Lipschitz Spaces via Commutators of Riesz Transforms on Stratified Lie Groups

In this paper, we provide a deeper study of Riesz transforms and fractional integral operators on a stratified Lie group $\mathcal{G}$ via their connections to Lipschitz spaces and Triebel–Lizorkin spaces. We prove that Lipschitz spaces can be characterised by the boundedness of commutators of Riesz transforms and commutators of fractional integral operators from $L^{p}(\mathcal{G})$ spaces to Triebel–Lizorkin spaces or $L^{q}(\mathcal{G})$ spaces, under suitable conditions on the indices. Our results extend these characterisations from the Euclidean setting to the stratified Lie group $\mathcal{G}$. Compared to the Euclidean case, our proofs overcome the barrier of relying on Fourier transforms and provide new approaches for studying other problems related to the boundedness of operators on stratified Lie groups.