Matrix weighted norm inequalities for commutators and paraproducts with matrix symbols

Abstract

Abstract. Let B be a locally integrable matrix function, W a matrix A p weight with 1 < p < ∞, and T be any of the Riesz transforms. We will characterize the boundedness of the commu-in terms of the membership of B in a natural matrix weighted BMO space. To do this, we will characterize the boundedness of dyadic paraproducts on L p (W ) via a new matrix weighted Carleson embedding theorem. Finally, we will use some of the ideas from these proofs to (among other things) obtain quantitative weighted norm inequalities for these operators and also use them to prove sharp L 2 bounds for the Christ/Goldberg matrix weighted maximal function associated with matrix A 2 weights.

Boundedness of Commutators and H $${}^1$$ 1 -BMO Duality in the Two Matrix Weighted Setting

“…Like [9,10], the ideas and techniques in this paper are "dyadic" in nature and are very different than the more classical ideas and techniques in [3]. However, since the techniques in [9,10] are obviously scalar weighted techniques, we will not draw from them in this paper, but instead heavily rely on the ideas developed in two recent preprints, the first being [13] by the author, H. K. Kwon, and Sandra Pott, and the second being [12] by the author. It should be commented, however, that we will in fact use the papers [9,10] as a sort of "guiding light" for recasting the various matrix weighted BMO conditions in [12,13] into two matrix weighted conditions.…”

“…Similarly, we will require a number of different equivalent versions of BMO p W,U throughout the paper. Surprisingly, many of these various versions in the matrix weighted setting have already appeared in [12,13] in the special cases where either U = W or when one of the matrix weights W or U is the identity.…”

“…However, since the techniques in [9,10] are obviously scalar weighted techniques, we will not draw from them in this paper, but instead heavily rely on the ideas developed in two recent preprints, the first being [13] by the author, H. K. Kwon, and Sandra Pott, and the second being [12] by the author. It should be commented, however, that we will in fact use the papers [9,10] as a sort of "guiding light" for recasting the various matrix weighted BMO conditions in [12,13] into two matrix weighted conditions. Also note that with this in mind, one can think of this paper as a kind of "two weight unification" of some of the ideas and results in [12,13].…”

“…It should be commented, however, that we will in fact use the papers [9,10] as a sort of "guiding light" for recasting the various matrix weighted BMO conditions in [12,13] into two matrix weighted conditions. Also note that with this in mind, one can think of this paper as a kind of "two weight unification" of some of the ideas and results in [12,13].…”

“…In fact, the original definition of BMO p W,W is a very natural and important BMO condition (after using Lemma 2.1 twice) to consider when formulating and proving a T 1 theorem regarding the boundedness of matrix kernelled CZOs on L p (W ) when W is a matrix A p weight for 1 < p < ∞ (see [12] for a precise statement and proof of such a T 1 theorem). Interestingly note that unlike in the scalar setting, BMO p W,W does not reduce to the classical unweighted John-Nirenberg space BMO (see [13] for more information. )…”