Abstract. Let R be the vector of Riesz transforms on R n , and let µ, λ ∈ A p be two weights on R n , 1 < p < ∞. The two-weight norm inequality for the commutatoris shown to be equivalent to the function b being in a BM O space adapted to µ and λ. This is a common extension of a result of Coifman-Rochberg-Weiss in the case of both λ and µ being Lebesgue measure, and Bloom in the case of dimension one.
We characterize the boundedness of the commutators [b, T ] with biparameter Journé operators T in the two-weight, Bloom-type setting, and express the norms of these commutators in terms of a weighted little bmo norm of the symbol b. Specifically, if µ and λ are biparameter Ap weights, ν := µ 1/p λ −1/p is the Bloom weight, and b is in bmo(ν), then we prove a lower bound and testing condition b bmo(ν)where R 1 k and R 2 l are Riesz transforms acting in each variable. Further, we prove that for such symbols b and any biparameter Journé operators T the commutator [b, T ] : L p (µ) → L p (λ) is bounded. Previous results in the Bloom setting do not include the biparameter case and are restricted to Calderón-Zygmund operators. Even in the unweighted, p = 2 case, the upper bound fills a gap that remained open in the multiparameter literature for iterated commutators with Journé operators. As a by-product we also obtain a much simplified proof for a one-weight bound for Journé operators originally due to R. Fefferman. Some of these operatorsHaar coefficient in one variable and averaging in the other variable, such as b, h Q1 × ½ Q2 /|Q 2 | . Since, ultimately, we wish to use some type of H 1 − BM O duality, the goal will be to "separate out" b from the inner product O(b, f ), g . If O(b, f ) involves full Haar coefficients of b, we use duality with product BMO and obtainis the operator we are left with after separating out b, and S D is the full biparameter dyadic square function. If O(b, f ) involves terms of the form b, h Q1 × ½ Q2 /|Q 2 | , we use duality with little bmo, and obtain something of the formwhere S D1 is the dyadic square function in the first variable. Obviously this is replaced with S D2 if the Haar coefficient on b is in the second variable. 3. Then the next goal is to show thatwhere O 1,2 will be operators satisfying a one-weight bound of the type L p (w) → L p (w). These operators will usually be a combination of the biparameter square functions in Section 3. Once we have this, we are done.
We prove that the multiplier algebra of the Drury-Arveson Hardy space H 2 n on the unit ball in ރ n has no corona in its maximal ideal space, thus generalizing the corona theorem of L. Carleson to higher dimensions. This result is obtained as a corollary of the Toeplitz corona theorem and a new Banach space result: the Besov-Sobolev space B p has the "baby corona property" for all 0 and 1 < p < 1. In addition we obtain infinite generator and semi-infinite matrix versions of these theorems.
In this paper we consider the reproducing kernel thesis for boundedness and compactness for various operators on Bergman-type spaces. In particular, the results in this paper apply to the weighted Bergman space on the unit ball, and more generally to weighted Fock spaces.2000 Mathematics Subject Classification. 32A36, 32A, 47B05, 47B35.
In this paper, we establish the two weight commutator of Calderón-Zygmund operators in the sense of Coifman-Weiss on spaces of homogeneous type, by studying the weighted Hardy and BMO space for A 2 weight and by proving the sparse operator domination of commutators. The main tool here is the Haar basis and the adjacent dyadic systems on spaces of homogeneous type, and the construction of a suitable version of a sparse operator on spaces of homogeneous type. As applications, we provide a two weight commutator theorem (including the high order commutator) for the following Calderón-Zygmund operators: Cauchy integral operator on R, Cauchy-Szegö projection operator on Heisenberg groups, Szegö projection operators on a family of unbounded weakly pseudoconvex domains, Riesz transform associated with the sub-Laplacian on stratified Lie groups, as well as the Bessel Riesz transforms (one-dimension and high dimension).to the subspace of functions {F b } that are boundary values of functions F ∈ H 2 (U n ). The associated Cauchy-Szegö kernel is as follows.Then it is natural to study the following question: is there a setting, by which the characterisation of two weight commutators and the related BMO space for Calderón-Zygmund operators T can be obtained, that can be applied to Calderón-Zygmund operators such as the Bessel Riesz transform, the Cauchy-Szegö projection operator on Heisenberg groups, and many other examples?To address this question we work in a general setting: spaces of homogeneous type introduced by Coifman and Weiss in the early 1970s, in [9], see also [10]. We say that (X, d, µ) is a space of homogeneous type in the sense of Coifman and Weiss if d is a quasi-metric on X and µ is a nonzero measure satisfying the doubling condition. A quasi-metric d on a set X is a function d : X × X −→ [0, ∞) satisfying (i) d(x, y) = d(y, x) ≥ 0 for all x, y ∈ X; (ii) d(x, y) = 0 if and only if x = y; and (iii) the quasi-triangle inequality: there is a constant A 0 ∈ [1, ∞) such that for all x, y, z ∈ X,
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