Convolution equation and operators on the euclidean motion group

Let G = R² SO(2) be the Euclidean motion group, let g be the Lie algebra of G and let U(g) be the universal enveloping algebra of g. Then U(g) is an infinite dimensional, linear associative and non-commutative algebra consisting of invariant differential operators on G. The Dirac measure on G is represented by δG, while the convolution product of functions or measures on G is represented by . Among other notable results, it is demonstrated that for each u in U(g), there is a distribution E on G such that the convolution equation u E = δG is solved by method of convolution. Further more, it is established that the (convolution) operator A^ : C^∞c(G)→ C^∞(G),, which is defined as A^ f = f Tⁿδ(t) extends to a bounded linear operator on L²(G), for f∈ C^∞c(G), the space of infinitely differentiable functions on G with compact support. Furthermore, we demonstrate that the left convolution operator LT denoted as LTf = T f commutes with left translation, for T∈ D⁽G).