Uniform boundedness and compactness for the commutator of an extension of Riesz transform on stratified Lie groups

Abstract

Let G{\mathcal{G}} be a stratified Lie group, and let {Xj}1≤j≤n1{\left\{{X}_{j}\right\}}_{1\le j\le {n}_{1}} be a basis of the left-invariant vector fields of degree one on G{\mathcal{G}} and Δ=−∑j=1n1Xj2\Delta =-{\sum }_{j=1}^{{n}_{1}}{X}_{j}^{2} be the sub-Laplacian of G{\mathcal{G}}. Given that 0≤α<Q0\le \alpha \lt {\mathbb{Q}}, this article studies the commutators [b,XjΔ−1+α2]m,j=1,…,n1{\left[b,{X}_{j}{\Delta }^{-\frac{1+\alpha }{2}}]}^{m},\hspace{0.33em}j=1,\ldots ,{n}_{1} of order m∈Nm\in {\mathbb{N}} and establishes their uniform two-weight boundedness from Lp(μp){L}^{p}\left({\mu }^{p}) to Lq(wq){L}^{q}\left({w}^{q}) for any 0<α<Q0\lt \alpha \lt {\mathbb{Q}} and 1q=1p−αQ\frac{1}{q}=\frac{1}{p}-\frac{\alpha }{{\mathbb{Q}}} via BMOν1⁄m(G){{\rm{BMO}}}_{{\nu }^{1/m}}\left({\mathcal{G}}) space, assuming that μ,w∈Ap,q\mu ,w\in {A}_{p,q} and ν=μw\nu =\frac{\mu }{w}. Based on this, we also give the characterization of VMO(G){\rm{VMO}}\left({\mathcal{G}}) space with respect to the uniform weighted compactness of [b,XjΔ−1+α2]\left[b,{X}_{j}{\Delta }^{-\tfrac{1+\alpha }{2}}] for 0≤α<Q0\le \alpha \lt {\mathbb{Q}}. As a consequence of our results, the corresponding boundedness and compactness for commutators of Riesz transforms on G{\mathcal{G}} can be recovered as α→0\alpha \to 0.

Keywords

• stratified lie groups
• riesz transform
• bmo space
• vmo space
• 42b20
• 42b35
• 43a15
• 22e30