Dixmier–Malliavin factorization theorem - Imported Knowlpedia Content

Statement

Let $G$ be a finite-dimensional Lie group. The Dixmier–Malliavin factorization theorem states that every $f\in C_c^\infty(G)$ is a finite sum of convolutions:

f=\sum_{j=1}^{N}a_j*b_j,\qquad a_j,b_j\in C_c^\infty(G).

Consequently, if $G$ acts continuously on a Fréchet space $E$ , every smooth vector is a finite sum

v=\sum_{j=1}^{N}\pi(f_j)v_j

with $f_j\in C_c^\infty(G)$ and $v_j\in E$ . Hence the smooth-vector space equals the Gårding subspace, not merely its closure.

Support control

The function factorization is local in one factor: a neighborhood $U$ of the identity may be fixed in advance and the factors chosen so that one factor in each convolution has support in $U$ , while the other has support controlled by that of $f$ . This strengthened statement is Theorem 3.1 of the original paper Dixmier–Malliavin, Theorem 3.1.

From functions to vectors

The vector form is a direct part of the Dixmier–Malliavin theorem: the smooth vectors of a continuous Fréchet representation are the finite sums of vectors $\pi(f)w$ , with $f\in C_c^\infty(G)$ and $w\in E$ . It does not require an individual compactly supported $f$ satisfying $\pi(f)v=v$ , which need not exist. Associativity for the integrated operator, $\pi(a*b)=\pi(a)\pi(b)$ , is compatible with factorization once the vector theorem has been established. In a unitary representation it gives

\mathcal H^\infty=\operatorname{span}\{\pi(f)w:f\in C_c^\infty(G),\ w\in\mathcal H\}.

Distinction from density

Gårding's argument only shows that smoothed vectors form a dense smooth subspace. Dixmier–Malliavin identifies every smooth vector as a finite algebraic sum of smoothed vectors. It does not assert that every test function is a single convolution; finite sums are indispensable in general.

References

Jacques Dixmier and Paul Malliavin, Factorisations de fonctions et de vecteurs indéfiniment différentiables, Bulletin des Sciences Mathématiques 102 (1978), 307–330. Academic-hosted scan. Relevant: Theorem 3.1 and the representation-vector consequences.
Michael D. Francis, A Dixmier–Malliavin theorem for Lie groupoids, Journal of Lie Theory 32 (2022), 879–898. Publisher record. Relevant: Theorem 1.1 restates the classical Lie-group theorem with support control.