I am trying to understand Bruhat's generalized Schwartz functions over (Hausforff) locally compact Abelian groups [1], following this paper [2] by Osborne. There, the Schwartz-Bruhat space $\mathscr{S}(G)$ of complex functions over a (Hausdorff) locally compact Abelian group $G$ is defined based on the notion of differential operators on groups. I have trouble understanding how differential operators are defined in groups that are discrete; see questions 1 and 2 below.
For simplicity I only look at groups of the form
For simplicity (and as in the previous post ) I will always assume that $G$ is a group of the form
$$G=\mathbb{R}^a\times\mathbb{T}^b\times\mathbb{Z}^c\times F$$,
where $\mathbb{T}^m$ is an $m$-dimensional torus and $F$ an finite Abelian group given in the form $$F=\mathbb{Z}_{d_1}\times\cdots\times \mathbb{Z}_{d_e}.$$
For groups of the form $G$, the space $\mathscr{S}(G)$ is defined as follows: a function $f$ is in $\mathscr{S}(G)$ if $f$ is infinitely differentiable, and if $P(\partial)f\in L^2(G)$ for every polynomial (in the $\mathbb{R}^a\times \mathbb{Z}^c$ variables) differential operator $P(\partial)$ on $G$.
Question 1 What is a “polynomial differential operator $P(\partial)$'' over a group $G$? I am vaguely aware that groups of the form $G$ are the abelian Lie groups that are compactly-generated (cf. [3]). Therefore, I assume there must exist a robust-notion of differentiability of functions over these groups. Still I do not know how differential operators on function spaces over discrete groups such as $\mathbb{Z}^c$ or $F$ should be defined.
Question 2 Why does $P(\partial)$ have to be a polynomial differential operator only in the $\mathbb{R}^a\times \mathbb{Z}^c$ variables. I do not understand why $\mathbb{T}^b$ and $F$ are not mentioned. Does it perhaps something to do with compactness?
Related posts.
I have opened another post asking questions about an alternative equivalent way to define the Schwartz-Bruhat space via functions of rapid decay. Also, my first question seems to be related to this question.
Best Answer
I strongly recommend you to read the François Bruhat paper, that Osborne cites. For an arbitrary locally compact (not necessarily abelian) group $G$ Bruhat defines smooth function $\varphi:G\to{\mathbb C}$ as the one that can be locally represented as a composition $$ \varphi=\psi\circ\pi $$ where $\pi:G\to H$ is a continuous homomorphism into a Lie group $H$, and $\psi:H\to{\mathbb C}$ is a "usual" smooth function on $H$ (considered as a smooth manifold). (Actually, this is the definition for the so-called LP-groups, and the general definition becomes obvious after that.)
Bruhat describes the most important properties of the space ${\mathcal E}(G)$ of smooth functions on $G$ (it is not obvious, for example, that ${\mathcal E}(G)$ is always dense in ${\mathcal C}(G)$). As far as I remember, he gives also a definition of a differential operator on $G$, but I am not sure, anyway you can also take a look at my paper, where differential operators on $G$ are defined as linear maps $$ D:{\mathcal E}(G)\to {\mathcal E}(G) $$
which preserve the support of functions: $$ \text{supp}(D\varphi)\subseteq \text{supp}\ \varphi,\qquad \varphi\in {\mathcal E}(G). $$ ($\text{supp}\ \varphi$ is the set of points in $G$ where the germ of $\varphi$ is non-zero). One can prove (see again my paper) that every such operator has unique decomposition $$ D=\sum_{\alpha\in{\mathbb N}_I}\xi_\alpha\cdot\partial e_\alpha, $$ where $I$ is the set of indices (in general, infinite) for a given basis $\{e_i;\ i\in I\}$ in the tangent space $T_1(G)$ (=Lie algebra of $G$), ${\mathbb N}_I$ the set of multi-indices over $I$, $\partial e_\alpha$ are the corresponding translation-invariant differential operators on $G$, and $\xi_\alpha$ are some smooth functions on $G$ (defined by $D$).
$D$ is called a polynomial differential operator, if the coefficients $\xi_\alpha$ are polynomials, i.e. usual polynomials of finite sets of real characters $\chi:G\to{\mathbb R}$ (see Hewitt-Ross; obviously, each polynomial on ${\mathbb R}^a\times {\mathbb T}^b\times {\mathbb Z}^c \times F$ vanishes on ${\mathbb T}^b$ and $F$).
Using these notions one can define the Schwartz-Bruhat space ${\mathcal S}(H)$ on an abelian Lie group $H$ as consisting of smooth functions $\psi:H\to{\mathbb C}$ with compactly generated support and with the property $$ \sup_{x\in H}|D\psi(x)|<\infty, $$ for every polynomial differential operator $D$ on $H$. And if $G$ is an arbitrary locally compact abelian group (not necessarily a Lie group) then the space ${\mathcal S}(G)$ is defined as the direct limit (=union) of the spaces ${\mathcal S}(H)$ where $H$ runs through all Lie quotient groups of $G$: $$ {\mathcal S}(G)=\bigcup_H {\mathcal S}(H). $$ I believe this is equivalent to Osborne's definitions. By $K^m$ he means, I think, the $m$-th power of a set $K\subseteq G$, defined by induction $$ K^1=K,\qquad K^{m+1}=K^m\cdot K, $$ where $A\cdot B=\{a\cdot b;\ a\in A,\ b\in B\}$ (this is for your new post).