Commutators in the two-weight setting

Abstract

Abstract. Let R be the vector of Riesz transforms on R n , and let µ, λ ∈ A p be two weights on R n , 1 < p < ∞. The two-weight norm inequality for the commutatoris shown to be equivalent to the function b being in a BM O space adapted to µ and λ. This is a common extension of a result of Coifman-Rochberg-Weiss in the case of both λ and µ being Lebesgue measure, and Bloom in the case of dimension one.

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

“…On the other hand, in the very recent preprint [10], the authors extended the results in [3] to all CZOs on R d for d > 1. Given these results, it is natural to try to prove two matrix weighted norm inequalities for commutators [T, B] where T is a CZO and B is a locally integrable M n (C) valued function.…”

“…Before we state our results, let us rewrite Bloom's BMO condition in a way that naturally extends to the matrix weighted setting. First, by multiple uses of the A p property and Hölder's inequality, it is easy to We can now state the main result of the paper Note that sufficiency in Theorem 1.1 is new even in the scalar setting in the sense that [10] proves that b ∈ BMO ν if all of the Riesz transforms R j for j = 1, . .…”

“…For a dyadic grid D let BMO ν,D be the canonical dyadic version of BMO ν . In the scalar weighted setting, sufficiency in Theorem 1.1 for general CZOs was proved in [10] using the known (see [17]…”

“…It should be noted that various different equivalent versions of BMO ν were needed in [10] to prove their main result. Similarly, we will require a number of different equivalent versions of BMO p W,U throughout the paper.…”

“…Let us briefly comment on the ideas and techniques used in this paper. 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.…”