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.

In 1985, Bloom characterized the boundedness of the commutator [b, H] as a map between a pair of weighted L p spaces, where both weights are in A p . The characterization is in terms of a novel BM O condition. We give a 'modern' proof of this result, in the case of p = 2. In a subsequent paper, this argument will be used to generalize Bloom's result to all Calderón-Zygmund operators and dimensions. 1 2 . 2000 Mathematics Subject Classification. Primary .

No abstract

We develop a Radon transform on Banach spaces using Gaussian measure and prove that if a bounded continuous function on a separable Banach space has zero Gaussian integral over all hyperplanes outside a closed bounded convex set in the Hilbert space corresponding to the Gaussian measure then the function is zero outside this set.