Compactness of Riesz transform commutator on stratified Lie groups

By Peng Chen, Xuan Thinh Duong, Ji Li, Qingyan Wu. Journal of Functional Analysis, 2018

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Abstract Let G be a stratified Lie group and { X j } 1 ≤ j ≤ n a basis for the left-invariant vector fields of degree one on G . Let Δ = ∑ j = 1 n X j 2 be the sub-Laplacian on G . The jth Riesz transform on G is defined by R j : = X j ( − Δ ) − 1 2 , 1 ≤ j ≤ n . In this paper, we provide a concrete construction of the “twisted truncated sector” which is related to the pointwise lower bound of the kernel of R j on G . Then we obtain the characterisation of compactness of the commutators of R j with a function b ∈ VMO ( G ) , the space of functions with vanishing mean oscillation on G .

Subfields: Applied Mathematics

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