Triebel–Lizorkin spaces with variable smoothness and integrability on Lie groups of polynomial growth

In this paper we characterize Triebel–Lizorkin spaces \(F_{p(\cdot ),q(\cdot )}^{\alpha (\cdot )}\) with variable smoothness and integrability on Lie groups of polynomial growth. We show that such spaces can be well defined under some conditions. By molecular decomposition we give the relationship between the spaces \(F_{p(\cdot ),q(\cdot )}^{\alpha (\cdot )}\) and \(f_{p(\cdot ),q(\cdot )}^{\alpha (\cdot )}\).