Schwartz–Bruhat space on a locally compact abelian group

Definition

Let $G$ be a locally compact abelian group. Its Schwartz–Bruhat space $\mathcal S(G)$ consists of functions obtained as follows: choose an open, compactly generated subgroup $H\subseteq G$ and a compact subgroup $K\subseteq H$ such that $H/K$ is an elementary group, pull a Schwartz function on $H/K$ back along the quotient map, and extend it by zero outside $H$ . Here an elementary group has the form $\mathbb R^n\times\mathbb Z^m\times\mathbb T^r\times F$ , with $F$ finite. Its Euclidean factor uses the ordinary Schwartz space, while the discrete directions are rapidly decreasing and the torus directions are smooth. The locally convex topology is the corresponding inductive-limit topology.

Structural role

The construction is independent of the auxiliary pair $(H,K)$ : enlarging $H$ or shrinking $K$ gives compatible transition maps. Translation, reflection, multiplication by characters, and convolution preserve $\mathcal S(G)$ . After compatible Haar measures are chosen, the Fourier transform is a topological isomorphism

\mathcal S(G)\longrightarrow \mathcal S(\widehat G),

where $\widehat G$ is the Pontryagin dual. This Fourier-invariant construction is the principal reason to use the Schwartz–Bruhat space rather than compactly supported continuous functions.

Standard cases

For $G=\mathbb R^n$ , one recovers the ordinary Schwartz space. If $G$ is discrete, the space consists of rapidly decreasing functions on each finitely generated subgroup, extended by zero. If $G$ is compact, every test function factors through a compact Lie quotient; thus $\mathcal S(G)$ is generally larger than the locally constant functions unless $G$ is totally disconnected.

Conventions and scope

The notations $\mathcal S(G)$ , $\mathcal D(G)$ , and “Bruhat space” vary across sources. The quotient-based definition is essential: a general locally compact abelian group need not itself be a Lie group or admit coordinates in which derivatives and polynomial weights can be written directly. Bruhat’s construction and its Fourier invariance are developed in Bruhat, pp. 60–61.

References

François Bruhat, “Distributions sur un groupe localement compact et applications à l’étude des représentations des groupes $p$ -adiques,” Bulletin de la Société Mathématique de France 89 (1961), 43–75. DOI record. Relevant: pp. 60–61 on rapidly decreasing test functions and Fourier transformation.