Higher order Journé commutators and characterizations of multi-parameter BMO

Abstract

We characterize L p boundedness of iterated commutators of multiplication by a symbol function and tensor products of Riesz and Hilbert transforms. We obtain a two-sided norm estimate that shows that such operators are bounded on L p if and only if the symbol belongs to the appropriate multi-parameter BMO class. We extend our results to a much more intricate situation; commutators of multiplication by a symbol function and paraproduct-free Journé operators. We show that the boundedness of these commutators is also determined by the inclusion of their symbol function in the same multi-parameter BMO class. In this sense the tensor products of Riesz transforms are a representative testing class for Journé operators.Previous results in this direction do not apply to tensor products and only to Journé operators which can be reduced to Calderón-Zygmund operators. Upper norm estimate of Journé commutators are new even in the case of no iterations. Lower norm estimates for iterated commutators only existed when no tensor products were present. In the case of one dimension, lower estimates were known for products of two Hilbert transforms, and without iterations. New methods using Journé operators are developed to obtain these lower norm estimates in the multiparameter real variable setting.

Bilinear Calderón–Zygmund theory on product spaces

“…The theory of commutator estimates is extremely vast and important. By bounding commutators of shifts Ou, Petermichl and Strouse [55] proved that [b, T ] is L 2 bounded, when T is a bi-parameter SIO as in [50] and free of paraproducts. This is the important base case for more complicated multiparameter commutator estimates -i.e., the proof method using the DMOs lends itself to more complicated estimates.…”

“…Proof of Theorem 1.5 and commutator decompositions. The main challenge in going from [55] to [32], apart from the Bloom setting, appeared to be that the various paraproducts include non-cancellative Haar functions and have a more complicated structure than the shifts. In [32] everything was reduced to a so called remainder term, which entailed always expanding bf in the bi-parameter sense.…”

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…”

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

“…As the somewhat lengthy kernel estimates and testing conditions of T are not explicitly needed here, we refer to [24] for the remaining details. Using the dyadic representation theorem Ou, Petermichl and Strouse proved in [25] that [b, T ] : L 2 (R n+m ) → L 2 (R n+m ), when T is a paraproduct free bi-parameter singular integral and b is a little BMO function. This is the important base case for more complicated multi-parameter commutator estimates involving product BMO and iterated commutators of the form [T 1 , [b, T 2 ]] -see again [25] and Dalenc-Ou [7].…”

“…Using the dyadic representation theorem Ou, Petermichl and Strouse proved in [25] that [b, T ] : L 2 (R n+m ) → L 2 (R n+m ), when T is a paraproduct free bi-parameter singular integral and b is a little BMO function. This is the important base case for more complicated multi-parameter commutator estimates involving product BMO and iterated commutators of the form [T 1 , [b, T 2 ]] -see again [25] and Dalenc-Ou [7]. For the earlier deep commutator lower bounds in the Hilbert and Riesz settings see Ferguson-Lacey [10] and Lacey-Petermichl-Pipher-Wick [19].…”