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

Abstract

Abstract. In this paper we characterize the two matrix weighted boundedness of commutators with any of the Riesz transforms (when both are matrix A p weights) in terms of a natural two matrix weighted BMO space. Furthermore, we identify this BMO space when p = 2 as the dual of a natural two matrix weighted H 1 space, and use our commutator result to provide a converse to Bloom's matrix A 2 theorem, which as a very special case proves Buckley's summation condition for matrix A 2 weights. Finally, we use our results to prove a matrix weighted John-Nirenberg inequality, and we also briefly discuss the challenging question of extending our results to the matrix weighted vector BMO setting.

Dyadic product BMO in the Bloom setting

“…Conversely, for every bounded linear functional ℓ on H 1 D (U, V, p) there is B ∈ BMO prod,D (U, V, p) with ℓ = ℓ B . We note that our proof of Theorem 1.4 trivially works also in the one-parameter setting, thus answering to the positive the question posed in [20] about the extension of the one-parameter two-matrix weighted H 1 -BMO duality proved there from p = 2 to arbitrary exponents 1 < p < ∞.…”

“…, which is the direct biparameter analog of the one-parameter two-matrix weighted H 1 norm from [20]. In this context, we prove the following result.…”

“…This is the direct biparameter analog of the one-parameter two-matrix weughted H 1 norm defined in [20]. It is easy to check that (H 1 D (U, V, p), ⋅…”

“…where the supremum is taken over all cubes Q in R n with sides parallel to the coordinate hyperplanes, and all reducing operators are taken with respect to exponent p. These matrix-weighted BMO spaces and variants of them were studied extensively in [20], [22], culminating in [24] which gave a complete theory of such spaces. In particular, [24] established a matrix-valued analog of (1.2), (1.5)…”

“…, where all implied constants depend only on n, d, p, [U] Ap(R n ) and [V ] Ap(R n ) . Moreover, work [20] established in addition an H 1 -BMO duality type result in the matrix-weighted setting for p = 2.…”