Boundedness of Journé operators with matrix weights

Dyadic product BMO in the Bloom setting

“…In fact, it is not hard to see that for any 1 < p < ∞, one can choose the reducing operator W x 1 ,F in a way that is measurable in x 1 . A proof of that can be found in the appendix of [12].…”

“…For a detailed exposition of reducing operators we refer for example to [12]. Here we note just the following estimates that we will need in the sequel.…”

“…Here we note just the following estimates that we will need in the sequel. A proof of part (1) can be found, for example, in [12]. A proof of part (2) can be found in [22].…”

“…the other estimate being symmetric. We will rely on the ideas from the proof of [12,Lemma 3.6]. First of all, note that there exists a measurable subset A of R m such that R m ∖ A = 0 and…”

“…Remark 4.3. Observe that if W is a d × d-matrix valued biparameter A 2 weight defined on R n+m , then the operators R 1 j R 2 k (which are paraproduct-free) are bounded in L p (W ) for 1 ≤ j ≤ n and 1 ≤ k ≤ m (see [12], and also [29] in the case p = 2 and [10] in the case of general p for the one-parameter results).…”