Besov spaces with variable smoothness and integrability on Lie groups of polynomial growth (Q2328214)

Let \(G\) be a connected Lie group of polynomial growth. Let \(X_1,\dots , X_k\) be left-invariant vector fields on \(G\) satisfiying Hörmander's condition and \(\mathcal{L}\) be the corresponding positive sub-Laplacian given by the sum of squares. The authors consider Besov spaces \(B^{\alpha(\cdot)}_{p(\cdot), q(\cdot)}(G)\) on \(G\) with variable smoothness \(\alpha(\cdot)\) and integrability \(p(\cdot)\) related to the sub-Laplacian \(\mathcal{L}\). It is proved that the function spaces are well defined in the sense that their definitions are independent of the choice of basis functions under some specific assumptions. Moreover, the authors show some embeddings between the spaces with the same parameter \(p(\cdot)\).