Bloom Type Upper Bounds in the Product BMO Setting

Abstract

We prove some Bloom type estimates in the product BMO setting. More specifically, for a bounded singular integral Tn in R n and a bounded singular integral Tm in R m we prove thatwhere p ∈ (1, ∞), µ, λ ∈ Ap and ν := µ 1/p λ −1/p is the Bloom weight. Here T 1 n is Tn acting on the first variable, T 2 m is Tm acting on the second variable, Ap stands for the bi-parameter weights of R n × R m and BMO prod (ν) is a weighted product BMO space.2010 Mathematics Subject Classification. 42B20.

Product Space Singular Integrals with Mild Kernel Regularity

“…For example, we prove the following two-weight estimate for commutators. The result (1) extends [29] and the result (2) extends [19] and [28].…”

“…These Bloomstyle two-weight estimates have recently been one of the main lines of development concerning commutators, see e.g. [1,2,18,19,25,26,28,29] for a non-exhaustive list.…”

Two-weight commutator estimates: general multi-parameter framework

“…The paraproduct free assumption can be removed according to [18]. We prove results of this type in the two-weight Bloom case generalising [36] and (1.3). As a byproduct, we get explicit proof of unweighted multi-parameter estimates of [40].…”

“…They also recently showed the corresponding lower bound in [36] using the median method. In [8] Dalenc and Ou extended [12] by proving that for all one-parameter CZOs…”

“…This requires that when our CZOs are not paraproduct free, they are at most bi-parameter -otherwise the partial paraproducts would not be amenable to sparse domination methods. This restriction comes from the methods of [34,36], which we adapt here. We have not found a way to estimate certain terms without relying on these methods.…”