Two Weight Commutators on Spaces of Homogeneous Type and Applications

Abstract

In this paper, we establish the two weight commutator of Calderón-Zygmund operators in the sense of Coifman-Weiss on spaces of homogeneous type, by studying the weighted Hardy and BMO space for A 2 weight and by proving the sparse operator domination of commutators. The main tool here is the Haar basis and the adjacent dyadic systems on spaces of homogeneous type, and the construction of a suitable version of a sparse operator on spaces of homogeneous type. As applications, we provide a two weight commutator theorem (including the high order commutator) for the following Calderón-Zygmund operators: Cauchy integral operator on R, Cauchy-Szegö projection operator on Heisenberg groups, Szegö projection operators on a family of unbounded weakly pseudoconvex domains, Riesz transform associated with the sub-Laplacian on stratified Lie groups, as well as the Bessel Riesz transforms (one-dimension and high dimension).to the subspace of functions {F b } that are boundary values of functions F ∈ H 2 (U n ). The associated Cauchy-Szegö kernel is as follows.Then it is natural to study the following question: is there a setting, by which the characterisation of two weight commutators and the related BMO space for Calderón-Zygmund operators T can be obtained, that can be applied to Calderón-Zygmund operators such as the Bessel Riesz transform, the Cauchy-Szegö projection operator on Heisenberg groups, and many other examples?To address this question we work in a general setting: spaces of homogeneous type introduced by Coifman and Weiss in the early 1970s, in [9], see also [10]. We say that (X, d, µ) is a space of homogeneous type in the sense of Coifman and Weiss if d is a quasi-metric on X and µ is a nonzero measure satisfying the doubling condition. A quasi-metric d on a set X is a function d : X × X −→ [0, ∞) satisfying (i) d(x, y) = d(y, x) ≥ 0 for all x, y ∈ X; (ii) d(x, y) = 0 if and only if x = y; and (iii) the quasi-triangle inequality: there is a constant A 0 ∈ [1, ∞) such that for all x, y, z ∈ X,

Commutators on Weighted Morrey Spaces on Spaces of Homogeneous Type

“…We mainly combine the ideas in [12] and [35] to prove our main result. We also point out that to obtain the above theorem, we provide an equivalent characterisation of VMO(X), which is stated in Lemma 2.4 below, and is of independent interest.…”

“…We rst recall another version of the homogeneous condition (formulated in [12]): there exist positive constants ≤ A ≤ A such that for any ball B := B(x , r) ⊂ X, there exist balls B := B(y , r) such that A r ≤ d(x , y ) ≤ A r, and for all (x, y) ∈ (B × B), K(x, y) does not change sign and…”

“…Note that by the doubling condition there exist two constants C and C such that C V(y, x) ≤ V(x, y) ≤ C V(y, x). As in [12], we assume that for any Calderón-Zygmund operator T as in De nition 1.1 with β(t) → as t → , the following "non-degenerate" condition holds: there exist positive constants c andĀ such that for every x ∈ X and r > , there exists y ∈ B(x,Ār) \ B(x, r), satisfying…”

“…[12]). Let T be the Calderón-Zygmund operator as in De nition 1.1 and satisfy the homogeneous condition as in (1.6).…”

Weighted Estimates for Iterated Commutators of Riesz Potential on Homogeneous Groups

“…We review the concept of sparse family given in [12] on ordinary spaces of homogeneous description in the sense of Coifman and Weiss [13], which is also suitable in the case of homogeneous groups. Definition 5.…”