nLab compactly supported distribution

Idea

A compactly supported distribution is a distribution whose support of a distribution is a compact subset.

This implies that compactly supported distributions may be evaluated not just on bump functions, but in fact on the larger space of all smooth functions. Indeed it turns out that the compactly supported distributions exhaust the continuous linear functionals on the space of smooth functions (Hörmander 90, theorem 2.3.1).

Therefore and since Schwartz’s notation for the space $C^\infty(X)$ of all smooth functions on the given smooth manifold is $\mathcal{E}(X)$, the space of compactly supported distributions is often denoted by $\mathcal{E}'(X)$ (Hörmander 90, p. 45).

Definition

As continuous linear duals to smooth functions

Definition

(topological vector space of smooth functions on a Cartesian space)

For $n \in \mathbb{R}^n$, write $C^\infty(\mathbb{R}^n)$ for the $\mathbb{R}$-vector space of smooth functions $\mathbb{R}^n \to \mathbb{R}$.

Then the vector space $C^\infty(\mathbb{R}^n)$ becomes a Fréchet vector space induced by the family of seminorms

$$\array{ C^\infty(\mathbb{R}^n) &\overset{p_{K}^\alpha}{\longrightarrow}& [0,\infty) \\ \Phi &\mapsto& \underset{x \in K}{sup} {\vert \partial^\alpha \Phi(x) \vert} } \,,$$

indexed by $K \subset \mathbb{R}^n$ a compact subset and $\alpha \in \mathbb{N}^n$ a tuple encoding the degrees of the partial derivative $\partial^\alpha$.

Hence the seminorm $p_K^\alpha$ sends a smooth function $\Phi$ to the supremum over the points $x \in K$ of the absolute value of the partial derivative $\partial^\alpha \Phi$; and the open subsets defined thereby are the unions of translations of the base open balls

$$B^\circ_{K,\alpha,\epsilon}(0) = \left\{ v \in C^\infty\mathbb{R}^n \;\vert\; p_K^\alpha(v) \lt \epsilon \right\}$$

for $\epsilon \in (0,\infty)$.

We write

$$\mathcal{E}(\mathbb{R}^n) \in TopVect_{\mathbb{R}}$$

for the resulting Fréchet topological vector space.

Definition

(compactly supported distibutions as continuous linear duals to bump functions)

The space $\mathcal{E}'(\mathbb{R}^n)$ of compactly supported distributions on $\mathbb{R}^n$ is the dual space

$$\mathcal{E}'(\mathbb{R}^n) \;\coloneqq\; \left(\mathcal{E}(\mathbb{R}^n)\right)^\ast$$

to the topological vector space of smooth functions from def. .

e.g. (Klainerman 08, p. 9)

This means the following

Proposition

(characterization of continuity for compactly supported distributions)

A linear function

$$u \;\colon\; C^\infty(\mathbb{R}^n) \longrightarrow \mathbb{R}$$

is continuous with respect to the topology of def. , hence is a compactly supported distribution (def. )

$$\mathcal{E}(\mathbb{R}^n) \longrightarrow \mathbb{R}$$

precisely if the following condition holds

• there exists a compact subset $K \subset \mathbb{R}^n$ and $k \in \mathbb{N}$ and $C \in (0,\infty)$ such that

  (1)$$\underset{\Phi \in C^\infty(\mathbb{R}^n)}{\forall} \left( \vert u(\Phi) \vert \;\leq\; C \underset{ {\vert \alpha \vert \leq k} }{\sum} \underset{x \in K}{sup} \vert \partial^\alpha \Phi \vert \right)$$

(see also Hörmander 90, (2.3.2) and theorem 2.3.1)

Proof

By this prop. the continuity of $L$ is equivalent to there being an inhabited finite set $\{ (K_1, \alpha_1), \cdots, (K_r, \alpha_r) \}$ and $C \in (0,\infty)$ such that

(2)$${\vert u(\Phi)\vert} \;\leq\; C \underset{i = 1, \cdots, n}{max} \left( \underset{x \in K_i}{sup} {\vert \partial^{\alpha_i} \Phi \vert} \right)$$

We need to see that this is equivalent to a bound of the form (1).

In one direction, assume that (2) holds.

The union $K \coloneqq \underset{i = 1, \cdots, n}{\cup} K_i$ is still a compact subset (this prop.). Hence (2) implies that

$$\begin{aligned} {\vert u(\Phi)\vert} & \leq C \underset{i = 1, \cdots, n}{max} \underset{x \in K}{sup} {\vert \partial^{\alpha_i} \Phi \vert} \end{aligned}$$

and hence with $k \coloneqq \underset{i = 1, \cdots, n}{max} {\vert \alpha_i\vert}$ that

$${\vert u(\Phi)\vert} \;\leq\; C \underset{{\vert \alpha\vert} \leq k}{\sum} \underset{x \in K}{sup} {\vert \partial^{\alpha} \Phi \vert} \,,$$

which is of form (1).

Conversely, assume that a bound of the form (1) holds. Then take the finite set of pairs $(K_i, \alpha_i)$ where all $K_i \coloneqq K$ and with all ${\vert\alpha_i\vert} \leq k$. With $N$ denoting the number of $n$-tuples $\alpha$ with ${\vert \alpha \vert} \leq k$ we then get a bound as in (2) with coefficient $N C$.

As smooth linear duals to smooth functions

Given that distributions are concerned with smooth functions it is sometimes more natural to regard them not as continuous linear functionals as in def. , but as smooth linear functionals. Indeed this turns out to be equivalent (prop. below), if one considers an ambient context of suitably generalized smooth spaces, namely diffeological spaces or more generally smooth sets or formal smooth sets. We will write $\mathbf{H}$ for any of these categories of generalized smooth spaces.

We may canonially regard any smooth manifold such as the Cartesian space $\mathbb{R}^n$ as an object of $\mathbf{H}$. For $X \in \mathbf{H}$ any object, we write $[X,\mathbb{R}]$ for the mapping space (the internal hom). The underlying set is $C^\infty(X)$. If $X$ itself has $\mathbb{R}$-linear structure, we write

$$[X,\mathbb{R}]_{\mathbb{R}} \hookrightarrow [[X,\mathbb{R}], \mathbb{R}]$$

for the subobject of $\mathbb{R}$-linear maps.

Concretely, for $U$ a smooth manifold (or just a Cartesian space), then the sheaf $[X,\mathbb{R}]$ assigns (see at closed monoidal structure on presheaves for details)

$$[X,\mathbb{R}](U) = C^\infty(U \times X)$$

and $[X,\mathbb{R}](U) \subset C^\infty(U \times X)$ is the subset of those functions $\Phi_{(-)}(-)$ such that for all $u \in U$ the function $\Phi_u \colon X \to \mathbb{R}$ is $\mathbb{R}$-linear. The global elements $\Gamma(-)$ of the mapping space constitute the ordinary hom set

$$\Gamma [X,\mathbb{R}] \simeq \mathbf{H}(X,\mathbb{R}) \,.$$

Proposition

(compactly supported distributions are the smooth linear functionals)

For $n \in \mathbb{N}$, there is a natural bijection between the underlying sets of compactly supported distributions on $\mathbb{R}^n$ (def. ) and the $\mathbb{R}$-linear mapping space formed in the category $\mathbf{H}$ of either diffeological space or smooth sets or formal smooth sets:

$$\widetilde{(-)} \;\colon\; \mathcal{E}'(\mathbb{R}^n) \overset{\simeq}{\longrightarrow} \mathbf{H}([\mathbb{R}^n, \mathbb{R}], \mathbb{R})_{\mathbb{R}}$$

given by sending $\mu \in \mathcal{E}'(\mathbb{R}^n)$ to the natural transformation which on a test space $U \in$ CartSp takes a smoothly $U$-parameterized function $\Phi_{(-)}(-) \colon U \times \mathbb{R}^n \to \mathbb{R}$ to its evaluation in $\mu$ pointwise in $U$:

$$\tilde \mu(\Phi_{(-)})(u) \;\coloneqq\; \langle \mu, \Phi_{u}\rangle \,.$$

(Moerdijk-Reyes 91, chapter II, prop. 3.6)

Proof

First consider this for the case that $\mathbf{H} =$ SmoothSet (which immediately subsumes the case that $\mathbf{H} =$ DiffelogicalSpace).

To see that $\widetilde{(-)}$ is well defined, we need to check that the function

$$\array{ U &\overset{ \tilde \mu(\Phi_{(-)})}{\longrightarrow}& \mathbb{R} \\ u &\mapsto& \langle \mu, \Phi_u\rangle }$$

is smooth. But this follows immediately since $\langle \mu,-\rangle$ by definition is linear and continuous (Hörmander 90, theorem 2.1.3).

To see that $\widetilde{(-)}$ is indeed a bijection for each $U$ it remains that every $\mathbb{R}$-linear smooth functional (morphisms of smooth sets) of the form

$$A \;\colon\; [\mathbb{R}^n,\mathbb{R}] \longrightarrow \mathbb{R}$$

when restricted on global elements to a function of sets

$$A(\ast) \;\colon\; C^\infty(\mathbb{R}^n) \longrightarrow \mathbb{R}$$

is continuous with respect to the topological vector space structure from def. on the left.

Now by definition of the internal hom $A$ is actually “path-smooth” (this def.) and hence the statement is implied by this prop.

Finally to see that this argument generalizes to $\mathbf{H} =$ FormalSmoothSet observe that the Weil algebra of every infinitesimally thickened point is a quotient ring of an algebra of smooth functions on some Cartesian space (by the Hadamard lemma). The previous argument now applies to representatives under this quotient coprojection and one checks that it is independent of the representative chosen.

Properties

Fourier-Laplace transform

Proposition/Definition

For $n \in \mathbb{N}$, let $u \in \mathcal{E}'(\mathbb{R}^n)$ be a compactly supported distribution on Cartesian space $\mathbb{R}^n$. Then its Fourier transform of distributions is the function

$$\array{ \mathbb{R}^n &\overset{\hat u}{\longrightarrow}& \mathbb{R} \\ \zeta &\mapsto& u\left(e^{-i\langle -,\zeta \rangle} \right) }$$

where on the right we have the application of $u$, regarded as a linear function $u \colon C^\infty(\mathbb{R}^n) \to \mathbb{R}$, to the exponential function applied to the canonical inner product $\langle -,-\rangle$ on $\mathbb{R}^n$.

This same formula makes sense more generally for complex numbers $\zeta \in \mathbb{C}^n$. This is then called the Fourier-Laplace transform of $u$, still denoted by the same symbol:

$$\array{ \mathbb{C}^n &\overset{\hat u}{\longrightarrow}& \mathbb{R} \\ \zeta &\mapsto& u\left(e^{-i\langle -,\zeta \rangle} \right) }$$

This is an entire analytic function on $\mathbb{C}^n$.

(Hörmander 90, theorem 7.1.14)

Theorem

(Paley-Wiener-Schwartz theorem)

For $n \in \mathbb{N}$ the vector space $C^\infty_c(\mathbb{R}^n)$ of compactly supported smooth functions (bump functions) on Euclidean space $\mathbb{R}^n$ is (algebraically and topologically) isomorphic, via the Fourier-Laplace transform (prop. ), to the space of entire functions $F$ on $\mathbb{C}^n$ which satisfy the following estimate: there is a positive real number $B$ such that for every integer $N \gt 0$ there is a real number $C_N$ such that:

$$\underset{\xi \in \mathbb{C}^n}{\forall} \left( {\Vert F(\xi) \Vert} \le C_N \left( 1 + {\vert \xi\vert } \right)^{-N} e^{B \, |Im(\xi)|} \right) \,.$$

More generally, the space of compactly supported distributions on $\mathbb{R}^n$ of order $N$ is isomorphic via Fourier transform of distributions to those entire functions on $\mathbb{C}^n$ for which there exists positive real numbers $C$ and $B$ such that

$$\underset{\xi \in \mathbb{C}^n}{\forall} \left( {\Vert F(\xi) \Vert} \le C_N (1 + {\vert \xi\vert })^{N} e^{ B \; |Im(\xi)|} \right) \,.$$

(Notice that the Fourier transform of a compactly supported distribution is guaranteed to be a smooth function, by this prop..)

(e.g. Hoermander 90, theorem 7.3.1)

Proposition

(decay of Fourier transform of compactly supported functions)

A compactly supported distribution $u \in \mathcal{E}'(\mathbb{R}^n)$ is a non-singular distribution (def. ), hence given by a compactly supported function $b \in C^\infty_{cp}(\mathbb{R}^n)$ via $u(f) = \int b(x) f(x) dvol(x)$, precisely if its Fourier transform $\hat u$ (this def.) satisfies the following decay property:

For all $N \in \mathbb{N}$ there exists $C_N \in \mathbb{R}_+$ such that for all $\xi \in \mathbb{R}^n$ we have that the absolute value ${\vert \hat v(\xi)\vert}$ of the Fourier transform at that point is bounded by

(3)$${\vert \hat v(\xi)\vert} \;\leq\; C_N \left( 1 + {\vert \xi\vert} \right)^{-N} \,.$$

(e.g. Hoermander 90, around (8.1.1))

Singular support and Wave front set

Definition

(singular support of a compactly supported distribution)

For $n \in \mathbb{N}$ and $u \in \mathcal{E}'(\mathbb{R}^n)$ a compactly supported distribution, its singular support is the subset of the Cartesian space $\mathbb{R}^n$ of those points which have no neighbourhood on which $u$ restricts to a non-singular distribution:

$$supp_{sing}(u) \;\coloneqq\; \left\{ u \in \mathbb{R}^n \,\vert\, \not \left( \underset{U \underset{\text{nbhd}}{\supset} \{x\}}{\exists} \left( u\vert_U \in C^\infty_{cp}(U) \right) \right) \right\} \,.$$

By prop. the singular support of a distribution (def. ) consists of those points around which the Fourier transform of the distribution receives large high-frequency (“UV”) contributions. But in fact prop. allows to say more precisely which high frequency Fourier modes make the distribution singular at a given point. These are said to be part of the wave front of the distribution, and the collection of all of them is called the wave front set of the distribution:

Definition

(wavefront set)

For $n \in \mathbb{N}$ let $u \in \mathcal{E}'(\mathbb{R}^n)$ be a compactly supported distribution.

For $b \in C^\infty_{cp}(\mathbb{R}^n)$ a compactly supported smooth function, write $b u \in \mathcal{E}'(\mathbb{R}^n)$ for the corresponding product (this example).

For $x\in supp(b) \subset \mathbb{R}^n$, we say that a unit covector $\xi \in S((\mathbb{R}^n)^\ast)$ is regular if there exists a neighbourhood $U \subset S((\mathbb{R}^n)^\ast)$ of $\xi$ in the unit sphere such that for all $c \xi' \in (\mathbb{R}^n)^\ast$ with $c \in \mathbb{R}_+$ and $\xi' \in U \subset S((\mathbb{R}^n)^\ast)$ the decay estimate (3) is valid for the Fourier transform $\widehat{b u}$ of $b u$; at $c \xi'$. Otherwise $\xi$ is non-regular. Write

$$\Sigma(b u) \;\coloneqq\; \left\{ \xi \in S((\mathbb{R}^n)^\ast) \;\vert\; \xi \, \text{non-regular} \right\}$$

for the set of non-regular covectors of $b u$.

The wave front set at $x$ is the intersection of these sets as $b$ ranges over bump functions whose support includes $x$:

$$\Sigma_x(u) \;\coloneqq\; \underset{ { b \in C^\infty_{cp}(\mathbb{R}^n) } \atop { x \in supp(b) } }{\cap} \Sigma(b u) \,.$$

Finally the wave front set of $u$ is the subset of the sphere bundle $S(T^\ast \mathbb{R}^n)$ which over $x \in \mathbb{R}^n$ consists of $\Sigma_x(U) \subset T^\ast_x \mathbb{R}^n$:

$$WF(u) \;\coloneqq\; \underset{x \in \mathbb{R}^n}{\cup} \Sigma_x(u) \;\subset\; S(T^\ast \mathbb{R}^n)$$

Usually this is considered as the full conical set inside the cotangent bundle generated by the unit covectors under multiplication with positive real numbers.

(Hörmander 90, def. 8.1.2)

Hence the wave front set of a compactly supported distribution consists of all those directions of wave vectors along which the Fourier transform of the distribution is not a rapidly decreasing function.

Proposition

(empty wave front set corresponds to ordinary smooth functions)

The wave front set (def. ) of a compactly supported distribution is empty precisely if the distribution is non-singular (example ).

This is effectively the Paley-Wiener-Schwartz theorem (Hörmander 90, below (8.1.1)).

Examples

Definition

(non-singular compactly supported distributions)

For $n \in \mathbb{N}$, a compactly supported smooth function $b \in C^\infty_{cp}(\mathbb{R}^n)$ (a bump function) induces a compactly supported distribution

$$\int_{\mathbb{R}^n} (-) b dvol_{\mathbb{R}^n} \;\colon\; C^\infty(\mathbb{R}^n) \longrightarrow \mathbb{R}$$

by integration of smooth functions against $b dvol$.

This construction defines a linear inclusion

$$C^\infty_{cp}(\mathbb{R}^n) \hookrightarrow \mathcal{E}'(\mathbb{R}^n) \,.$$

The compactly supported distributions arising this way are called the non-singular distributions.

• microcausal functional

Related concepts

• compactly supported continuous function

• bump function

References

Traditional

Textbook accounts include

• Lars Hörmander, section 2.3 of The analysis of linear partial differential operators, vol. I, Springer 1983, 1990

Lecture notes include

• Sergiu Klainerman, Analysis 2008 (pdf)

In terms of smooth toposes

Discussion of compactly supported distributions in terms morphisms out of internal homs in a smooth topos is in

• Ieke Moerdijk, Gonzalo Reyes, around prop. 3.6 in chapter II of Models for Smooth Infinitesimal Analysis, Springer 1991

and specifically for the Cahiers topos of formal smooth sets in

• Anders Kock, Gonzalo Reyes, Some calculus with extensive quantities: wave equation, Theory and Applications of Categories , Vol. 11, 2003, No. 14, pp 321-336 (TAC)

• Anders Kock, Commutative monads as a theory of distributions (arxiv:1108.5952)

using results of

• Alfred Frölicher, Andreas Kriegl, section 5 of Linear spaces and differentiation theory, Wiley 1988 (pdf)

and following the general conception of “intensive and extensive” in

• William Lawvere, Introduction to Categories in Continuum Physics, Lectures given at a Workshop held at SUNY, Buffalo 1982. Lecture Notes in Mathematics 1174. 1986

Generalization to non-compactly supported distributions is in

• Anders Kock, Gonzalo Reyes, Categorical distribution theory; heat equation (arXiv:math/0407242)

and sheaf theoretic discussion of distributions as morphisms of smooth spaces is in

• Frédéric Paugam, section 3.2 of Towards the mathematics of quantum field theory, 2012 (pdf)