Weighted little bmo and two-weight inequalities for Journé commutators

Abstract

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.

Bloom-Type Inequality for Bi-Parameter Singular Integrals: Efficient Proof and Iterated Commutators

“…Proof of Lemma 3.1. The operators A i (b, ·) (but in a somewhat different form) are already discussed in [14]. To aid the reader we note that the proofs essentially write themselves if one knows certain weighted H 1 -BMO type duality estimates.…”

“…(4.7) for an example of the resulting simple decomposition. In [14] everything was always reduced to a so called remainder term, which essentially entails expanding bf in the bi-parameter sense in all of the above situations. However, this remainder term has a particularly nice structure only when there are no non-cancellative Haar functions (the shift case) -otherwise it can lead to some difficult tail terms.…”

Bloom Type Upper Bounds in the Product BMO Setting

“…that T 1 and T 2 are bi-parameter CZOs in R n 1 ×R n 2 and R n 3 ×R n 4 , respectively. Then according to Ou-Petermich-Strouse [35] and Holmes-Petermichl-Wick [16] we have…”

“…Here T 1 is a partial adjoint of T in the first slot, T 1 (f 1 ⊗ f 2 ), g 1 ⊗ g 2 = T (g 1 ⊗ f 2 ), f 1 ⊗ g 2 , and BMO prod is the product BMO of Chang and Fefferman [6,7]. Holmes-Petermichl-Wick [16] proved the first bi-parameter Bloom type estimate…”

Two Weight Commutators on Spaces of Homogeneous Type and Applications

“…In the last section we also provide a new proof of the lower bound of two weight commutator in the product setting for little bmo space on spaces of homogeneous type. Note that in R n × R m , this was first studied by [22] by using the Fourier transform for the Riesz transform kernel.…”