∈ R max such that 2dR⊆ Ω and˜ dis maximal with this property. By using Journ´e’s Lemma [8] in the form presented in [13], we have that for every ǫ>0 X R∈Rd max |R| . 2ǫd|Ω|. (33) Our initial sum in IIis now smalle”