A reproducing kernel associated with vector fields

Let G be a Lie group, X(G) the space of infinitely differentiable vector fields on G, C (G) the space of smooth complex-valued functions on G and let g denote the Lie algebra of the group G.In this work, we consider a continuous evaluation functional on X(G) together with an inner product on this space, with the aim of constructing a positive definite reproducing kernel defined on C (G) C (G).This construction enables us to endow X(G) with the structure of a reproducing kernel Hilbert space.Since g can be viewed as a Lie subalgebra of X(G), we also induce a structure of reproducing kernel Hilbert space on g.Subsequently, we define a linear transformation from g into F(C (G)), the space of complex-valued functions on C (G) in order to establish both an inversion formula and the transform of a reproducing kernel associated with vector fields on G.