Lower bound of Riesz transform kernels and commutator theorems on stratified nilpotent Lie groups

Abstract

We provide a study of the Riesz transforms on stratified nilpotent Lie groups, and obtain a certain version of the pointwise lower bound of the Riesz transform kernel. Then we establish the characterisation of the BMO space on stratified nilpotent Lie groups via the boundedness of the commutator of the Riesz transforms and the BMO function. This extends the well-known Coifman, Rochberg, Weiss theorem from Euclidean space to the setting of stratified nilpotent Lie groups. In particular, these results apply to the wellknown example of the Heisenberg group. As an application, we also study the curl operator on the Heisenberg group and stratified nilpotent Lie groups, and establish the div-curl lemma with respect to the Hardy space on stratified nilpotent Lie groups.

Two Weight Commutators on Spaces of Homogeneous Type and Applications

“…(1.6) Note that in R n , this "non-degenrated" condition was first proposed in [24], and a similar assumption on the behaviour of the kernel lower bound was proposed in [32]. On stratified Lie groups, a similar condition of the Riesz transform kernel lower bound was verified in [14]. Then we have the following lower bound.…”

“…Recent remarkable results were achieved by Holmes-Lacey-Wick [21] giving the characterisation of weighted BMO space on R n in terms of boundedness of commutators of Riesz transforms, and by Lerner-Ombrosi-Rivera-Ríos [32,33] in terms of boundedness of commutators of Calderón-Zygmund operators with homogeneous kernels Ω( x |x| ) 1 |x| n , and Hytönen [25] in terms of boundedness of commutators of a more general version of Calderón-Zygmund operators and weighted BMO functions on R n . Meanwhile, two weight commutator has also been studied extensively in different settings, see for example [13,15,20,27].…”

“…We point out that such verification for Cauchy integral operator C A is direct. The verifications of Cauchy-Szegö projection operator on Heisenberg group, the Szegö projection operator on a family of weakly pseudoconvex domains and the Riesz transforms associated with Sub-Laplacian on stratified Lie groups can be derived based on the results in [44, Chapter XII], [18] and [14], respectively. The verification for Riesz transforms associated with Bessel operator ∆ λ on R + for λ > 0 can be derived from the result in [40], while for λ ∈ (−1/2, 0) is new here.…”

“…Proposition 7.4. Theorem 1.6 holds for the Riesz transform X j (−∆) − 1 2 (1 ≤ j ≤ n) with the underlying setting (G, ρ, dg), where ρ is the homogeneous norm on G (see [14]).…”

“…Proof. For the Riesz transform kernel, we have the following lower bound estimate, obtained in [14]:…”