Lipschitz spaces via commutators of singular integrals on stratified Lie groups
In this thesis, we establish characterisations of Lipschitz spaces on a stratified Lie group G using commutators of Riesz transforms and commutators of fractional integrals. We obtain several equivalence relationships between Lipschitz spaces [formula] and the boundedness of these commutators from Lp(G) spaces (1 < p < ∞) into Triebel–Lizorkin spaces [formula], and from Lp(G) spaces into Lq(G) spaces with suitable conditions on the indices depending on the singular integrals. Moreover, we give a characterisation of BMO spaces on stratified Lie groups using commutators of fractional integrals.
Our results extend the characterisations from the Euclidean setting to the stratified Lie group G. The proofs that overcome the barrier of using Fourier transforms offer new approaches for studying other problems related to the boundedness of operators on stratified Lie groups.