$$
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% CHAPTER 1: DIFFERENTIAL DUALITY
% ============================================
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% CHAPTER 3: ULTRA CALCULUS
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$$
Distributions
The cube measures \(\delta(x; c)\) E0016 are finitely supported
signed measures. The point distributions \(\delta(x; \xi)\) are
their smooth counterparts, supported at a single point and
parameterized by symmetric probes E0027.
(1) Definition (Smooth functions and distributions).
- \(\mathcal{E}(X) := C^\infty(X, \IR)\) with the standard
Fréchet topology (for \(X\) finite-dimensional).
- \(\mathcal{E}'(X)\) is the space of compactly supported
distributions (continuous dual of \(\mathcal{E}(X)\)).
- \(\mathcal{E}'(X; x)\) is the subspace of distributions
supported at \(x\).
- \(\mathcal{E}'(X; x)_{\leq k}\) is its subspace of order
\(\leq k\).
(2) Definition (Point distributions).
-
For \(\xi \in ST_k(X; x)\), the point distribution
\(\delta(x; \xi)\) is
\[
\langle \delta(x; \xi), f \rangle := D(f; x; \xi),
\qquad f \in C^k(X, \IR).
\]
In particular \(\delta(x; 1) = \delta(x)\) is evaluation
at \(x\).
-
The map \(\xi \mapsto \delta(x; \xi)\) is a linear
isomorphism \(ST_k(X; x) \xrightarrow{\sim}
\mathcal{E}'(X; x)_{\leq k}\) (Schwartz structure theorem).
(3) Proposition (Measures embed in distributions).
Restriction along \(\mathcal{E}(X) \subseteq F(X)\) embeds the
finitely supported measures into the distributions as the
order-\(0\) distributions with finite support. In particular,
every cube measure \(\delta(x; c)\) E0016 is a distribution.
Proof.
Evaluation \(f \mapsto f(z)\) at a point \(z\) is continuous on
\(\mathcal{E}(X)\) and extends to \(C^0(X, \IR)\), so a finitely
supported measure \(\mu = \sum_i \lambda_i \delta(z_i)\) defines
a distribution of order \(0\). The map is injective since the
\(\delta(z_i)\) are linearly independent as functionals on
\(\mathcal{E}(X)\).
Proof.
\(\langle \phi_* \delta(x; \xi), f \rangle =
D(\phi^* f; x; \xi) = D(f; y; D_+(\phi; x)\, \xi) =
\langle \delta(y; D_+(\phi; x)\, \xi), f \rangle\) by the
smooth adjunction E0028.
(5) Proposition (Filtration and principal symbol).
\(D_+(\phi; x)\) preserves the degree filtration, and on the
associated graded it is the symmetric power of the first
derivative:
\[
\gr_r\, D_+(\phi; x)
= \SYM^r(D(\phi; x)) : \SYM^r(E) \to \SYM^r(F).
\]
Proof.
A partition \(\pi \in \Part(r)\) contributes in degree
\(|\pi| \leq r\), with equality only for the partition into
singletons, whose term is \(\prod_i D(\phi; x; v_i)\).
Used by: